stress caused by coupling between the temperature and displacement fields are plotted in **Figure 8** to illustrate the variation of the bio-thermal stress σ11 along *x*‐axis for BEM, FDM and FEM, where the boundary of the special case problem has been discretized into 21, 42 and 84 boundary elements (bes). The validity, accuracy and efficiency of our proposed technique have been confirmed by a graphical comparison of the three different boundary elements (21, 42 and 84) with those obtained using the FDM results of Shen and Zhang [49] and FEM results of Torvi and Dale [50] for the special case under consideration, the increase of BEM boundary elements leads to improve the accuracy and efficiency of the BEM, also, it can

**Figure 6.**

**Figure 7.**

**254**

*Variation of the fluid pressure p along x*‐*axis.*

*Recent Developments in the Solution of Nonlinear Differential Equations*

*Variation of the bio-thermal stress σ***11** *along x*‐*axis.*

7.Numerical findings confirm the validity, efficiency and accuracy of the proposed BEM technique.

*Qext* external heat source *S*0 scattering coefficient *T* soft tissue temperature

*DOI: http://dx.doi.org/10.5772/intechopen.96268*

*x* traction derivative *u* solid displacement *u f* fluid displacement *V* ¼ *V f* þ *Vs* bulk volume *V f* fluid volume *Vs* solid volume

*Wb* blood perfusion rate

∈ linear strain tensor *ζ* fluid volume variation

*ρ* ¼ *ρs*ð Þþ 1 � *ϕ ϕρ f* bulk density

*ρ f* blood density *σ* total stress tensor

*V* porosity

Ω considered region

*τ* time

B stress-temperature coefficients

*ρs* ¼ *ϕρ f* mass density of soft tissue

*τq* phase lag for heat flux

*τT* phase lag for temperature gradient

*i* ½ � *k* continuous polynomial shape functions

½ � *k* discontinuous polynomial shape functions

1 Jamoum University College, Umm Al-Qura University, Makkah, Saudi Arabia

2 Faculty of Computers and Informatics, Suez Canal University, Ismailia, Egypt

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

*b* arterial blood temperature

*Boundary Element Modeling and Simulation Algorithm for Fractional Bio-Thermomechanical…*

*T ^*

*Te*

*φα*

*ψβ j*

**257**

*ϕ* ¼ *V f*

**Author details**

Mohamed Abdelsabour Fahmy1,2

<EMAIL_ADDRESS>
provided the original work is properly cited.

\*Address all correspondence to<EMAIL_ADDRESS>

### **Nomenclature**

*Boundary Element Modeling and Simulation Algorithm for Fractional Bio-Thermomechanical… DOI: http://dx.doi.org/10.5772/intechopen.96268*

#### **Author details**

7.Numerical findings confirm the validity, efficiency and accuracy of the

8.The proposed technique can be applied to a wide variety of fractional

10.The presence of fractional order parameter in the current study plays a

treatment of the bio-thermomechanical soft tissue models.

transfer, mechanics, neurophysiology, biology and clinicians.

significant role in all the physical quantities during modeling and simulation

11.From the research that has been performed, it is possible to conclude that the proposed BEM is an easier, effective, predictable, and stable technique in the

12. It can be concluded from this chapter that Biot's equations for the dynamic response of poroelastic media can be combined with the bio-heat transfer models to describe the fractional bio-thermomechanical interactions of

13.Current numerical results for our complex and general problem may provide interesting information for researchers and scientists in bioengineering, heat

9.For open boundary problems of soft tissues, such as the considered problem, the BEM users need only to deal with real geometry boundaries. But for these problems, FDM and FEM use artificial boundaries, which are far away from the real soft tissues. Also, these artificial boundaries are also becoming a big challenge for FDM users and FEM users. So, BEM becomes the best method

bio-thermomechanical problems in anisotropic soft tissues.

*Recent Developments in the Solution of Nonlinear Differential Equations*

proposed BEM technique.

for the considered problem.

in medicine and healthcare.

anisotropic soft tissues.

*A* ¼ *ϕ*ð Þ 1 þ *Q=R* Biot's coefficient

shape factor

*F* bulk body forces ^*gD* Dirichlet datum ^*gN* Neumann datum *K* dynamic permeability

*m* iterative parameter *p* pore pressure *P*0ð Þ*τ* heating power

*Qmet* metabolic heat source

*x*~ linear elastostatics operator *Γ* considered boundary *ΓD* Dirichlet boundary *ΓN* Neumann boundary *C* specific heat of soft tissue

*Cb* specific heat of the blood *Cp jkl* specific heat of the blood

K thermal conductivity of soft tissue

*Q*, *R* solid–fluid coupling parameters

**Nomenclature**

*Be*

**256**

Mohamed Abdelsabour Fahmy1,2

1 Jamoum University College, Umm Al-Qura University, Makkah, Saudi Arabia

2 Faculty of Computers and Informatics, Suez Canal University, Ismailia, Egypt

\*Address all correspondence to<EMAIL_ADDRESS><EMAIL_ADDRESS>
© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Cho YI. Advances in Heat Transfer: Bioengineering Heat Transfer. San Diego: Academic Press Inc.; 1992

[2] Pennes HH. Analysis of tissue and arterial blood temperatures in the resting human forearm. J. Appl. Physiol. 1948;1:93–122.

[3] Askarizadeh H, Ahmadikia H. Analytical study on the transient heating of a two-dimensional skin tissue using parabolic and hyperbolic bioheat transfer equations. Appl. Math. Modell. 2015;39:3704–3720

[4] Li X., Li C., Xue Z., Tian X. Analytical study of transient thermomechanical responses of dual-layer skin tissue with variable thermal material properties. Int. J. Therm. Sci. 2018;124: 459–466

[5] Ahmadikia H., Fazlali R., Moradi A. Analytical solution of the parabolic and hyperbolic heat transfer equations with constant and transient heat flux conditions on skin tissue. Int. Commun. Heat Mass Transfer 2012;39:121–130.

[6] Shih TC, Yuan P, Lin WL, Kou HS. Analytical analysis of the Pennes bioheat transfer equation with sinusoidal heat flux condition on skin surface. Med. Eng. Phys. 2007;29:946–953.

[7] Lua WQ, Liub J, Zenga Y. Simulation of the thermal wave propagation in biological tissues by the dual reciprocity boundary element method. Eng. Anal. Boundary Elem. 1998;22:167–174.

[8] Shen W, Zhang J, Yang F. Modeling and numerical simulation of bioheat transfer and biomechanics in soft tissue. Math. Comput. Modell. 2005;41: 1251–1265.

[9] Zhang Y. Generalized dual-phase lag bioheat equations based on nonequilibrium heat transfer in living

biological tissues. Int. J. Heat Mass Transfer 2009;52:4829–4834.

[10] Bourantas GC, Loukopoulos VC, Burganos VN, and Nikiforidis GC. A meshless point collocation treatment of transient bioheat problems. Int. J. Numer. Methods Biomed. Eng. 2014;30: 587–601

problem of a rotating functionally graded anisotropic solids with and without energy dissipation. Numer. Heat Transfer, Part A 2013;63:713–733.

*DOI: http://dx.doi.org/10.5772/intechopen.96268*

[25] Ng EYK, Tan HM, Ooi EH.

[26] Majchrzak E, Turchan L. The general boundary element method for 3D dual-phase lag model of bioheat transfer, Eng. Anal. Boundary Elem.

Boundary Element Solution of

for the Simulation of SAR and Temperature Increase in Biological Tissues. IEEE Trans. Magn. 2012;48:

[28] Deng ZS, Liu J. Modeling of multidimensional freezing problem during cryosurgery by the dual reciprocity boundary element method. Eng. Anal. Boundary Elem. 2004;28:97–108

Fluid Flow 2009;19:25–38.

(1992;114:358–365.

2002.

118:384–392.

441–458.

[30] Chan CL. Boundary Element Method Analysis for the Bioheat Transfer Equation. J. Biomech. Eng.

[32] Wang CY, Achenbach JD.

dimensional time-harmonic

[31] Wrobel LC. The boundary element method, Applications in thermos-fluids and acoustics. vol. 1. New York: Wiley;

Elastodynamic fundamental solution for anisotropic solids. Geophys. J. Int. 1994;

[33] Wang CY, Achenbach JD. Three-

elastodynamic Green's functions for anisotropic solids. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 1995;449:

2009;35:987–997.

*Boundary Element Modeling and Simulation Algorithm for Fractional Bio-Thermomechanical…*

2015;50:76–82.

691–694.

Boundary element method with bioheat equation for skin burn injury. Burns

[27] Bottauscio O, Chiampi M, Zilberti L.

Electromagnetic and Bioheat Equations

[29] Partridge PW, Wrobel LC. A coupled dual reciprocity BEM/genetic algorithm for identification of blood perfusion parameters. Int. J. Numer. Methods Heat

thermoelastic problems: Applications to

structures. Saarbrücken, Germany: LAP Lambert Academic Publishing; 2017.

[20] Fahmy MA. Shape design sensitivity and optimization for two-temperature generalized magneto-thermoelastic problems using time-domain DRBEM. J. Therm. Stresses 2018;41:119–138.

[21] Fahmy MA. Shape design sensitivity

[22] Pei RZ, Jing L, Cheng WC, Xue JP. Boundary element method (BEM) for solving normal or inverse bio-heat transfer problem of biological bodies with complex shape. J. Therm. Sci. 4

[23] Ooi EH, Ang WT, Ng EYK. A boundary element model for

[24] Zhou J., Chen JK, Zhang Y. Simulation of Laser-Induced Thermotherapy Using a Dual-

Reciprocity Boundary Element Model with Dynamic Tissue Properties. IEEE Trans. Biomed. Eng. 2010;57:238–245.

investigating the effects of eye tumor on the temperature distribution inside the human eye. Comput. Biol. Med. 2009;

and optimization of anisotropic functionally graded smart structures using bicubic B-splines DRBEM. Eng. Anal. Boundary Elem. 87 2018,87:

[18] Fahmy MA. Computerized boundary element solutions for

functionally graded anisotropic

[19] Fahmy MA. Boundary element computation of shape sensitivity and

optimization: applications to functionally graded anisotropic structures. Saarbrücken, Germany: LAP Lambert Academic Publishing;

2017.

27–35.

1995;4:117–124.

39:667–677.

**259**

[11] Fahmy MA. A time-stepping DRBEM for the transient magnetothermo-visco-elastic stresses in a rotating non-homogeneous anisotropic solid. Eng. Anal. Boundary Elem. 2012; 36:335–345.

[12] Fahmy MA. Transient magnetothermoviscoelastic plane waves in a non-homogeneous anisotropic thick strip subjected to a moving heat source. Appl. Math. Modell. 2012;36:4565–4578.

[13] Fahmy MA. The effect of rotation and inhomogeneity on the transient magneto-thermoviscoelastic stresses in an anisotropic solid. ASME J. Appl. Mech. 2012;79:051015.

[14] Fahmy MA. Transient magnetothermo-elastic stresses in an anisotropic viscoelastic solid with and without a moving heat source. Numer. Heat Transfer, Part A 2012;61:633–650.

[15] Fahmy MA. Implicit-Explicit time integration DRBEM for generalized magneto-thermoelasticity problems of rotating anisotropic viscoelastic functionally graded solids. Eng. Anal. Boundary Elem. 2013;37:107–115.

[16] Fahmy MA. Generalized magnetothermo-viscoelastic problems of rotating functionally graded anisotropic plates by the dual reciprocity boundary element method. J. Therm. Stresses 2013;36:1–20.

[17] Fahmy MA. A three-dimensional generalized magneto-thermo-viscoelastic *Boundary Element Modeling and Simulation Algorithm for Fractional Bio-Thermomechanical… DOI: http://dx.doi.org/10.5772/intechopen.96268*

problem of a rotating functionally graded anisotropic solids with and without energy dissipation. Numer. Heat Transfer, Part A 2013;63:713–733.

**References**

1948;1:93–122.

2015;39:3704–3720

459–466

[1] Cho YI. Advances in Heat Transfer: Bioengineering Heat Transfer. San Diego: Academic Press Inc.; 1992

*Recent Developments in the Solution of Nonlinear Differential Equations*

biological tissues. Int. J. Heat Mass Transfer 2009;52:4829–4834.

[10] Bourantas GC, Loukopoulos VC, Burganos VN, and Nikiforidis GC. A meshless point collocation treatment of transient bioheat problems. Int. J. Numer. Methods Biomed. Eng. 2014;30:

[11] Fahmy MA. A time-stepping DRBEM for the transient magnetothermo-visco-elastic stresses in a rotating non-homogeneous anisotropic solid. Eng. Anal. Boundary Elem. 2012;

[12] Fahmy MA. Transient magnetothermoviscoelastic plane waves in a non-homogeneous anisotropic thick strip subjected to a moving heat source. Appl. Math. Modell. 2012;36:4565–4578.

[13] Fahmy MA. The effect of rotation and inhomogeneity on the transient magneto-thermoviscoelastic stresses in an anisotropic solid. ASME J. Appl.

[14] Fahmy MA. Transient magnetothermo-elastic stresses in an anisotropic viscoelastic solid with and without a moving heat source. Numer. Heat Transfer, Part A 2012;61:633–650.

[15] Fahmy MA. Implicit-Explicit time integration DRBEM for generalized magneto-thermoelasticity problems of rotating anisotropic viscoelastic functionally graded solids. Eng. Anal. Boundary Elem. 2013;37:107–115.

[16] Fahmy MA. Generalized magnetothermo-viscoelastic problems of

rotating functionally graded anisotropic plates by the dual reciprocity boundary element method. J. Therm. Stresses

[17] Fahmy MA. A three-dimensional generalized magneto-thermo-viscoelastic

2013;36:1–20.

Mech. 2012;79:051015.

587–601

36:335–345.

[2] Pennes HH. Analysis of tissue and arterial blood temperatures in the resting human forearm. J. Appl. Physiol.

[3] Askarizadeh H, Ahmadikia H. Analytical study on the transient heating of a two-dimensional skin tissue using parabolic and hyperbolic bioheat transfer equations. Appl. Math. Modell.

[4] Li X., Li C., Xue Z., Tian X. Analytical study of transient thermomechanical responses of dual-layer skin tissue with variable thermal material properties. Int. J. Therm. Sci. 2018;124:

[5] Ahmadikia H., Fazlali R., Moradi A. Analytical solution of the parabolic and hyperbolic heat transfer equations with

conditions on skin tissue. Int. Commun. Heat Mass Transfer 2012;39:121–130.

[6] Shih TC, Yuan P, Lin WL, Kou HS. Analytical analysis of the Pennes bioheat transfer equation with sinusoidal heat flux condition on skin surface. Med.

[7] Lua WQ, Liub J, Zenga Y. Simulation of the thermal wave propagation in biological tissues by the dual reciprocity boundary element method. Eng. Anal. Boundary Elem. 1998;22:167–174.

[8] Shen W, Zhang J, Yang F. Modeling and numerical simulation of bioheat transfer and biomechanics in soft tissue.

[9] Zhang Y. Generalized dual-phase lag

nonequilibrium heat transfer in living

Math. Comput. Modell. 2005;41:

bioheat equations based on

1251–1265.

**258**

constant and transient heat flux

Eng. Phys. 2007;29:946–953.

[18] Fahmy MA. Computerized boundary element solutions for thermoelastic problems: Applications to functionally graded anisotropic structures. Saarbrücken, Germany: LAP Lambert Academic Publishing; 2017.

[19] Fahmy MA. Boundary element computation of shape sensitivity and optimization: applications to functionally graded anisotropic structures. Saarbrücken, Germany: LAP Lambert Academic Publishing; 2017.

[20] Fahmy MA. Shape design sensitivity and optimization for two-temperature generalized magneto-thermoelastic problems using time-domain DRBEM. J. Therm. Stresses 2018;41:119–138.

[21] Fahmy MA. Shape design sensitivity and optimization of anisotropic functionally graded smart structures using bicubic B-splines DRBEM. Eng. Anal. Boundary Elem. 87 2018,87: 27–35.

[22] Pei RZ, Jing L, Cheng WC, Xue JP. Boundary element method (BEM) for solving normal or inverse bio-heat transfer problem of biological bodies with complex shape. J. Therm. Sci. 4 1995;4:117–124.

[23] Ooi EH, Ang WT, Ng EYK. A boundary element model for investigating the effects of eye tumor on the temperature distribution inside the human eye. Comput. Biol. Med. 2009; 39:667–677.

[24] Zhou J., Chen JK, Zhang Y. Simulation of Laser-Induced Thermotherapy Using a Dual-Reciprocity Boundary Element Model with Dynamic Tissue Properties. IEEE Trans. Biomed. Eng. 2010;57:238–245.

[25] Ng EYK, Tan HM, Ooi EH. Boundary element method with bioheat equation for skin burn injury. Burns 2009;35:987–997.

[26] Majchrzak E, Turchan L. The general boundary element method for 3D dual-phase lag model of bioheat transfer, Eng. Anal. Boundary Elem. 2015;50:76–82.

[27] Bottauscio O, Chiampi M, Zilberti L. Boundary Element Solution of Electromagnetic and Bioheat Equations for the Simulation of SAR and Temperature Increase in Biological Tissues. IEEE Trans. Magn. 2012;48: 691–694.

[28] Deng ZS, Liu J. Modeling of multidimensional freezing problem during cryosurgery by the dual reciprocity boundary element method. Eng. Anal. Boundary Elem. 2004;28:97–108

[29] Partridge PW, Wrobel LC. A coupled dual reciprocity BEM/genetic algorithm for identification of blood perfusion parameters. Int. J. Numer. Methods Heat Fluid Flow 2009;19:25–38.

[30] Chan CL. Boundary Element Method Analysis for the Bioheat Transfer Equation. J. Biomech. Eng. (1992;114:358–365.

[31] Wrobel LC. The boundary element method, Applications in thermos-fluids and acoustics. vol. 1. New York: Wiley; 2002.

[32] Wang CY, Achenbach JD. Elastodynamic fundamental solution for anisotropic solids. Geophys. J. Int. 1994; 118:384–392.

[33] Wang CY, Achenbach JD. Threedimensional time-harmonic elastodynamic Green's functions for anisotropic solids. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 1995;449: 441–458.

[34] Biot MA. Theory of propagation of elastic waves in a fluid- saturated porous solid. I. Low frequency range. J. Acoust. Soc. Am. 1956;28:168–178.

[35] Biot MA. Theory of propagation of elastic waves in a fluid- saturated porous solid. II. Higher frequency range. J. Acoust. Soc. Am. 1956;28:179–191.

[36] Schanz M. Wave propagation in viscoelastic and poroelastic continua. Lecture notes in applied mechanics. vol 2. New York: Springer; 2001.

[37] Xiaoyun Jiang and Haitao Qi, Thermal wave model of bioheat transfer with modified Riemann–Liouville fractional derivative, Journal of Physics A: Mathematical and Theoretical 2012; 45:485101

[38] Bonnet G, Auriault JL. Dynamics of saturated and deform- able porous media: homogenization theory and determination of the solid-liquid coupling coefficients. Physics of finely divided matter, Berlin: Springer Verlag; 1985. p. 306–316.

[39] Bonnet G. Basic singular solutions for a poroelastic medium in the dynamic range, J. Acoust. Soc. Am. 1987;82:1758– 1763.

[40] Messner M, Schanz M. A regularized collocation boundary element method for linear poroelasticity. Comput. Mech. 2011;47:669–680.

[41] Fahmy MA. Boundary Element Modeling and Simulation of Biothermomechanical Behavior in Anisotropic Laser-Induced Tissue Hyperthermia. Engineering Analysis with Boundary Elements. 2019;101: 156–164. doi: 10.1016/j. enganabound.2019.01.006

[42] Steinbach O. Numerical approximation methods for elliptic boundary value problems. New York: Springer; 2008.

[43] Kielhorn L. A time-domain symmetric Galerkin BEM for viscoelastodynamics. In: Brenn G, Holzapfel GA, Schanz M, Steinbach O, eds. Computation in engineering and science. Austria: Graz University of Technology, Institute of Applied Mechanics; 2009.

thermoviscoelastic Problems in Porous Smart Structures Using NURBS and Genetic Algorithm. International Journal of Thermophysics. 2021;42:29.

*DOI: http://dx.doi.org/10.5772/intechopen.96268*

*Boundary Element Modeling and Simulation Algorithm for Fractional Bio-Thermomechanical…*

[52] Fahmy MA. A novel BEM for modeling and simulation of 3T nonlinear generalized anisotropic micropolar-thermoelasticity theory with memory dependent derivative. CMES-Computer Modeling in Engineering &

Sciences. 2021;126:175–199.

[53] Fahmy MA. A new boundary element algorithm for modeling and simulation of nonlinear thermal stresses in micropolar FGA composites with temperature-dependent properties. Advanced Modeling and Simulation in Engineering Sciences. 2021 (In Press).

[54] Fahmy MA. A new BEM for fractional nonlinear generalized porothermoelastic wave propagation problems, Computers, Materials and

[55] Fahmy MA. A new boundary element technique for one- and two-

biothermomechanical behavior of anisotropic biological tissues. 14th Virtual Congress WCCM & ECCOMAS

2020. Virtual congress 11–15

Continua. 2021 (In Press).

temperature models of

January 2021.

**261**

[44] Lubich C. Convolution quadrature and discretized operational calculus. I. Numerische Mathematik 1988;52:129– 145.

[45] Lubich C. Convolution quadrature and discretized operational calculus II. Numerische Mathematik 1988;52:413– 425.

[46] Freund RW, Nachtigal NM. An implementation of the QMR method based on coupled two-term recurrences," SIAM J. Sci. Comput. 1994;15:313–337.

[47] Gilchrist MD, Murphy JG, Parnell W, Pierrat B. Modelling the slight compressibility of anisotropic soft tissue. Int. J. Solids Struct. 2014;51:3857– 3865.

[48] Morrow DA, Haut Donahue TL, Odegard GM, Kaufman KR. Transversely isotropic tensile material properties of skeletal muscle tissue. J. Mech. Behav. Biomed. Mater. 2010;3: 124–129.

[49] Shen W, Zhang J. Modeling and numerical simulation of bioheat transfer and biomechanics in soft tissue. Math. Comput. Modell. 2005;41:1251–1265.

[50] Torvi DA, Dale JD. A finite element model of skin subjected to a flash fire. J. Biomech. Eng. 1994;116: 250–255.

[51] Fahmy MA, Shaw S, Mondal S, Abouelregal AE, Lotfy Kh., Kudinov IV, Soliman AH. Boundary Element Modeling for Simulation and Optimization of Three-Temperature Anisotropic Micropolar Magneto*Boundary Element Modeling and Simulation Algorithm for Fractional Bio-Thermomechanical… DOI: http://dx.doi.org/10.5772/intechopen.96268*

thermoviscoelastic Problems in Porous Smart Structures Using NURBS and Genetic Algorithm. International Journal of Thermophysics. 2021;42:29.

[34] Biot MA. Theory of propagation of elastic waves in a fluid- saturated porous solid. I. Low frequency range. J. Acoust.

*Recent Developments in the Solution of Nonlinear Differential Equations*

[43] Kielhorn L. A time-domain symmetric Galerkin BEM for viscoelastodynamics. In: Brenn G, Holzapfel GA, Schanz M, Steinbach O, eds. Computation in engineering and science. Austria: Graz University of Technology, Institute of Applied

[44] Lubich C. Convolution quadrature and discretized operational calculus. I. Numerische Mathematik 1988;52:129–

[45] Lubich C. Convolution quadrature and discretized operational calculus II. Numerische Mathematik 1988;52:413–

[46] Freund RW, Nachtigal NM. An implementation of the QMR method

recurrences," SIAM J. Sci. Comput.

[48] Morrow DA, Haut Donahue TL,

Transversely isotropic tensile material properties of skeletal muscle tissue. J. Mech. Behav. Biomed. Mater. 2010;3:

[49] Shen W, Zhang J. Modeling and numerical simulation of bioheat transfer and biomechanics in soft tissue. Math. Comput. Modell. 2005;41:1251–1265.

[50] Torvi DA, Dale JD. A finite element model of skin subjected to a flash fire. J. Biomech. Eng. 1994;116: 250–255.

[51] Fahmy MA, Shaw S, Mondal S, Abouelregal AE, Lotfy Kh., Kudinov IV,

Optimization of Three-Temperature Anisotropic Micropolar Magneto-

Soliman AH. Boundary Element Modeling for Simulation and

Odegard GM, Kaufman KR.

based on coupled two-term

[47] Gilchrist MD, Murphy JG, Parnell W, Pierrat B. Modelling the slight compressibility of anisotropic soft tissue. Int. J. Solids Struct. 2014;51:3857–

1994;15:313–337.

Mechanics; 2009.

145.

425.

3865.

124–129.

[35] Biot MA. Theory of propagation of elastic waves in a fluid- saturated porous solid. II. Higher frequency range. J. Acoust. Soc. Am. 1956;28:179–191.

[36] Schanz M. Wave propagation in viscoelastic and poroelastic continua. Lecture notes in applied mechanics. vol

2. New York: Springer; 2001.

45:485101

1985. p. 306–316.

2011;47:669–680.

156–164. doi: 10.1016/j. enganabound.2019.01.006

Springer; 2008.

**260**

[42] Steinbach O. Numerical approximation methods for elliptic boundary value problems. New York:

1763.

[37] Xiaoyun Jiang and Haitao Qi, Thermal wave model of bioheat transfer with modified Riemann–Liouville fractional derivative, Journal of Physics A: Mathematical and Theoretical 2012;

[38] Bonnet G, Auriault JL. Dynamics of saturated and deform- able porous media: homogenization theory and determination of the solid-liquid coupling coefficients. Physics of finely divided matter, Berlin: Springer Verlag;

[39] Bonnet G. Basic singular solutions for a poroelastic medium in the dynamic range, J. Acoust. Soc. Am. 1987;82:1758–

[40] Messner M, Schanz M. A regularized collocation boundary element method for linear poroelasticity. Comput. Mech.

[41] Fahmy MA. Boundary Element Modeling and Simulation of Biothermomechanical Behavior in Anisotropic Laser-Induced Tissue Hyperthermia. Engineering Analysis with Boundary Elements. 2019;101:

Soc. Am. 1956;28:168–178.

[52] Fahmy MA. A novel BEM for modeling and simulation of 3T nonlinear generalized anisotropic micropolar-thermoelasticity theory with memory dependent derivative. CMES-Computer Modeling in Engineering & Sciences. 2021;126:175–199.

[53] Fahmy MA. A new boundary element algorithm for modeling and simulation of nonlinear thermal stresses in micropolar FGA composites with temperature-dependent properties. Advanced Modeling and Simulation in Engineering Sciences. 2021 (In Press).

[54] Fahmy MA. A new BEM for fractional nonlinear generalized porothermoelastic wave propagation problems, Computers, Materials and Continua. 2021 (In Press).

[55] Fahmy MA. A new boundary element technique for one- and twotemperature models of biothermomechanical behavior of anisotropic biological tissues. 14th Virtual Congress WCCM & ECCOMAS 2020. Virtual congress 11–15 January 2021.

**Chapter 16**

*Fritz Schwarz*

computer algebra

**1. Introduction**

until its end.

**263**

**Abstract**

Solving Second-Order Differential

The subject of this article are linear and quasilinear differential equations of

second order that may be decomposed into a first-order component with guaranteed solution procedure for obtaining closed-form solutions. These are homogeneous or inhomogeneous linear components, special Riccati components, Bernoulli, Clairaut or d'Alembert components. Procedures are described how they may be determined and how solutions of the originally given second order equation may be obtained from them. This makes it possible to solve new classes of differential equations and opens up a new area of research. Applying decomposition to linear *inhomogeneous* equations a simple procedure for determining a special solution follows. It is not based on the method of *variation of constants* of Lagrange, and consequently does not require the knowledge of a fundamental system. Algorithms based on these results are implemented in the computer algebra system ALLTYPES

**Keywords:** ordinary differential equations, decomposition, exact solutions,

The history of differential equations begins shortly after the establishment of the analysis by Newton and Leibniz in the 17th century. A brief overview of its first hundred years can be found in Appendix A of Ince's book [1]. These early investigations were mainly limited to first-order equations, associated with the names Riccati, Bernoulli and Euler. Starting in the early 18th century special linear

A more systematic search for solution methods was initiated by the results of Galois for solving algebraic equations in the early 19th century. Inspired by these results, Picard and Pessiot in Paris founded a solution theory for *linear* differential equations, known as Picard-Vessiot theory or differential Galois theory. A good introduction into their work and its extensions by Loewy may be found in the books [2, 3]. Completely independent of these activities Sophus Lie in Leipzig founded the so-called symmetry analysis for solving *nonlinear* differential Equations [4, 5]. Its main weaknesses are that most differential equations have no symmetries and therefore it cannot be applied. Furthermore, there are many differential equations with fairly simple closed form solutions that have no symmetries. That was essentially the status in the early twentieth century, which did not fundamentally change

which is available on the website www.alltypes.de.

equations of higher order were also investigated.

Equations by Decomposition

#### **Chapter 16**

## Solving Second-Order Differential Equations by Decomposition

*Fritz Schwarz*

#### **Abstract**

The subject of this article are linear and quasilinear differential equations of second order that may be decomposed into a first-order component with guaranteed solution procedure for obtaining closed-form solutions. These are homogeneous or inhomogeneous linear components, special Riccati components, Bernoulli, Clairaut or d'Alembert components. Procedures are described how they may be determined and how solutions of the originally given second order equation may be obtained from them. This makes it possible to solve new classes of differential equations and opens up a new area of research. Applying decomposition to linear *inhomogeneous* equations a simple procedure for determining a special solution follows. It is not based on the method of *variation of constants* of Lagrange, and consequently does not require the knowledge of a fundamental system. Algorithms based on these results are implemented in the computer algebra system ALLTYPES which is available on the website www.alltypes.de.

**Keywords:** ordinary differential equations, decomposition, exact solutions, computer algebra

#### **1. Introduction**

The history of differential equations begins shortly after the establishment of the analysis by Newton and Leibniz in the 17th century. A brief overview of its first hundred years can be found in Appendix A of Ince's book [1]. These early investigations were mainly limited to first-order equations, associated with the names Riccati, Bernoulli and Euler. Starting in the early 18th century special linear equations of higher order were also investigated.

A more systematic search for solution methods was initiated by the results of Galois for solving algebraic equations in the early 19th century. Inspired by these results, Picard and Pessiot in Paris founded a solution theory for *linear* differential equations, known as Picard-Vessiot theory or differential Galois theory. A good introduction into their work and its extensions by Loewy may be found in the books [2, 3]. Completely independent of these activities Sophus Lie in Leipzig founded the so-called symmetry analysis for solving *nonlinear* differential Equations [4, 5]. Its main weaknesses are that most differential equations have no symmetries and therefore it cannot be applied. Furthermore, there are many differential equations with fairly simple closed form solutions that have no symmetries. That was essentially the status in the early twentieth century, which did not fundamentally change until its end.

In this situation, a new solution method based on *decompositions* was proposed [6]. Essentially a decomposition means to find a *component* of lower order such that the original equation may be represented as a differential polynomial in terms of this component. Its existence is based on the following observation. Let *F x*, *y*, *y*0 , *y*00 ð Þ¼ 0 be a second-order differential equation for a function *y* depending on a variable *x*, and *ω*ð Þ¼ *x*, *y*,*C*1,*C*2 0 its general solution depending on two undetermined constants. It describes a two-parameter family of curves in the *x* � *y*-plane. If *C*1 and *C*2 are constrained by a relation *φ*ð Þ¼ *C*1,*C*2 0 the resulting expression for *ω* contains effectively a single parameter *C*. It describes a family of curves that may obey a first-order differential equation called a component. Its solutions are also solutions of the originally given second-order equation.

Every second-order equation has an infinite number of first-order components corresponding to the choice of *φ*ð Þ *C*1,*C*2 . Any such component has the form

$$F(\mathbf{x}, \mathbf{y}, \mathbf{y}', \mathbf{y}'') = f(\mathbf{x}, \mathbf{y}, \mathbf{z}, \mathbf{z}', \mathbf{C}) (\mathbf{z} \equiv \mathbf{g}(\mathbf{x}, \mathbf{y}, \mathbf{y}', \mathbf{C})). \tag{1}$$

Equations of this form appear in numerous applications, as can be seen in the collections of solved examples quoted above. The following proposition has been proved in [6], it is the basis for generating quasilinear first-order components; as

Proposition 1 Let a second-order quasilinear Eq. (2) be given. A first-order

ð Þ �1 *k*

*k* � �ð Þ1 *k*

Proposition 2. Let a second-order quasilinear Eq. (2) be given. In order that it has a linear first-order component *z* � *y*0 þ *a x*ð Þ*y* þ *b x*ð Þ the coefficients *a x*ð Þ and *b x*ð Þ

ð Þ �1 *k*

*K*

*k*¼0

ð Þ *ay* þ *b k* � � ! *z* � *y*0 ð þ *ay* þ *b*Þ ¼ 0*:*

The coefficients *a* and *b* are solutions of a first-order algebro-differential system.

*a*0 � *a*2 þ *p a*ð Þ¼ , *b*, *x* 0, *b*0 � *ab* þ *q a*ð Þ¼ , *b*, *x* 0, *ri*ð Þ¼ *a*, *b*, *x* 0 (7)

for *i* ¼ 1, 2 … ; *p a*ð Þ , *b*, *x* , *q a*ð Þ , *b*, *x* and *ri*ð Þ *a*, *b*, *x* are polynomials in *a* and *b*, and rational in *x*; their maximal degree in *a* and *b* is *K*. The *ri*ð Þ *a*, *b*, *x* generate an ideal

*Proof.* Substituting *r* ¼ *a x*ð Þ*y* þ *b x*ð Þ into (3) yields (5). At this point *y* is considered as an undetermined function. Therefore the left-hand side of (5) is represented as a partial fraction in *y*. Equating its coefficients to zero yields sufficient conditions in order that (5) vanishes and *z* is a component of (2). The first order ode's for *a* and *b* in the determining system (7) originate from the coefficients of first and zeroth degree in *y* of (5). The polynomials in *a* and *b*, i.e. *p a*ð Þ , *b*, *x* , *q a*ð Þ , *b*, *x* and *ri*ð Þ *a*, *b*, *x* in (7) originate from the powers of *ay* þ *b* and the rational coefficients *ck*ð Þ *x*, *y* in (5), i.e. exclusively from the nonlinearities of (2). Substitution of *y*0 ¼ *z* � *ay* � *b* into (4) yields (6). As a result, the sums at the left-hand side of (6) are a polynomial

The proof may be found in Section 2 of [6]. As a first application linear firstorder components of the form *z* � *y*0 þ *a x*ð Þ*y* þ *b x*ð Þ are searched for, i.e. with the above notation *r x*ð Þ¼ , *y a x*ð Þ*y* þ *b x*ð Þ; its coefficients *a* and *b* are solutions of the socalled *determining system*, they may be in any field extension of ð Þ *x* . The following

*ck*ð Þ *x*, *y r*

*r k* � � ! *z* � *y*0 ð Þ¼ þ *r* 0*:* (4)

*k* ¼ 0*:* (3)

*ck*ð Þ *x*, *y* ð Þ *ay* þ *b k* ¼ 0*:* (5)

(6)

*K*

*k*¼0

Then the original second-order equation can be decomposed as

usual *y*0 � *dy*

have to satisfy

*z*0 � *az* þX

Its general form is

*Iab* ∈ ð Þ *x* ½ � *a*, *b* .

**265**

*K*

*k*¼1

*dx* and *D* � *d*

*z*0 � *ryz* þX

*K*

*k*¼1

proposition describes how they may be obtained.

Then (2) may be written as follows

*a*0 � *a*2 � �*y* þ *b*0 � *ab* �X

in *z* the coefficients of which may depend explicitly on *y*.

*ck*ð Þ *x*, *y* ð Þ *z* � *ay* � *b k* � �ð Þ1 *k*

*dx*.

*DOI: http://dx.doi.org/10.5772/intechopen.94993*

component *z* � *y*0 þ *r x*ð Þ , *y* exists if *r x*ð Þ , *y* satisfies

*Solving Second-Order Differential Equations by Decomposition*

*rx* � *rry* �X

*ck*ð Þ *x*, *y* ð Þ *z* � *r*

Its meaning may be described as follows. If *z* � *g x*, *y*, *y*0 ð Þ ,*C* is substituted into *f x*, *y*, *z*, *z*0 ð Þ ,*C* the second-order equation on the left-hand side is obtained. The constant *C* does not necessarily occur in *f* and *g*, the same is true for *y* and its occurence in *f*.

Solving a second-order equation by decomposition involves two steps. First a decomposition of a certain type has to be found. Then the first order equation has to be solved in order to get the solutions of the original second-order equation. Of particular interest are those components the solution of which can always be determined. These are linear homogeneous and inhomogeneous components, special Riccati components, Bernoulli, Clairaut or d'Alembert components.

In this article equations of second order for an unknown function *y* depending on *x* with leading term *y*00 or *y*0 *y*00 are considered. They are assumed to be linear in *y*00, polynomial in the derivatives *y*0 , and rational in *y* and *x*. Equations of this kind are fairly common in applications, therefore many special examples of them are given in the collections by Kamke [7], Murphy [8], Polyanin [9], Sachdev [10] and Zwillinger [11]. Many interesting applications of such differential equations can be found in the textbooks by MacCluer et al. [12] and Swift and Wirkus [13].

In the following Section 2 equations with leading term *y*00 are considered, and possible linear or Bernoulli components are determined. For linear inhomogeneous equations it is shown how decomposition leads to a new procedure for determining a special solution without first having to know a fundamental system. Equations with leading term *y*0 *y*00 and possible components of Clairaut or d'Alembert type are the subject of Section 3. Most of the examples do not have Lie symmetries, so decomposition is the only way to solve them. The last Section 4 discusses various possible generalizations of the decomposition method, on the one hand more general equations to be solved, on the other hand more general first-order components.

#### **2. Equations with leading term** *y*00

Equations that are linear in the highest derivative *y*00, but may contain powers of *y*0 with coefficients that are rational in *y* and *x* are considered in this section. Moreover it is assumed that they are primitive, i.e. the leading coefficient is unity. Their general form is

$$\left\|\mathbf{y}\right\|^{\prime} + \sum\_{k=0}^{K} c\_{k}(\mathbf{x}, \boldsymbol{\mathcal{y}}) \mathbf{y}^{\prime k} = \mathbf{0} \\ \text{with} \\ c\_{k}(\mathbf{x}, \boldsymbol{\mathcal{y}}) \in \mathbb{Q}(\boldsymbol{x}, \boldsymbol{\mathcal{y}}), \quad K \in \mathbb{N}. \tag{2}$$

*Solving Second-Order Differential Equations by Decomposition DOI: http://dx.doi.org/10.5772/intechopen.94993*

In this situation, a new solution method based on *decompositions* was proposed [6]. Essentially a decomposition means to find a *component* of lower order such that the original equation may be represented as a differential polynomial in terms of

, *y*00 ð Þ¼ 0 be a second-order differential equation for a function *y* depending

Every second-order equation has an infinite number of first-order components

, *y*00 ð Þ¼ *f x*, *y*, *z*, *z*0 ð Þ ,*C z* � *g x*, *y*, *y*0 ð Þ ð Þ ,*C :* (1)

*y*00 are considered. They are assumed to be linear in *y*00,

, and rational in *y* and *x*. Equations of this kind are

*y*00 and possible components of Clairaut or d'Alembert type are

*ck*ð Þ *x*, *y y*0*k* ¼ 0with*ck*ð Þ *x*, *y* ∈ ð Þ *x*, *y* , *K* ∈ *:* (2)

this component. Its existence is based on the following observation. Let

*Recent Developments in the Solution of Nonlinear Differential Equations*

on a variable *x*, and *ω*ð Þ¼ *x*, *y*,*C*1,*C*2 0 its general solution depending on two undetermined constants. It describes a two-parameter family of curves in the *x* � *y*-plane. If *C*1 and *C*2 are constrained by a relation *φ*ð Þ¼ *C*1,*C*2 0 the resulting expression for *ω* contains effectively a single parameter *C*. It describes a family of curves that may obey a first-order differential equation called a component. Its solutions are also solutions of the originally given second-order equation.

corresponding to the choice of *φ*ð Þ *C*1,*C*2 . Any such component has the form

Riccati components, Bernoulli, Clairaut or d'Alembert components.

Its meaning may be described as follows. If *z* � *g x*, *y*, *y*0 ð Þ ,*C* is substituted into *f x*, *y*, *z*, *z*0 ð Þ ,*C* the second-order equation on the left-hand side is obtained. The constant *C* does not necessarily occur in *f* and *g*, the same is true for *y* and its occurence in *f*. Solving a second-order equation by decomposition involves two steps. First a decomposition of a certain type has to be found. Then the first order equation has to be solved in order to get the solutions of the original second-order equation. Of particular interest are those components the solution of which can always be determined. These are linear homogeneous and inhomogeneous components, special

In this article equations of second order for an unknown function *y* depending

fairly common in applications, therefore many special examples of them are given in the collections by Kamke [7], Murphy [8], Polyanin [9], Sachdev [10] and Zwillinger [11]. Many interesting applications of such differential equations can be

In the following Section 2 equations with leading term *y*00 are considered, and possible linear or Bernoulli components are determined. For linear inhomogeneous equations it is shown how decomposition leads to a new procedure for determining a special solution without first having to know a fundamental system. Equations

Equations that are linear in the highest derivative *y*00, but may contain powers of

found in the textbooks by MacCluer et al. [12] and Swift and Wirkus [13].

the subject of Section 3. Most of the examples do not have Lie symmetries, so decomposition is the only way to solve them. The last Section 4 discusses various possible generalizations of the decomposition method, on the one hand more general equations to be solved, on the other hand more general first-order components.

*y*0 with coefficients that are rational in *y* and *x* are considered in this section. Moreover it is assumed that they are primitive, i.e. the leading coefficient is unity.

*F x*, *y*, *y*0

on *x* with leading term *y*00 or *y*0

with leading term *y*0

Their general form is

**264**

polynomial in the derivatives *y*0

**2. Equations with leading term** *y*00

*y*00 þX *K*

*k*¼0

*F x*, *y*, *y*0

Equations of this form appear in numerous applications, as can be seen in the collections of solved examples quoted above. The following proposition has been proved in [6], it is the basis for generating quasilinear first-order components; as usual *y*0 � *dy dx* and *D* � *d dx*.

Proposition 1 Let a second-order quasilinear Eq. (2) be given. A first-order component *z* � *y*0 þ *r x*ð Þ , *y* exists if *r x*ð Þ , *y* satisfies

$$r\_{\mathbf{x}} - rr\_{\mathbf{y}} - \sum\_{k=0}^{K} (-\mathbf{1})^k c\_k(\mathbf{x}, \mathbf{y}) r^k = \mathbf{0}.\tag{3}$$

Then the original second-order equation can be decomposed as

$$\left(z'-r\_{\mathcal{I}}z+\sum\_{k=1}^{K}c\_{k}(x,y)\left((z-r)^{k}-(-1)^{k}r^{k}\right)\right)(z\equiv y'+r)=0.\tag{4}$$

The proof may be found in Section 2 of [6]. As a first application linear firstorder components of the form *z* � *y*0 þ *a x*ð Þ*y* þ *b x*ð Þ are searched for, i.e. with the above notation *r x*ð Þ¼ , *y a x*ð Þ*y* þ *b x*ð Þ; its coefficients *a* and *b* are solutions of the socalled *determining system*, they may be in any field extension of ð Þ *x* . The following proposition describes how they may be obtained.

Proposition 2. Let a second-order quasilinear Eq. (2) be given. In order that it has a linear first-order component *z* � *y*0 þ *a x*ð Þ*y* þ *b x*ð Þ the coefficients *a x*ð Þ and *b x*ð Þ have to satisfy

$$(a'-a^2)y + b' - ab - \sum\_{k=0}^{K} (-1)^k c\_k(x, y)(ay + b)^k = 0. \tag{5}$$

Then (2) may be written as follows

$$\left( \left( z' - az + \sum\_{k=1}^{K} c\_k(x, y) \left( (z - ay - b)^k - (-1)^k (ay + b)^k \right) \right) \left( x \equiv y' + ay + b \right) = 0. \tag{6}$$

The coefficients *a* and *b* are solutions of a first-order algebro-differential system. Its general form is

$$a' - a^2 + p(a, b, \mathfrak{x}) = 0, \; b' - ab + q(a, b, \mathfrak{x}) = 0, \; r\_i(a, b, \mathfrak{x}) = 0 \tag{7}$$

for *i* ¼ 1, 2 … ; *p a*ð Þ , *b*, *x* , *q a*ð Þ , *b*, *x* and *ri*ð Þ *a*, *b*, *x* are polynomials in *a* and *b*, and rational in *x*; their maximal degree in *a* and *b* is *K*. The *ri*ð Þ *a*, *b*, *x* generate an ideal *Iab* ∈ ð Þ *x* ½ � *a*, *b* .

*Proof.* Substituting *r* ¼ *a x*ð Þ*y* þ *b x*ð Þ into (3) yields (5). At this point *y* is considered as an undetermined function. Therefore the left-hand side of (5) is represented as a partial fraction in *y*. Equating its coefficients to zero yields sufficient conditions in order that (5) vanishes and *z* is a component of (2). The first order ode's for *a* and *b* in the determining system (7) originate from the coefficients of first and zeroth degree in *y* of (5). The polynomials in *a* and *b*, i.e. *p a*ð Þ , *b*, *x* , *q a*ð Þ , *b*, *x* and *ri*ð Þ *a*, *b*, *x* in (7) originate from the powers of *ay* þ *b* and the rational coefficients *ck*ð Þ *x*, *y* in (5), i.e. exclusively from the nonlinearities of (2). Substitution of *y*0 ¼ *z* � *ay* � *b* into (4) yields (6). As a result, the sums at the left-hand side of (6) are a polynomial in *z* the coefficients of which may depend explicitly on *y*.

It is important to represent the left side of (5) as a partial fraction in *y*, only in this way the structure of the system (7) is assured.

#### **2.1 Linear equations**

If *K* ¼ 1, *c*1ð Þ¼ *x*, *y c*1ð Þ *x* and *c*0ð Þ¼ *x*, *y c*0ð Þ *x y* þ *cr*ð Þ *x* the above proposition contains the decomposition of linear equations as a special case as shown next.

Corollary 1 Let *K* ¼ 1, *c*1ð Þ¼ *x*, *y c*1ð Þ *x* , *c*0ð Þ¼ *x*, *y c*0ð Þ *x y* þ *cr*ð Þ *x* and the linear inhomogeneous second-order equation

$$
\sigma'' + c\_1(\mathbf{x})y' + c\_0(\mathbf{x})y + c\_r(\mathbf{x}) = \mathbf{0} \tag{8}
$$

Integration yields the general solution

*DOI: http://dx.doi.org/10.5772/intechopen.94993*

*Solving Second-Order Differential Equations by Decomposition*

*y* ¼ *C*1*xex* þ *C*2*xex*

special solution is determined as shown next. Example 2 Consider the equation

Here *c*1 ¼ � 1

the special solution *a* ¼ ffiffiffi

*z* � *y*0 þ ffiffiffi

*y*0 ¼ � cos

and the component

Its general solution

**2.2 Quasilinear equations**

**267**

This is also the general solution of Eq. (11).

ð

exp ð Þ �*x*

but one of the special cases of Section 4.9 (a), ... (e) in [7]. Then it may be advantageous to assume that all integration constants in (9) are zero and only a

*y*00 � 1

3 *x* ffiffiffi *x* � � p , it yields

*x* p tan 2

2 3 *x* ffiffiffi *x* p � �*y* þ

One more integration yields a special solution of (12).

*z* ¼ *y*0 þ 2*y* þ *C*exp ð Þþ �2*x*

*y* ¼ *C*1 exp ð Þþ �2*x C*2*x* exp ð Þþ �2*x*

*x* p � � ð ffiffiffi

*b*0 � 1 2*x* þ ffiffiffi *x* p tan

Its special solution leads to the component

2 3 *x* ffiffiffi

*x* p tan

It may occur that a fundamental system of a second-order equation is rather complicated. Usually this is the case when the Riccati equation for *a* in (9) does not have a special rational solution and the usual algorithms for solving it do not apply,

2*x*, *c*0 ¼ *x* and *cr* ¼ 1. The Riccati equation *a*0 � *a*2 � 1

2 3 ffiffiffi *x* p � � � � *b* � 1 ¼ 0*:*

*x* p

3 *x* ffiffiffi *x* � � p 2

The application of Corollary 1 is particularly convenient if the coefficients *c*1 and *c*0 are constant and the solutions of the algebraic equation *a*2 � *c*1*a* þ *c*0 ¼ 0 are also solutions of the Riccati equation for *a*. The following example is of this type.

Example 3 The equation *y*00 þ 4*y*0 þ 4*y* ¼ cosh ð Þ *x* has coefficients *c*1 ¼ *c*0 ¼ 4 and *cr* ¼ � cosh ð Þ *x* . The solution of ð Þ *a* � 2 2 ¼ 0 is *a* ¼ 2. It leads to *b*0 þ 2*b* ¼ cosh ð Þ *x*

> 1 3

is also the general solution of the given second-order equation. □

The most interesting applications of Proposition 2 relate to nonlinear equations, of course. They differ from the linear case mainly by the occurence of the ideal *Iab*

sinh ð Þ� *x*

9 cosh ð Þ� *x*

5

cos 2

ffiffiffi *x* p cos 2

3 *x* ffiffiffi *x* � � p

> ð cos

ð cos

2 3 *x* ffiffiffi *x* p � � *dx*ffiffiffi

> 2 3

sinh ð Þ *x :*

4

9 sinh ð Þ *x*

2 3 *x* ffiffiffi *x* p � � *dx*ffiffiffi

*dx*

*x*2 þ *x*2 � *x* log ð Þ� *x* 1 � �*e*

2*x y*0 þ *xy* þ 1 ¼ 0*:* (12)

*x:*

2*x a* � *x* ¼ 0 has

*x* p *:*

*x* p *dx:* □

be given. A first-order component *z* � *y*0 þ *a x*ð Þ*y* þ *b x*ð Þ exists if *a* and *b* are solutions of the determining system

$$a' - a^2 + c\_1(\mathfrak{x})a - c\_0(\mathfrak{x}) = \mathbf{0} \text{ and } b' + (c\_1(\mathfrak{x}) - a)b - c\_r(\mathfrak{x}) = \mathbf{0}.\tag{9}$$

If it is satisfied Eq. (8) may be written as

$$(z' + (c\_1 - a)z)(z \equiv y' + ay + b) = \mathbf{0}.\tag{10}$$

*Proof.* The system (9) follows from (5) for the given special values of *K* and the coefficients *ck*. Then reduction of (8) w.r.t. *z* yields (10). □

It is remarkable that in the case of linear equations the algebraic conditions *ri*ð Þ *a*, *b*, *x* are missing, i.e. they are the most significant contributions originating from possible nonlinearities in (2).

For linear *homogeneous* ode's, i.e. for *cr* ¼ 0 and *b* ¼ 0, Loewy decompositions have been shown to be an effective method for determining a fundamental system [3]. It is based on a factorization of the linear differential operator corresponding to the given equation over its base field, i.e. restricting the coefficients of the factors to the field of the coefficients of the given second-order equation. This restriction does not apply in the above corollary, the coefficients may be in any field extension.

For linear *inhomogeneous* equations in addition to a fundamental system a special solution has to be found. The above corollary avoids the usual method of variation of constants that somehow appears like an *ad hoc* method. The method described in the above corollary requires only a special solution of a Riccati equation and subsequently solving a linear first-order equation in order to obtain the general solution of the second-order Eq. (8). The following example applies this procedure.

Example 1 The equation

$$y'' - y' - \frac{1}{x}y = (x+1)e^x \tag{11}$$

is Equation 2.109 in Kamke's collection [7]. Here *c*1 ¼ �1, *c*0 ¼ � 1 *x* and *cr* ¼ �ð Þ *x* þ 1 *ex*. The Riccati equation *a*0 � *a*2 � *a* þ 1 *x* ¼ 0 has the special solution *a* ¼ �1 � 1 *x*. From *b*0 þ 1 *x b* ¼ ð Þ *x* þ 1 *ex* follows *b* ¼ *C x* þ 1 *x x*ð Þ 2 � *x* þ 1 *ex* and leads to the component

$$z = \mathbf{y}' - \left(\mathbf{1} + \frac{\mathbf{1}}{\varkappa}\right)\mathbf{y} + \frac{\mathbf{1}}{\varkappa}\left(\mathbf{C} + \left(\varkappa^2 - \varkappa + \mathbf{1}\right)\mathbf{e}^{\varkappa}\right).$$

*Solving Second-Order Differential Equations by Decomposition DOI: http://dx.doi.org/10.5772/intechopen.94993*

Integration yields the general solution

It is important to represent the left side of (5) as a partial fraction in *y*, only in

If *K* ¼ 1, *c*1ð Þ¼ *x*, *y c*1ð Þ *x* and *c*0ð Þ¼ *x*, *y c*0ð Þ *x y* þ *cr*ð Þ *x* the above proposition contains the decomposition of linear equations as a special case as shown next. Corollary 1 Let *K* ¼ 1, *c*1ð Þ¼ *x*, *y c*1ð Þ *x* , *c*0ð Þ¼ *x*, *y c*0ð Þ *x y* þ *cr*ð Þ *x* and the linear

be given. A first-order component *z* � *y*0 þ *a x*ð Þ*y* þ *b x*ð Þ exists if *a* and *b* are

*a*0 � *a*2 þ *c*1ð Þ *x a* � *c*0ð Þ¼ *x* 0 and *b*0 þ ð Þ *c*1ð Þ� *x a b* � *cr*ð Þ¼ *x* 0*:* (9)

*Proof.* The system (9) follows from (5) for the given special values of *K* and the coefficients *ck*. Then reduction of (8) w.r.t. *z* yields (10). □ It is remarkable that in the case of linear equations the algebraic conditions *ri*ð Þ *a*, *b*, *x* are missing, i.e. they are the most significant contributions originating

For linear *homogeneous* ode's, i.e. for *cr* ¼ 0 and *b* ¼ 0, Loewy decompositions

have been shown to be an effective method for determining a fundamental system [3]. It is based on a factorization of the linear differential operator corresponding to the given equation over its base field, i.e. restricting the coefficients of the factors to the field of the coefficients of the given second-order equation. This restriction does not apply in the above corollary, the coefficients may

For linear *inhomogeneous* equations in addition to a fundamental system a special solution has to be found. The above corollary avoids the usual method of variation of constants that somehow appears like an *ad hoc* method. The method described in the above corollary requires only a special solution of a Riccati equation and subsequently solving a linear first-order equation in order to obtain the general solution of the second-order Eq. (8). The following example applies this

*y*00 � *y*0 � 1

is Equation 2.109 in Kamke's collection [7]. Here *c*1 ¼ �1, *c*0 ¼ � 1

*x b* ¼ ð Þ *x* þ 1 *ex* follows *b* ¼ *C*

*y* þ 1 *x*

1 *x*

*cr* ¼ �ð Þ *x* þ 1 *ex*. The Riccati equation *a*0 � *a*2 � *a* þ 1

*z* ¼ *y*0 � 1 þ

*x y* ¼ ð Þ *x* þ 1 *e*

*x* þ 1

*C* þ *x*2 � *x* þ 1 *e x :*

*x* (11)

*x* ¼ 0 has the special solution

*x x*ð Þ 2 � *x* þ 1 *ex* and leads to

*x* and

*y*00 þ *c*1ð Þ *x y*0 þ *c*0ð Þ *x y* þ *cr*ð Þ¼ *x* 0 (8)

*z*0 ð Þ þ ð Þ *c*1 � *a z z* � *y*0 ð þ *ay* þ *b*Þ ¼ 0*:* (10)

this way the structure of the system (7) is assured.

*Recent Developments in the Solution of Nonlinear Differential Equations*

inhomogeneous second-order equation

solutions of the determining system

from possible nonlinearities in (2).

be in any field extension.

Example 1 The equation

*x*. From *b*0 þ 1

procedure.

*a* ¼ �1 � 1

**266**

the component

If it is satisfied Eq. (8) may be written as

**2.1 Linear equations**

$$y = C\_1 \varkappa e^{\varkappa} + C\_2 \varkappa e^{\varkappa} \int \exp\left(-\varkappa\right) \frac{d\varkappa}{\varkappa^2} + \left(\varkappa^2 - \varkappa \log\left(\varkappa\right) - 1\right) e^{\varkappa} \Lambda$$

This is also the general solution of Eq. (11).

It may occur that a fundamental system of a second-order equation is rather complicated. Usually this is the case when the Riccati equation for *a* in (9) does not have a special rational solution and the usual algorithms for solving it do not apply, but one of the special cases of Section 4.9 (a), ... (e) in [7]. Then it may be advantageous to assume that all integration constants in (9) are zero and only a special solution is determined as shown next.

Example 2 Consider the equation

$$
\lambda y'' - \frac{1}{2\varkappa} y' + \varkappa y + \mathbf{1} = \mathbf{0}.\tag{12}
$$

Here *c*1 ¼ � 1 2*x*, *c*0 ¼ *x* and *cr* ¼ 1. The Riccati equation *a*0 � *a*2 � 1 2*x a* � *x* ¼ 0 has the special solution *a* ¼ ffiffiffi *x* p tan 2 3 *x* ffiffiffi *x* � � p , it yields

$$b' - \left(\frac{1}{2\mathfrak{x}} + \sqrt{\mathfrak{x}}\tan\left(\frac{2}{3}\sqrt{\mathfrak{x}}\right)\right)b - 1 = \mathbf{0}.$$

Its special solution leads to the component

$$z \equiv y' + \sqrt{\pi} \tan\left(\frac{2}{3}\pi\sqrt{\pi}\right)y + \frac{\sqrt{\pi}}{\cos\left(\frac{2}{3}\pi\sqrt{\pi}\right)} \left[\cos\left(\frac{2}{3}\pi\sqrt{\pi}\right)\frac{d\pi}{\sqrt{\pi}}.\right]$$

One more integration yields a special solution of (12).

$$y\_0 = -\cos\left(\frac{2}{3}x\sqrt{\mathfrak{x}}\right) \left[\frac{\sqrt{\mathfrak{x}}}{\cos\left(\frac{2}{3}x\sqrt{\mathfrak{x}}\right)^2} \right] \cos\left(\frac{2}{3}x\sqrt{\mathfrak{x}}\right) \frac{d\mathfrak{x}}{\sqrt{\mathfrak{x}}}d\mathfrak{x}.\tag{7}$$

The application of Corollary 1 is particularly convenient if the coefficients *c*1 and *c*0 are constant and the solutions of the algebraic equation *a*2 � *c*1*a* þ *c*0 ¼ 0 are also solutions of the Riccati equation for *a*. The following example is of this type.

Example 3 The equation *y*00 þ 4*y*0 þ 4*y* ¼ cosh ð Þ *x* has coefficients *c*1 ¼ *c*0 ¼ 4 and *cr* ¼ � cosh ð Þ *x* . The solution of ð Þ *a* � 2 2 ¼ 0 is *a* ¼ 2. It leads to *b*0 þ 2*b* ¼ cosh ð Þ *x* and the component

$$z = y' + 2y + C \exp\left(-2x\right) + \frac{1}{3} \sinh\left(x\right) - \frac{2}{3} \sinh\left(x\right).$$

Its general solution

$$y = C\_1 \exp\left(-2\pi\right) + C\_2 \varkappa \exp\left(-2\pi\right) + \frac{5}{9} \cosh\left(\varkappa\right) - \frac{4}{9} \sinh\left(\varkappa\right)$$

is also the general solution of the given second-order equation. □

#### **2.2 Quasilinear equations**

The most interesting applications of Proposition 2 relate to nonlinear equations, of course. They differ from the linear case mainly by the occurence of the ideal *Iab*

in (7), which defines algebraic conditions *ri*ð Þ¼ *a*, *b*, *x* 0 for the coefficients of a possible component. Furthermore, the first-order ode's for *a* and *b* are modified due to the nonlinearity by additional terms. The structure of the determining system (7) suggests the following solution procedure.

*a*0 þ 1

*DOI: http://dx.doi.org/10.5772/intechopen.94993*

*z*0 � 1 *y z*2 � 1

be given explicitly.

example. Here *c*2 ¼ 1

*y*00 � 1 *y y*02 þ *k y*

Example 6 The equation

*y*

undetermined function then *a* ¼ Ð

*y*0 þ *q x*ð Þ*y*

*y z*2 þ

exists. Defining *Q x*ð Þ� ,*C*1

*y* ¼ exp �

¼ *z*0 � 1

component yields

mentioned above.

**269**

, *c*1 ¼ *p x*ð Þ *y*

ð

ð

*q x*ð Þ*dx* þ *C* � �*z* � *k*

Ð

*Q x*ð Þ ,*C*1 *dx* � � *k*

� � *z* � *y*0 þ

*z* � *y*0 þ *ay* þ *b* exists if *a* and *b* are solutions of the system

with solution *a* ¼ � log ð Þþ *x C*, *b* ¼ � 2

*y* ¼ 2

of Eq. (13), it does not have a Lie symmetry.

*y* log ð Þ *x y* � *Cy* þ

� �*z*

3 *xx* exp ð Þ *C*1*x*

*x* ¼ 0, *b*0

*Solving Second-Order Differential Equations by Decomposition*

þ *ab* þ

2 3

� � *z* � *y*0 � ð Þ log ð Þ� *x C y* � 2

2

is obtained. Integration of the first-order component leads to the general solution

ð

Here the question arises how exceptional are the equations that have a oneparameter family of linear components of the first order and thus have a general solution in closed form. The following example is a generalization of the previous one. A family of second order equations is constructed whose general solution can

with undetermined coefficients *p x*ð Þ and *q x*ð Þ generalizes the preceding

*a*0 � *q x*ð Þ¼ 0, *b*0 þ *ab* þ *p x*ð Þ*a* ¼ 0, *b b*ð Þ¼ þ *p x*ð Þ 0*:*

The result may be described as follows. If *p x*ð Þ¼ *k* is a constant, and *q x*ð Þ is an

*y z*

ð

This is the general solution of Eq. (14). □ It turns out that a behavior similar to that in the previous example often applies, i.e. first-order linear components often exist not only for isolated equations, but for entire families, which are parameterized by indefinite functions. This explains the

Bernoulli equations are another class of first-order ode's with guaranteed closed form general solutions. In addition to a term linear in *y* they contain a nonlinearity *yn* where *n* is an integer; *n* ¼ 1 or *n* ¼ 0 correspond to linear homogeneous or linear

�

existence of families of solvable equations as those given in the collections

exp ð

3 *a* ¼ 0, *b*2 þ

exp ð Þ �*C*1*x*

� �

2 3 *b* ¼ 0

3 from which the decomposition

� �

*dx xx* þ *C*2

*yy*00 � *y*02 þ *p x*ð Þ*y*0 þ *q x*ð Þ*y*2 ¼ 0 (14)

and *c*0 ¼ *q x*ð Þ*y*. A first-order linear component

*q x*ð Þ*dx* þ *C*, *b* ¼ �*k* and the decomposition

ð

*Q x*ð Þ ,*C*1 *dx*Þ þ *C*2 � �*:*

*q x*ð Þ*dx* þ *C*1 integration of the first-order

*q x*ð Þ*dx* þ *C* � �*y* � *k* � �

¼ 0

3

¼ 0

At first the algebraic system *ri*ð Þ¼ *a*, *b*, *x* 0 is established and a Gröbner basis for the ideal *Iab* is generated. Usually it may be determined rather efficiently.

If it is inconsistent a linear component does not exist in any field extension. This applies to a generic nonlinear equation of the form (2).

If the ideal *Iab* is finite-dimensional each solution that satisfies the two firstorder ode's yields a component that may be integrated and leads to a one-parameter family of solutions of the given second-order equation.

Finally, the algebraic equations may generate a relation between *a* and *b*; substitution into the first-order differential equations may lead to one of the above cases, or to a solution depending on a parameter. In the latter case a one-parameter family of linear components exists, integrating the corresponding equation yields the general solution of the given second-order equation containing two undetermined constants.

Subsequently this proceeding will be illustrated by several examples. They show that all of the alternatives mentioned actually exist.

Example 4 Consider the equation

$$y'' + \left. xy'^2 + (\varkappa - \mathbf{1})yy' + \frac{\varkappa}{\varkappa + \mathbf{1}}y' - y^2 - \frac{\mathbf{1}}{\varkappa + \mathbf{1}}y = \mathbf{0}.$$

Its coefficients *c*2 ¼ *x*, *c*1 ¼ ð Þ *x* � 1 *y* þ *x x*þ1 and *c*0 ¼ �*y*2 � *y x*þ1 result in the system

$$a' - a^2 - 2abx + \frac{a\varkappa}{\varkappa + 1} + b(\varkappa - 1) = 0, \; b' - ab - b^2\varkappa + \frac{b\varkappa}{\varkappa + 1} + \frac{1}{\varkappa + 1} = 0,$$

$$a^2 - \frac{\varkappa - 1}{\varkappa}a - \frac{1}{\varkappa} = 0.$$

The single algebraic equation has the solutions *a* ¼ � 1 *x* and *a* ¼ 1 and the decompositions

$$\left(z' + \varkappa z^2 + \frac{\varkappa^3 y + 2\varkappa^2 y + \varkappa^2 + \varkappa y + \varkappa + 1}{\varkappa^2 + \varkappa} z\right) \left(z \equiv y' - \frac{1}{\varkappa} y\right) = 0,$$

$$\left(z' + \varkappa z^2 - \frac{\varkappa^2 y + 2\varkappa y + y + 1}{\varkappa + 1} z\right) (z \equiv y' + y) = 0$$

follow. Integration of the two components yields the two one-parameter families *y* ¼ *C*exp ð Þ �*x* and *y* ¼ *Cx* of solution curves. It is not obvious how the general solution of the second-order equation involving two constants is supported by them. □

The most interesting, of course, are equations that allow a one-parameter family of linear components and whose integration gives the general solution. The next example is of this type.

Example 5 Consider the equation

$$yy'' - {y'}^2 + \frac{2}{3}y' - \frac{1}{x}y^2 = 0\tag{13}$$

with *K* ¼ 2 and the coefficients *c*2 ¼ � 1 *y* , *c*1 ¼ 2 3*y* and *c*0 ¼ � *y x*; they generate the determining system

*Solving Second-Order Differential Equations by Decomposition DOI: http://dx.doi.org/10.5772/intechopen.94993*

$$a' + \frac{1}{\mathfrak{x}} = 0, \ b' + ab + \frac{2}{\mathfrak{z}}a = 0, \ b^2 + \frac{2}{\mathfrak{z}}b = 0$$

with solution *a* ¼ � log ð Þþ *x C*, *b* ¼ � 2 3 from which the decomposition

$$\left( \left( z' - \frac{1}{\mathcal{Y}} z^2 - \frac{1}{\mathcal{Y}} \left( \log \left( x \right) \mathcal{Y} - \mathcal{C} \boldsymbol{y} + \frac{2}{3} \right) \mathbf{z} \right) \left( \mathbf{z} \equiv \mathcal{Y} - \left( \log \left( \mathbf{x} \right) - \mathcal{C} \boldsymbol{y} - \frac{2}{3} \right) \mathbf{z} \right) = \mathbf{0}$$

is obtained. Integration of the first-order component leads to the general solution

$$y = \frac{2}{3} \mathfrak{x}^{\mathfrak{x}} \exp\left(\mathbf{C}\_1 \mathfrak{x}\right) \left(\int \exp\left(-\mathbf{C}\_1 \mathfrak{x}\right) \frac{d\mathfrak{x}}{\mathfrak{x}^{\mathfrak{x}}} + \mathbf{C}\_2\right),$$

of Eq. (13), it does not have a Lie symmetry.

Here the question arises how exceptional are the equations that have a oneparameter family of linear components of the first order and thus have a general solution in closed form. The following example is a generalization of the previous one. A family of second order equations is constructed whose general solution can be given explicitly.

Example 6 The equation

in (7), which defines algebraic conditions *ri*ð Þ¼ *a*, *b*, *x* 0 for the coefficients of a possible component. Furthermore, the first-order ode's for *a* and *b* are modified due to the nonlinearity by additional terms. The structure of the determining system

the ideal *Iab* is generated. Usually it may be determined rather efficiently.

At first the algebraic system *ri*ð Þ¼ *a*, *b*, *x* 0 is established and a Gröbner basis for

If it is inconsistent a linear component does not exist in any field extension. This

Subsequently this proceeding will be illustrated by several examples. They show

*x x* þ 1

þ *b x*ð Þ¼ � 1 0, *b*0 � *ab* � *b*2

*x* þ 1 *z*

follow. Integration of the two components yields the two one-parameter families

2 3 *y*0 � 1

> *y* , *c*1 ¼ 2 3*y*

*y* ¼ *C*exp ð Þ �*x* and *y* ¼ *Cx* of solution curves. It is not obvious how the general solution of the second-order equation involving two constants is supported by them. □ The most interesting, of course, are equations that allow a one-parameter family of linear components and whose integration gives the general solution. The next

*a* � 1

*x* ¼ 0*:*

*z*

*a*2 � *x* � 1 *x*

*x*3*y* þ 2*x*2*y* þ *x*2 þ *xy* þ *x* þ 1 *x*2 þ *x*

The single algebraic equation has the solutions *a* ¼ � 1

*z*0 þ *xz*2 � *x*2*y* þ 2*xy* þ *y* þ 1

*yy*00 � *y*02 þ

*y*0 � *y*2 � 1

*x*þ1 and *c*0 ¼ �*y*2 � *y*

*x* þ 1

*x* þ

*z* � *y*0 � 1

*z* � *y*0 ð Þ¼ þ *y* 0

and *c*0 ¼ � *y*

*y* ¼ 0*:*

*bx x* þ 1 þ 1 *x* þ 1 ¼ 0,

*x* and *a* ¼ 1 and the

*x y*

*x y*2 ¼ 0 (13)

*x*; they generate the

¼ 0,

*x*þ1 result in the system

If the ideal *Iab* is finite-dimensional each solution that satisfies the two firstorder ode's yields a component that may be integrated and leads to a one-parameter

Finally, the algebraic equations may generate a relation between *a* and *b*; substitution into the first-order differential equations may lead to one of the above cases, or to a solution depending on a parameter. In the latter case a one-parameter family of linear components exists, integrating the corresponding equation yields

the general solution of the given second-order equation containing two

(7) suggests the following solution procedure.

undetermined constants.

*a*0 � *a*2 � 2*abx* þ

*z*0 þ *xz*2 þ

example is of this type.

determining system

**268**

Example 5 Consider the equation

with *K* ¼ 2 and the coefficients *c*2 ¼ � 1

decompositions

applies to a generic nonlinear equation of the form (2).

*Recent Developments in the Solution of Nonlinear Differential Equations*

family of solutions of the given second-order equation.

that all of the alternatives mentioned actually exist.

Its coefficients *c*2 ¼ *x*, *c*1 ¼ ð Þ *x* � 1 *y* þ *x*

*ax x* þ 1

*y*00 þ *xy*02 þ ð Þ *x* � 1 *yy*0 þ

Example 4 Consider the equation

$$(yy'' - y'^2 + p(\mathbf{x})y' + q(\mathbf{x})y^2 = \mathbf{0} \tag{14}$$

with undetermined coefficients *p x*ð Þ and *q x*ð Þ generalizes the preceding example. Here *c*2 ¼ 1 *y* , *c*1 ¼ *p x*ð Þ *y* and *c*0 ¼ *q x*ð Þ*y*. A first-order linear component *z* � *y*0 þ *ay* þ *b* exists if *a* and *b* are solutions of the system

$$a' - q(\mathbf{x}) = \mathbf{0}, \ b' + ab + p(\mathbf{x})a = \mathbf{0}, \ b(b + p(\mathbf{x})) = \mathbf{0}.$$

The result may be described as follows. If *p x*ð Þ¼ *k* is a constant, and *q x*ð Þ is an undetermined function then *a* ¼ Ð *q x*ð Þ*dx* þ *C*, *b* ¼ �*k* and the decomposition

$$\begin{aligned} y'' - \frac{1}{\mathcal{Y}} \mathsf{y}'^2 + \frac{k}{\mathcal{Y}} \mathsf{y}' + q(\mathsf{x}) \mathsf{y} \\ &= \left( \mathsf{z}' - \frac{1}{\mathcal{Y}} \mathsf{z}^2 + \left( \int q(\mathsf{x}) d\mathsf{x} + \mathsf{C} \right) \mathsf{z} - \frac{k}{\mathcal{Y}} \mathsf{z} \right) \left( \mathsf{z} \equiv \mathsf{y}' + \left( \int q(\mathsf{x}) d\mathsf{x} + \mathsf{C} \right) \mathsf{y} - k \right) = \mathsf{0} \end{aligned}$$

exists. Defining *Q x*ð Þ� ,*C*1 Ð *q x*ð Þ*dx* þ *C*1 integration of the first-order component yields

$$y = \exp\left(-\int Q(\mathbf{x}, \mathbf{C}\_1) d\mathbf{x}\right) \left(k \int \exp\left(\int Q(\mathbf{x}, \mathbf{C}\_1) d\mathbf{x}\right) + \mathbf{C}\_2\right).$$

This is the general solution of Eq. (14). □ It turns out that a behavior similar to that in the previous example often applies, i.e. first-order linear components often exist not only for isolated equations, but for entire families, which are parameterized by indefinite functions. This explains the existence of families of solvable equations as those given in the collections mentioned above.

Bernoulli equations are another class of first-order ode's with guaranteed closed form general solutions. In addition to a term linear in *y* they contain a nonlinearity *yn* where *n* is an integer; *n* ¼ 1 or *n* ¼ 0 correspond to linear homogeneous or linear

inhomogeneous equations, respectively. Similar as for linear components, a special Bernoulli component guarantees a one-parameter set of solution curves of a given second-order equation, and a one-parameter family of such components guarantees the general solution of the latter. The main result of this section is the following proposition. *a*0 � 2*ab* þ 2*b* ¼ 0, *b*0

*Solving Second-Order Differential Equations by Decomposition*

1 2 *x*2 þ *C* � �*z* � � *z* � *y*0 þ *y*3 þ

*y* ¼ 1 ffiffiffiffiffiffi ð Þ2 p exp *C*1*x* þ 1

*y*3

has the only nonvanishing coefficient *c*2 ¼ � 3*y*2�1

ð *a*0 ð Þ � ð Þ *n* þ 1 *ab yn* þ *b*0 � *b*2 � �*y* � *na*2

� 1 *y*

ð *a*0 � 4*a*2 þ 2*ab* � �*y*3 þ *b*0

*b* ¼ *a*; Its solution *a* ¼ *b* ¼ � 1

*x*þ*C* ¼ 0 with general solution

possible in special cases.

Substitution into (15) yields

Example 9 Consider the equation

2

**271**

� 4*y y*2 þ 1 � � *a*2

It turns out that for *n* ¼ 3 this condition specializes to

2

*y* ¼

*y*00 þ

with *K* ¼ 2 and non-vanishing coefficients *c*2 ¼ *x*

*x y* � 1

p

follows. Integrating the right component yields the general solution

*y*00 þ *yy*00 � 3*y*2

6 *x*3 � � Ð exp �2*C*1*x* � 1

of Eq. (18). It does not have a Lie symmetry. □ The next example deals with a problem in hydrodynamics. The boundary layer at a circular cylinder immersed in the uniform flow of liquid is considered [14], see

*y*2*n*�1

þ 4*a*2 � 8*ab* þ 2*b*2 � �*y* � 4ð Þ *a* � *b* 2 *y*

*x*þ*C* leads to the Bernoulli equation *y*0 � � 1

*y*02 þ *y*0 þ *xy* ¼ 0 (22)

, *c*1 ¼ 1 and *c*0 ¼ *xy*.

*y*�1

After some simplifications the resulting system for *a* and *b* is *a*0 � 2*a*2 ¼ 0 and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � *C*1*x* � *C*2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *C*1*x* þ *C*2 p *:*

This is also the general solution of Eq. (20) □ In general it is *a priori* not known whether there exists a Bernoulli component of any order. If a component for small values of *n* cannot be found it is desirable to determine bounds for its possible existence. The next example shows that this is

*x*þ*C y*3 � 1

*y*2*n* þ 2*abyn*þ1 þ *b*2 *y*2 � � ¼ 0*:*

This system has the solution *a* ¼ 1, *b* ¼ 1

*z*2 þ *y*2 þ

*DOI: http://dx.doi.org/10.5772/intechopen.94993*

*z*0 � 1 *y*

also Eq. 6.210 of [7].

Example 8 The equation

� *x* ¼ 0, *a*2 � *a* ¼ 0*:*

2 *x*2 þ *C* from which the decomposition

*y*02 þ *y*02 ¼ 0 (20)

*y y*ð Þ 2þ1 . Substitution into (15) yields

*y*2 þ 1 ¼ 0*:*

2

(21)

¼ 0

2 *C*2

1 2 *x*2 þ *C* � �*y*

� �

3 *x*3 � �*dx* þ 1

� �1*=*2

Proposition 3 Let a second-order quasilinear Eq. (2) be given. In order that it has a first-order Bernoulli component *z* � *y*0 þ *a x*ð Þ*yn* þ *b x*ð Þ*y*, *n*∈ , the coefficients *a* and *b* have to satisfy

$$(a' - (n+1)ab)y^n + (b' - b^2)y - na^2y^{2n-1} - \sum\_{k=0}^{K} (-1)^k c\_k(x, y)(ay^n + by)^k = 0. \tag{15}$$

Then (2) may be written as follows

$$\begin{aligned} \left( z' - \left( nay^{n-1} + b \right) z + \sum\_{k=1}^{K} \left( \left( z - ay^n - by \right)^k + \left( -1 \right)^{k+1} \left( ay^n + by \right)^k \right) \right) \\ \qquad \left( z \equiv y' + ay^n + by \right) = 0. \end{aligned} \tag{16}$$

The coefficients *a* and *b* may be obtained from a first-order algebro-differential system; its general form is

$$a' - (n+1)ab + p(a,b,\mathfrak{x}) = 0, \quad b' - b^2 + q(a,b,\mathfrak{x}) = 0, \quad r\_i(a,b,\mathfrak{x}) = 0; \tag{17}$$

*p a*ð Þ , *b*, *x* , *q a*ð Þ , *b*, *x* and *ri*ð Þ *a*, *b*, *x* are polynomials in *a* and *b*, and rational in *x*; the maximal degree in *a* and *b* is *K*, they generate an ideal *Iab* in the ring ð Þ *x* ½ � *a*, *b* .

*Proof.* Substituting *r* ¼ *a x*ð Þ*yn* þ *b x*ð Þ*y* into (3) yields condition (15). Representing its left-hand side as partial fraction in the variable *y*, the coefficients of the various terms yield sufficient conditions for its vanishing. They form the algebro-differential system (17). The first order ode's for *a* and *b* originate from the coefficients of *n*th and first degree in *y*, respectively; *p a*ð Þ , *b*, *x* , *q a*ð Þ , *b*, *x* and *ri*ð Þ *a*, *b*, *x* originate from the coefficients *ck*ð Þ *x*, *y* and the powers of *ayn* þ *by*.

Substitution of *y*0 ¼ *z* � *ayn* � *by* into (4) yields (16). As a result, the sums at the left-hand side of (16) are a polynomial in *z* and *z*0 the coefficients of which may depend explicitly on *y*. □

The structure of the system (17) is similar as for linear components considered above, and consequently also the proceeding for its solution. The following examples applies the above proposition.

Example 7 The equation

$$\left(\mathbf{y}\mathbf{y}''-\mathbf{y}'^2+2\mathbf{y}^3\mathbf{y}'+\mathbf{x}\mathbf{y}^2=\mathbf{0}\tag{18}$$

with *K* ¼ 2 has coefficients *c*2 ¼ � 1 *y* , *c*1 ¼ 2*y*2 and *c*0 ¼ *xy*. For generic *n* the condition

$$(a' - (n - 1)ab)y'' + (b' - x)y - (n - 1)a^2y^{2n - 1} + 2ay^{n + 2} + 2by^3 = 0 \tag{19}$$

follows. The two equations *b*0 � *x* ¼ 0 and *b* ¼ 0 originating from the coefficients of *y* and *y*3, respectively, are inconsistent. In order for a Bernoulli component to exist, this inconsistency must be compensated by other coefficients for a suitable choice of *n*. To this end either *n* ¼ 1 or *n* ¼ 3 is required. The former leads to the inconsistency *x* ¼ 0, whereas the latter yields

*Solving Second-Order Differential Equations by Decomposition DOI: http://dx.doi.org/10.5772/intechopen.94993*

$$a' - 2ab + 2b = 0, \quad b' - x = 0, \quad a^2 - a = 0.$$

This system has the solution *a* ¼ 1, *b* ¼ 1 2 *x*2 þ *C* from which the decomposition

$$\left(z' - \frac{1}{\mathcal{Y}}z^2 + \left(\mathcal{y}^2 + \frac{1}{2}\mathbf{x}^2 + \mathcal{C}\right)\mathbf{z}\right)\left(z \equiv \mathcal{Y}' + \mathcal{Y}^3 + \left(\frac{1}{2}\mathbf{x}^2 + \mathcal{C}\right)\mathbf{y}\right) = \mathbf{0}$$

follows. Integrating the right component yields the general solution

$$y = \frac{1}{\sqrt{(2)} \exp\left(C\_1 \varkappa + \frac{1}{6} \varkappa^3\right) \left(\int \exp\left(-2C\_1 \varkappa - \frac{1}{3} \varkappa^3\right) d\varkappa + \frac{1}{2}C\_2\right)^{1/2}}$$

of Eq. (18). It does not have a Lie symmetry. □ The next example deals with a problem in hydrodynamics. The boundary layer at a circular cylinder immersed in the uniform flow of liquid is considered [14], see also Eq. 6.210 of [7].

Example 8 The equation

inhomogeneous equations, respectively. Similar as for linear components, a special Bernoulli component guarantees a one-parameter set of solution curves of a given second-order equation, and a one-parameter family of such components guarantees the general solution of the latter. The main result of this section is the following proposition. Proposition 3 Let a second-order quasilinear Eq. (2) be given. In order that it has a first-order Bernoulli component *z* � *y*0 þ *a x*ð Þ*yn* þ *b x*ð Þ*y*, *n*∈ , the coefficients *a*

*Recent Developments in the Solution of Nonlinear Differential Equations*

*y*2*n*�1 �X *K*

*z* � *ayn* ð Þ � *by k* þ �ð Þ1 *k*þ1

*z* � *y*0 þ *ayn* ð þ *by*Þ ¼ 0*:*

The coefficients *a* and *b* may be obtained from a first-order algebro-differential

*a*0 � ð Þ *n* þ 1 *ab* þ *p a*ð Þ¼ , *b*, *x* 0, *b*0 � *b*2 þ *q a*ð Þ¼ , *b*, *x* 0, *ri*ð Þ¼ *a*, *b*, *x* 0; (17)

*p a*ð Þ , *b*, *x* , *q a*ð Þ , *b*, *x* and *ri*ð Þ *a*, *b*, *x* are polynomials in *a* and *b*, and rational in *x*; the maximal degree in *a* and *b* is *K*, they generate an ideal *Iab* in the ring ð Þ *x* ½ � *a*, *b* .

Representing its left-hand side as partial fraction in the variable *y*, the coefficients of the various terms yield sufficient conditions for its vanishing. They form the algebro-differential system (17). The first order ode's for *a* and *b* originate from the

Substitution of *y*0 ¼ *z* � *ayn* � *by* into (4) yields (16). As a result, the sums at the left-hand side of (16) are a polynomial in *z* and *z*0 the coefficients of which may depend explicitly on *y*. □ The structure of the system (17) is similar as for linear components considered

*y*0 þ *xy*2 ¼ 0 (18)

*y*2*n*�1 þ 2*ayn*þ2 þ 2*by*3 ¼ 0 (19)

, *c*1 ¼ 2*y*2 and *c*0 ¼ *xy*. For generic *n* the

*Proof.* Substituting *r* ¼ *a x*ð Þ*yn* þ *b x*ð Þ*y* into (3) yields condition (15).

coefficients of *n*th and first degree in *y*, respectively; *p a*ð Þ , *b*, *x* , *q a*ð Þ , *b*, *x* and *ri*ð Þ *a*, *b*, *x* originate from the coefficients *ck*ð Þ *x*, *y* and the powers of *ayn* þ *by*.

above, and consequently also the proceeding for its solution. The following

*yy*00 � *y*02 þ 2*y*3

*y*

follows. The two equations *b*0 � *x* ¼ 0 and *b* ¼ 0 originating from the coefficients of *y* and *y*3, respectively, are inconsistent. In order for a Bernoulli component to exist, this inconsistency must be compensated by other coefficients for a suitable choice of *n*. To this end either *n* ¼ 1 or *n* ¼ 3 is required. The former leads to the

*ayn* ð Þ þ *by k* � � !

*k*¼0

ð Þ �1 *k*

*ck*ð Þ *x*, *y ayn* ð Þ þ *by k* ¼ 0*:*

(15)

(16)

and *b* have to satisfy

*a*0 ð Þ � ð Þ *n* þ 1 *ab yn* þ *b*0 � *b*2 � �*y* � *na*2

Then (2) may be written as follows

*K*

*k*¼1

*z*0 � *nayn*�1 þ *b* � �*z* þX

examples applies the above proposition.

with *K* ¼ 2 has coefficients *c*2 ¼ � 1

*a*0 ð Þ � ð Þ *n* � 1 *ab yn* þ *b*0 � *x* � �*y* � ð Þ *n* � 1 *a*2

inconsistency *x* ¼ 0, whereas the latter yields

Example 7 The equation

condition

**270**

system; its general form is

$$y^3y'' + yy'' - \mathfrak{B}y^2y'^2 + y'^2 = \mathbf{0} \tag{20}$$

has the only nonvanishing coefficient *c*2 ¼ � 3*y*2�1 *y y*ð Þ 2þ1 . Substitution into (15) yields

$$\begin{aligned} \left( (a' - (n+1)ab)y^n + (b' - b^2)y - na^2y^{2n-1} \\ -\left(\frac{1}{y} - \frac{4y}{y^2 + 1}\right) (a^2y^{2n} + 2aby^{n+1} + b^2y^2) = 0. \end{aligned} \tag{21}$$

It turns out that for *n* ¼ 3 this condition specializes to

$$((a' - 4a^2 + 2ab)y^3 + (b' + 4a^2 - 8ab + 2b^2)y - 4(a - b)^2 \frac{y}{y^2 + 1} = 0.1$$

After some simplifications the resulting system for *a* and *b* is *a*0 � 2*a*2 ¼ 0 and *b* ¼ *a*; Its solution *a* ¼ *b* ¼ � 1 2 *x*þ*C* leads to the Bernoulli equation *y*0 � � 1 2 *x*þ*C y*3 � 1 2 *x*þ*C* ¼ 0 with general solution

$$\mathcal{Y} = \frac{\sqrt{\mathbf{1} - \mathbf{C}\_1 \mathbf{x} - \mathbf{C}\_2}}{\sqrt{\mathbf{C}\_1 \mathbf{x} + \mathbf{C}\_2}}.$$

This is also the general solution of Eq. (20) □ In general it is *a priori* not known whether there exists a Bernoulli component of any order. If a component for small values of *n* cannot be found it is desirable to determine bounds for its possible existence. The next example shows that this is possible in special cases.

Example 9 Consider the equation

$$y'' + \frac{\varkappa}{y-1}y'^2 + y' + \varkappa y = 0\tag{22}$$

with *K* ¼ 2 and non-vanishing coefficients *c*2 ¼ *x y*�1 , *c*1 ¼ 1 and *c*0 ¼ *xy*. Substitution into (15) yields

$$(a' - (n+1)ab)y'' + (b' - b^2)y - na^2y^{2n-1} - xy + ay^n + by - \frac{x}{y-1}(ay^n + by)^2.$$

Expanding the last term into unique partial fractions by using the general formula

$$\frac{\mathcal{y}^n}{\mathcal{y}-k} = \sum\_{\nu=0}^{n-1} k^{n-\nu-1} \mathcal{y}^\nu + \frac{k^n}{\mathcal{y}-k} \tag{23}$$

The determining system for the two functions*f p*ð Þ and *g p*ð Þ may be obtained explicitly from (25) if the coefficients *c x*ð Þ , *y* and *ck*ð Þ *x*, *y* are known. Without restric-

equation obtained after separation w.r.t. *x*, and an algebraic system in *p*, *f p*ð Þ, *g p*ð Þ,

factorization is a suitable tool for solving them in many cases. If a solution has been obtained the corresponding component may be applied for generating the decomposition of the given equation explicitly. The following example uses this proceeding.

2*x y*02 � 1

0 *fp* � *f* 0 *p*2 þ *f*

*p* þ *fg* þ 2*fp* � *g*0

Transforming the left-hand sides into algebraic Gröbner bases in the term order

ð Þ *p* > *f p*ð Þ> *g p*ð Þ> *p*, the following two systems and their solutions are

*f* � *p* ¼ 0, *gp* þ *p*2 � 1 ¼ 0, ! *f* ¼ *p*, *g* ¼ 1

*xy*0 *z*

*xy*0 *z*

expression in the left-hand bracket of this decomposition and has to be discarded. Integration of the d'Alembert component *z* � *y*02 þ *yy*0 � *Cx* � 1 leads to the

� *C*1*p* log

� *C*1 log

Eq. (26) does not have a Lie symmetry. □

respectively. The former decomposition generates a Clairaut component. It yields

*p* þ *f* ¼ 0, *gp* þ *p*2 � 1 ¼ 0, ! *f* ¼ *C* 1

*y*0 *z*0 þ *xy*02 � *y*02 � 1

*y*0 *z*0 � *xy*02 þ *y*02 þ 1

!

!

general solution of (26) in a parameter representation

q

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *C*1 þ *p*2

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *C*1 þ *p*2

0

ð Þ *p* follows. It turns out that an algebraic Gröbner basis algorithm including

ð Þ *p* and *g*0

2*x yy*0 þ

2*x*. A linear or Bernoulli component does not exist. Proposition 4

1

2 � *fp* ¼ 0,

*p*2 � *gp* � 2*p*2 ¼ 0*:*

*p* � *p*,

*p* � *p:*

*y*0

*y*0

¼ 0,

¼ 0,

,

*:*

*p* , *g* ¼ 1

*z* � *y* � *xy*0 þ *y*0 � 1

� �

*y*0 *x* þ *y*0 � 1

p þ ð Þ *C*1*C*2 þ 1 *p*

p þ *C*1*C*2 þ 1

*z* � *y* � *C*

*C* � *C* of (26), *C* is an undetermined constant. Its parameter

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *C*1 þ *p*2 p þ *p* ffiffiffiffiffi *C*1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *C*1 þ *p*2 p þ *p* ffiffiffiffiffi *C*1

!

!

*p* does not solve it, it annihilates a lower-order factor of the

� �

ð Þ *p* may occur linearly in any

2*x* ¼ 0*:* (26)

2 *y*, *c*2 ¼ *x*�1

2*x* ,

tions on the coefficients*ck*ð Þ *x*, *y* the derivatives *f*

*DOI: http://dx.doi.org/10.5772/intechopen.94993*

*Solving Second-Order Differential Equations by Decomposition*

Example 10 Consider the equation

*g*0 *gp* þ *g*0

*f* 0 *gp* þ *f* 0 *p*2 � *f* 0 þ *f g*0

*f* 0

They lead to the decompositions

*zz*0 þ *xy*02 þ *y*02 þ 1

*zz*0 þ *y*02 þ *Cx* þ 1

*p*2 þ 1, *y* ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *C*1 þ *p*2 p

*C*1 þ *p*2 p *p*

the solution *y* ¼ *Cx* þ 1

*x* ¼ 1 *C*1

*y* ¼ � 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

solution *x* ¼ 1

**273**

*y*0 *y*00 þ 1

2 *yy*00 þ *x* � 1

Here *J* ¼ 1 and *K* ¼ 2, its nonvanishing coefficients are *c x*ð Þ¼ , *y* 1

*p*2 � *g*0 ¼ 0, *f*

*f* 0 ð Þ *p* and *g*0

*c*1 ¼ � *y*

*f* 0 ð Þ *p* > *g*0

obtained.

2*x* and *c*0 ¼ 1

leads to the system

leads to

$$\begin{aligned} (a' - (n+1)ab)y'' + (b' - b^2)y - na^2y^{2n-1} - xy + ay^n + by^n\\ -a^2x \sum\_{\nu=0}^{2n-1} y^\nu - 2abx \sum\_{\nu=0}^n y^\nu - \frac{(a+b)^2x}{y-1} = 0. \end{aligned}$$

The coefficients of the various terms yield a system for the unknowns *a*, *b* and *n*. There is always the subsystem *a*0 þ *a*2 þ *a* þ *x* ¼ 0, *a* þ *b* ¼ 0 independent of *n*, it originates from the coefficient of *y* and the term independent of *y*. Furthermore, the leading term of the first sum in the above equation requires *a* ¼ 0 for for any *n*≥2. These equations combined are inconsistent, i.e. the above Eq. (22) does not allow a Bernoulli component for any nonnegative natural number *n*. A similar reasoning exists for negative values of *n*. □

At the moment an algorithm for determining bounds for *n* is not known, it is not even clear whether the existence of bounds is decidable in general.

#### **3. Equations with leading term** *y*0 *y*00

Another important class of differential equations are those with leading term *y*0 *y*00, they are considered in this section. Their general form is

$$c\_{\lambda} \mathbf{y}\_{\lambda}^{\prime\prime} + c(\mathbf{x}, \mathbf{y}) \mathbf{y}^{\prime\prime} + \sum\_{k=0}^{K} c\_{k}(\mathbf{x}, \mathbf{y}) \mathbf{y}^{\prime k} = \mathbf{0} \text{ with } c(\mathbf{x}, \mathbf{y}), c\_{k}(\mathbf{x}, \mathbf{y}) \in \mathbb{Q}(\mathbf{x})[\mathbf{y}], \quad K \in \mathbb{N}. \tag{24}$$

Components of Clairaut or d'Alembert type *z* � *y* � *xf y*0 ð Þ� *g y*0 ð Þ may lead to partial or even general solutions in closed form, mostly in a parameter representation. The main result of this section is given in the following proposition.

Proposition 4 Let a second-order differential Eq. (24) be given. A first-order component *z* � *y* � *xf y*0 ð Þ� *g y*0 ð Þ exists if *f y*0 ð Þ and *g y*0 ð Þ satisfy

$$\begin{aligned} (p - f(p))(c(\mathbf{x}, \mathbf{x}f(p) + \mathbf{g}(p)) + p) \\ + (\mathbf{x}f'(p) + \mathbf{g}'(p)) \sum\_{k=0}^{K} c\_k(\mathbf{x}, \mathbf{x}f(p) + \mathbf{g}(p))p^k = \mathbf{0} \end{aligned} \tag{25}$$

where *p* � *y*0 has been defined. Representing the left hand side of (25) as a partial fraction w.r.t. *x* and equating the coefficients of the various terms to zero, a system of first-order quasilinear ode's for *f p*ð Þ and *g p*ð Þ is obtained; its degree in *f p*ð Þ and *g p*ð Þ is not higher than the degree in *y* of the coefficients *c x*ð Þ , *y* and *ck*ð Þ *x*, *y* .

*Proof.* Reduction of (24) w.r.t. *z* � *y* � *xf y*0 ð Þ� *g y*0 ð Þ leads to Eq. (25). Their properties follow directly from the assumptions about the coefficients *c x*ð Þ , *y* and *ck*ð Þ *x*, *y* in (24), and representing the left hand side of (25) as a partial fraction in *x*. *Solving Second-Order Differential Equations by Decomposition DOI: http://dx.doi.org/10.5772/intechopen.94993*

The determining system for the two functions*f p*ð Þ and *g p*ð Þ may be obtained explicitly from (25) if the coefficients *c x*ð Þ , *y* and *ck*ð Þ *x*, *y* are known. Without restrictions on the coefficients*ck*ð Þ *x*, *y* the derivatives *f* 0 ð Þ *p* and *g*0 ð Þ *p* may occur linearly in any equation obtained after separation w.r.t. *x*, and an algebraic system in *p*, *f p*ð Þ, *g p*ð Þ, *f* 0 ð Þ *p* and *g*0 ð Þ *p* follows. It turns out that an algebraic Gröbner basis algorithm including factorization is a suitable tool for solving them in many cases. If a solution has been obtained the corresponding component may be applied for generating the decomposition of the given equation explicitly. The following example uses this proceeding.

Example 10 Consider the equation

*a*0 ð Þ � ð Þ *n* þ 1 *ab yn* þ *b*0 � *b*2 � �*y* � *na*2

�*a*2 *x* 2 X*n*�1 *ν*¼0

**3. Equations with leading term** *y*0

*K*

*k*¼0

þ *x f*0

*y*00 þ *c x*ð Þ , *y y*00 þX

formula

*y*0

**272**

leads to

*y*2*n*�1 � *xy* þ *ayn* þ *by* � *x*

*kn*

*x y* � 1 ¼ 0*:*

*y*2*n*�1 � *xy* þ *ayn* þ *by*

Expanding the last term into unique partial fractions by using the general

*kn*�*ν*�1

*yν* þ

*yν* � ð Þ *a* þ *b* 2

The coefficients of the various terms yield a system for the unknowns *a*, *b* and *n*. There is always the subsystem *a*0 þ *a*2 þ *a* þ *x* ¼ 0, *a* þ *b* ¼ 0 independent of *n*, it originates from the coefficient of *y* and the term independent of *y*. Furthermore, the leading term of the first sum in the above equation requires *a* ¼ 0 for for any *n*≥2. These equations combined are inconsistent, i.e. the above Eq. (22) does not allow a Bernoulli component for any nonnegative natural number *n*. A similar reasoning exists for negative values of *n*. □ At the moment an algorithm for determining bounds for *n* is not known, it is not

*ν*¼0

*ν*¼0

*yn y* � *k* ¼ X*n*�1

*Recent Developments in the Solution of Nonlinear Differential Equations*

*a*0 ð Þ � ð Þ *n* þ 1 *ab yn* þ *b*0 � *b*2 � �*y* � *na*2

*yν* � 2*abx*X*n*

even clear whether the existence of bounds is decidable in general.

they are considered in this section. Their general form is

*y*00

Another important class of differential equations are those with leading term *y*0

Components of Clairaut or d'Alembert type *z* � *y* � *xf y*0 ð Þ� *g y*0 ð Þ may lead to partial or even general solutions in closed form, mostly in a parameter representa-

Proposition 4 Let a second-order differential Eq. (24) be given. A first-order

*K*

*k*¼0

where *p* � *y*0 has been defined. Representing the left hand side of (25) as a partial fraction w.r.t. *x* and equating the coefficients of the various terms to zero, a system of first-order quasilinear ode's for *f p*ð Þ and *g p*ð Þ is obtained; its degree in *f p*ð Þ and *g p*ð Þ is not higher than the degree in *y* of the coefficients *c x*ð Þ , *y* and *ck*ð Þ *x*, *y* . *Proof.* Reduction of (24) w.r.t. *z* � *y* � *xf y*0 ð Þ� *g y*0 ð Þ leads to Eq. (25). Their properties follow directly from the assumptions about the coefficients *c x*ð Þ , *y* and *ck*ð Þ *x*, *y* in (24), and representing the left hand side of (25) as a partial fraction in *x*.

tion. The main result of this section is given in the following proposition.

component *z* � *y* � *xf y*0 ð Þ� *g y*0 ð Þ exists if *f y*0 ð Þ and *g y*0 ð Þ satisfy

ð Þþ *p g*0 ð Þ *p* � �X

ð Þ *p* � *f p*ð Þ ð Þ *c x*ð , *xf p*ð Þþ *g p*ð ÞÞ þ *p*

*ck*ð Þ *x*, *y y*0*k* ¼ 0 with *c x*ð Þ , *y* ,*ck*ð Þ *x*, *y* ∈ ð Þ *x* ½ � *y* , *K* ∈ *:* (24)

*ck*ð Þ *x*, *xf p*ð Þþ *g p*ð Þ *pk* ¼ 0 (25)

*y* � 1 *ayn* ð Þ þ *by* 2

*y* � *k* (23)

*:*

*y*00,

$$y'y'' + \frac{1}{2}yy'' + \frac{\varkappa - 1}{2\varkappa}y'^2 - \frac{1}{2\varkappa}yy' + \frac{1}{2\varkappa} = 0. \tag{26}$$

Here *J* ¼ 1 and *K* ¼ 2, its nonvanishing coefficients are *c x*ð Þ¼ , *y* 1 2 *y*, *c*2 ¼ *x*�1 2*x* , *c*1 ¼ � *y* 2*x* and *c*0 ¼ 1 2*x*. A linear or Bernoulli component does not exist. Proposition 4 leads to the system

$$\mathbf{g'}\mathbf{g}p + \mathbf{g'}p^2 - \mathbf{g'} = \mathbf{0}, \quad f'\mathbf{f}p - f'p^2 + f^2 - f\mathbf{p} = \mathbf{0},$$

$$f'\mathbf{g}p + f'p^2 - f' + f\mathbf{g'}p + f\mathbf{g} + 2\mathbf{f}p - \mathbf{g'}p^2 - \mathbf{g}p - 2p^2 = \mathbf{0}.$$

Transforming the left-hand sides into algebraic Gröbner bases in the term order *f* 0 ð Þ *p* > *g*0 ð Þ *p* > *f p*ð Þ> *g p*ð Þ> *p*, the following two systems and their solutions are obtained.

$$f - p = 0, \text{g} \\ p + p^2 - \mathbf{1} = \mathbf{0}, \ \rightarrow \\ f = p, \mathbf{g} = \frac{1}{p} - p,$$

$$f'p + f = \mathbf{0}, \mathbf{g}p + p^2 - \mathbf{1} = \mathbf{0}, \ \rightarrow f = \mathbf{C} \\ \frac{1}{p}, \mathbf{g} = \frac{1}{p} - p.$$

They lead to the decompositions

$$\begin{aligned} \left( \mathbf{z} \mathbf{z}' + \frac{\mathbf{x} \mathbf{y}'^2 + \mathbf{y}'^2 + \mathbf{1}}{\mathbf{y}'} \mathbf{z}' + \frac{\mathbf{x} \mathbf{y}'^2 - \mathbf{y}'^2 - \mathbf{1}}{\mathbf{x} \mathbf{y}'} \mathbf{z} \right) \left( \mathbf{z} \equiv \mathbf{y} - \mathbf{x} \mathbf{y}' + \mathbf{y}' - \frac{\mathbf{1}}{\mathbf{y}'} \right) &= \mathbf{0}, \\\ \left( \mathbf{z} \mathbf{z}' + \frac{\mathbf{y}'^2 + \mathbf{C} \mathbf{x} + \mathbf{1}}{\mathbf{y}'} \mathbf{z}' - \frac{\mathbf{x} \mathbf{y}'^2 + \mathbf{y}'^2 + \mathbf{1}}{\mathbf{x} \mathbf{y}'} \mathbf{z} \right) \left( \mathbf{z} \equiv \mathbf{y} - \frac{\mathbf{C}}{\mathbf{y}'} \mathbf{x} + \mathbf{y}' - \frac{\mathbf{1}}{\mathbf{y}'} \right) &= \mathbf{0}, \end{aligned}$$

respectively. The former decomposition generates a Clairaut component. It yields the solution *y* ¼ *Cx* þ 1 *C* � *C* of (26), *C* is an undetermined constant. Its parameter solution *x* ¼ 1 *p*2 þ 1, *y* ¼ 2 *p* does not solve it, it annihilates a lower-order factor of the expression in the left-hand bracket of this decomposition and has to be discarded.

Integration of the d'Alembert component *z* � *y*02 þ *yy*0 � *Cx* � 1 leads to the general solution of (26) in a parameter representation

$$\begin{aligned} x &= \frac{1}{C\_1\sqrt{C\_1+p^2}} \left(\sqrt{C\_1+p^2} - C\_1p\log\frac{\sqrt{C\_1+p^2}+p}{\sqrt{C\_1}} + (C\_1C\_2+1)p\right), \\ y &= -\frac{1}{\sqrt{C\_1+p^2}} \left(p\sqrt{C\_1+p^2} - C\_1\log\frac{\sqrt{C\_1+p^2}+p}{\sqrt{C\_1}} + C\_1C\_2+1\right). \end{aligned}$$

Eq. (26) does not have a Lie symmetry. □

This example shows that solutions of a component must be tested to see if they meet the second order equation, otherwise they have to be discarded; this phenomenon seems to be quite common.

#### **4. Conclusions**

The structure of the determining systems for linear or Bernoulli components of a nonlinear Eq. (2) given in Propositions 2 or 3, respectively, show clearly its relation to the corresponding system for the decomposition of a linear equation. For a generic equation of the second order this appears to be the best possible result. The same applies to the verious solution steps given on page 6. The corresponding result for determining Clairaut and d'Alembert components given in Proposition 4 is less specific. However, it should be possible, to obtain more detailed results if special classes of second-order equations are considered. In general, this area is only at an early stage and a better understanding of the underlying mechanisms generating the solutions and also its limitations would be highly desirable.

There are numerous possible generalizations fairly obvious. On the one hand, this concerns the equations to be solved. More general function fields for its coefficients like e.g. algebraic or elementary functions may be allowed. Equations of order three or four would be interesting in many applications. The greatest challenge however is certainly to develop similar procedures for partial differential equations as it has been indicated in Section 5 of [6].

On the other hand, the component type offers space for extensions too. In principle all equations of first order, as described for example in Kamke's book [7], Part A, Section 4, are possible components. Components that guarantee at least a partial solution are of course particularly useful, the most important of them have been discussed in this article.

In order to apply decompositions to concrete problems the implementation of the procedures described in this article are available on the website www.alltypes.de [15].

Beyond that there are a number of general problems related to decompositions. For instance the question how rare are equations that allow a particular decomposition, Example 6 provides a partial answer. If two or more one-parameter families of solution curves are known as in Example 4, does it faciliate generating the general solution? The exact relation between Lie's symmetry analysis and solution by decompositions is another subject of interest.

**Author details**

provided the original work is properly cited.

Fraunhofer Gesellschaft, Institute SCAI, Schloss Birlinghoven, Germany

*Solving Second-Order Differential Equations by Decomposition*

*DOI: http://dx.doi.org/10.5772/intechopen.94993*

\*Address all correspondence to<EMAIL_ADDRESS>
© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Fritz Schwarz

**275**

#### **Acknowledgements**

Various helpful comments of a referee are gratefully acknowledged.

*Solving Second-Order Differential Equations by Decomposition DOI: http://dx.doi.org/10.5772/intechopen.94993*

This example shows that solutions of a component must be tested to see if they meet the second order equation, otherwise they have to be discarded; this phenom-

The structure of the determining systems for linear or Bernoulli components of a nonlinear Eq. (2) given in Propositions 2 or 3, respectively, show clearly its relation to the corresponding system for the decomposition of a linear equation. For a generic equation of the second order this appears to be the best possible result. The same applies to the verious solution steps given on page 6. The corresponding result for determining Clairaut and d'Alembert components given in Proposition 4 is less specific. However, it should be possible, to obtain more detailed results if special classes of second-order equations are considered. In general, this area is only at an early stage and a better understanding of the underlying mechanisms generating the

There are numerous possible generalizations fairly obvious. On the one hand, this concerns the equations to be solved. More general function fields for its coefficients like e.g. algebraic or elementary functions may be allowed. Equations of order three or four would be interesting in many applications. The greatest challenge however is certainly to develop similar procedures for partial differential equations

On the other hand, the component type offers space for extensions too. In principle all equations of first order, as described for example in Kamke's book [7], Part A, Section 4, are possible components. Components that guarantee at least a partial solution are of course particularly useful, the most important of them have

Various helpful comments of a referee are gratefully acknowledged.

In order to apply decompositions to concrete problems the implementation of the procedures described in this article are available on the website www.alltypes.de [15]. Beyond that there are a number of general problems related to decompositions. For instance the question how rare are equations that allow a particular decomposition, Example 6 provides a partial answer. If two or more one-parameter families of solution curves are known as in Example 4, does it faciliate generating the general solution? The exact relation between Lie's symmetry analysis and solution

solutions and also its limitations would be highly desirable.

*Recent Developments in the Solution of Nonlinear Differential Equations*

as it has been indicated in Section 5 of [6].

by decompositions is another subject of interest.

been discussed in this article.

**Acknowledgements**

**274**

enon seems to be quite common.

**4. Conclusions**

### **Author details**

Fritz Schwarz Fraunhofer Gesellschaft, Institute SCAI, Schloss Birlinghoven, Germany

\*Address all correspondence to<EMAIL_ADDRESS>
© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] E. L. Ince, Ordinary Differential Equations, Longmans, Green and Co., 1926 [Reprint by Dover Publications Inc., 1960].

[2] A. Magid, Lectures on differential Galois theory, AMS university lecture series, vol 7, AMS Press, Providence, 1994.

[3] F. Schwarz, Loewy Decomposition of Linear Differential Equations, Springer-Verlag, Wien, 2012 and Bulletin of Mathematical Sciences **3**, page 19–71 (2012); DOI 10.1007/s13373-012- 0026-7.

[4] S. Lie, Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen, Leipzig, 1891; reprinted by Chelsea Publishing Company, New York, 1967.

[5] F. Schwarz, Algorithmic Lie Theory for Solving Ordinary Differential Equations, Chapman & Hall/CRC, 2007.

[6] F. Schwarz, Decomposition of ordinary differential equations, Bulletin of MathematicalSciences **7**, 565–613 (2017); doi.org/10.1007/s13373-017- 0110-0.

[7] E. Kamke, Differentialgleichungen, Lösungsmethoden und Lösungen I. Gewöhnliche Differentialgleichungen, Akademische Verlagsgesellschaft, Leipzig, 1967.

[8] G. M. Murphy, Ordinary Differential Equations and Their Solutions, Van Nostrand Company, 1960.

[9] A. D. Polyanin, V. F. Zaitsev, Handbook of Ordinary Differential Equations, Exact Solutions, Methods and Problems, CRC Press, 2018.

[10] P. L. Sachdev, A Compendium on Nonlinear Ordinary Differential Equations, John Wiley & Sons, 1997.

[11] D. Zwillinger, Handbook of Differential Equations, Academic Press, 1998.

**Chapter 17**

**Abstract**

domains.

**1. Introduction**

*Q* and *Dαβ* ¼ *∂*∣*α*∣

**277**

*∂x α*1 1 , ⋯, *∂xαn n*

Sobolev-Morrey space *Wm*,1

equations, unbounded domain

Ω � ð Þ 0, *T* , being a cylinder in *n*þ1

*ut* � X ∣*α*∣ ≤ *m*, ∣*β*∣ ≤ *m*

> ⋯ *∂*∣*β*∣ *∂y β*1 1 , ⋯, *∂y βn n*

The Uniformly Parabolic

Morrey Spaces and Elliptic

*Tair Gadjiev and Konul Suleymanova*

*p*,*φ* ∘

Equations of Higher Order with

Discontinuous Data in Generalized

Equations in Unbounded Domains

We study the regularity of the solutions of the Cauchy-Dirichlet problem for linear uniformly parabolic equations of higher order with vanishing mean oscillation (VMO) coefficients. We prove continuity in generalized parabolic Morrey spaces *Mp*,*φ* of sublinear operators generated by the parabolic Calderon-Zygmund operator and by the commutator of this operator with bounded mean oscillation

**Keywords:** higher order parabolic equations, generalized Morrey spaces, sublinear operators, Calderon-Zygmund integrals, VMO, Cauchy-Dirichlet problem, elliptic

We consider the higher order linear Cauchy-Dirichlet problem in *Q* ¼

where *∂pQ* ¼ ð Þ ∂Ω � ½ � 0, *T* ∪ ð Þ Ω � f g *t* ¼ 0 stands for the parabolic boundary of

The unique strong solvability of this type problem was proved in [1]. In [2] the

, ∣*α*∣ ¼ P*n*

regularity of the solution in the Morrey spaces *Lp*,*λ n*þ1 � � with *p* ∈ð Þ 1, ∞ ,

ð Þ *Q* . Also we consider elliptic equation in unbounded

, Ω ⊂ *Rn* be a bounded domain 0 <*T* < ∞

*aαβ*ð Þ *x*, *t Dαβu x*ð Þ¼ , *t f x*ð Þ , *t* , *a:e:* in *Q* (1)

*u x*ð Þ¼ , *t* 0 on *∂pQ*, (2)

*k*¼1*βk*.

*k*¼1*αk*, *β* ¼ P*n*

(BMO) functions. We obtain strong solution belongs to the generalized

[12] B. D. MacCluer, P. S. Bourdon, T. L. Kriete, Differential Equations, Techniques, Theory, and Applications, American Mathematical Society, 2019.

[13] R. J. Swift, S. A. Wirkus, A Course in Ordinary Differential Equations, Chapman & Hall/CRC, 2007.

[14] K. Hiemenz, Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder, Dinglers Polytechnisches Journal 326(1911).

[15] F. Schwarz, ALLTYPES in the Web, ACM Communications in Computer Algebra **42**, No. 3, 185–187(2008).

#### **Chapter 17**

**References**

Inc., 1960].

1994.

0026-7.

0110-0.

Leipzig, 1967.

**276**

[1] E. L. Ince, Ordinary Differential Equations, Longmans, Green and Co., 1926 [Reprint by Dover Publications

*Recent Developments in the Solution of Nonlinear Differential Equations*

[11] D. Zwillinger, Handbook of

Kriete, Differential Equations,

Ordinary Differential Equations, Chapman & Hall/CRC, 2007.

einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder, Dinglers Polytechnisches Journal 326(1911).

1998.

Differential Equations, Academic Press,

[12] B. D. MacCluer, P. S. Bourdon, T. L.

Techniques, Theory, and Applications, American Mathematical Society, 2019.

[13] R. J. Swift, S. A. Wirkus, A Course in

[14] K. Hiemenz, Die Grenzschicht an

[15] F. Schwarz, ALLTYPES in the Web, ACM Communications in Computer Algebra **42**, No. 3, 185–187(2008).

[2] A. Magid, Lectures on differential Galois theory, AMS university lecture series, vol 7, AMS Press, Providence,

[3] F. Schwarz, Loewy Decomposition of Linear Differential Equations, Springer-Verlag, Wien, 2012 and Bulletin of Mathematical Sciences **3**, page 19–71 (2012); DOI 10.1007/s13373-012-

Differentialgleichungen mit bekannten infinitesimalen Transformationen, Leipzig, 1891; reprinted by Chelsea Publishing Company, New York, 1967.

[5] F. Schwarz, Algorithmic Lie Theory for Solving Ordinary Differential Equations, Chapman & Hall/CRC, 2007.

[7] E. Kamke, Differentialgleichungen, Lösungsmethoden und Lösungen I. Gewöhnliche Differentialgleichungen, Akademische Verlagsgesellschaft,

[8] G. M. Murphy, Ordinary Differential Equations and Their Solutions, Van

Nostrand Company, 1960.

[9] A. D. Polyanin, V. F. Zaitsev, Handbook of Ordinary Differential Equations, Exact Solutions, Methods and Problems, CRC Press, 2018.

[10] P. L. Sachdev, A Compendium on Nonlinear Ordinary Differential Equations, John Wiley & Sons, 1997.

[6] F. Schwarz, Decomposition of ordinary differential equations, Bulletin of MathematicalSciences **7**, 565–613 (2017); doi.org/10.1007/s13373-017-

[4] S. Lie, Vorlesungen über

## The Uniformly Parabolic Equations of Higher Order with Discontinuous Data in Generalized Morrey Spaces and Elliptic Equations in Unbounded Domains

*Tair Gadjiev and Konul Suleymanova*

#### **Abstract**

We study the regularity of the solutions of the Cauchy-Dirichlet problem for linear uniformly parabolic equations of higher order with vanishing mean oscillation (VMO) coefficients. We prove continuity in generalized parabolic Morrey spaces *Mp*,*φ* of sublinear operators generated by the parabolic Calderon-Zygmund operator and by the commutator of this operator with bounded mean oscillation (BMO) functions. We obtain strong solution belongs to the generalized

Sobolev-Morrey space *Wm*,1 *p*,*φ* ∘ ð Þ *Q* . Also we consider elliptic equation in unbounded domains.

**Keywords:** higher order parabolic equations, generalized Morrey spaces, sublinear operators, Calderon-Zygmund integrals, VMO, Cauchy-Dirichlet problem, elliptic equations, unbounded domain

#### **1. Introduction**

We consider the higher order linear Cauchy-Dirichlet problem in *Q* ¼ Ω � ð Þ 0, *T* , being a cylinder in *n*þ1 , Ω ⊂ *Rn* be a bounded domain 0 <*T* < ∞

$$u\_t - \sum\_{|\alpha| \le m, \atop |\beta| \le m} a\_{\alpha \beta}(\mathbf{x}, t) D^{\alpha \beta} u(\mathbf{x}, t) = f(\mathbf{x}, t), \text{ a.e. in } Q \tag{1}$$

$$|\vartheta| \le m$$

$$u(\mathbf{x}, t) = \mathbf{0} \text{ on } \partial\_p Q, \tag{2}$$

where *∂pQ* ¼ ð Þ ∂Ω � ½ � 0, *T* ∪ ð Þ Ω � f g *t* ¼ 0 stands for the parabolic boundary of *Q* and *Dαβ* ¼ *∂*∣*α*∣ *∂x α*1 1 , ⋯, *∂xαn n* ⋯ *∂*∣*β*∣ *∂y β*1 1 , ⋯, *∂y βn n* , ∣*α*∣ ¼ P*n k*¼1*αk*, *β* ¼ P*n k*¼1*βk*.

The unique strong solvability of this type problem was proved in [1]. In [2] the regularity of the solution in the Morrey spaces *Lp*,*λ n*þ1 � � with *p* ∈ð Þ 1, ∞ ,

*λ*∈ð Þ 0, *n* þ 2 and also its Hölder regularity was studied. In [3] Nakai extend these studies on generalized Morrey spaces *Mp*,*φ n*þ1 � � with a weight *φ* satisfying the integral condition

**2. Some notation and definition**

*DOI: http://dx.doi.org/10.5772/intechopen.96781*

*x* ¼ *x*00 ð Þ , *xn*, *t* ∈ *Dn*þ1

∣*Br*∣ ¼ *c* � *rn*; I*r x*0 ð Þ¼ , *t y*∈ *n*þ1 :j*x*0 � *y*0

*Sn* is the unit sphere in *n*þ1

We use the equivalent one

of the ellipsoids

E*r* j j�1Ð E*r*

**279**

oscillation [14]) if

∣ � ∣ is the Euclidean metric, ∣*x*∣ ¼ P*n*

I*r*ð Þ *x*, *τ* ∩ *Q* for each ð Þ *x*, *τ* ∈ *Q*, 2I*r*ð Þ¼I *x*, *τ* 2*r*ð Þ *x*, *τ* .

*Diu* ¼ *∂u ∂xi*

The following notations are used in this paper:

*x* ¼ *x*0 ð Þ , *t* , *y* ¼ *y*0 ð Þ , *τ* ∈ *n*þ1 ¼ *Rn* � *Rn*, *n*þ1

;

*Dαβu* ¼ *∂*∣*α*∣

the letter *C* is used for various positive constants.

*ρ*ð Þ¼ *x*, *t*

E*r*ð Þ¼ *x y*∈ *n*þ1 : *x*0 � *y*0 j j2