**5. Illustrative Examples**

In this section, to show the applicability and powerfulness of the introduced method, we present the solutions to five linear and nonlinear systems of fractional order differential equations.

**Example 1.** *We first considered the following linear system of fractional differential equations [7,8]:*

$$\begin{aligned} D^a u(t) &= u(t) + v(t) \\ D^a v(t) &= -u(t) + v(t) \end{aligned}$$

*subject to*

$$u(0) = 0, \ v(0) = 1.$$

*The exact solution of this system when α* = 1 *is known to be*

$$u(t) = e^t \sin t, \ v(t) = e^t \cos t.$$

This example was examined for *M* = 2, *k* = 0, and *α* = 0.9, 0.7, 0.5. When the obtained results were matched against the exact solution when *α* = 1, as demonstrated in Figure 1, we can clearly observe that when *α* approached 1, our results approached the exact solution. We also solved this problem by using Legendre polynomial operational matrix method (LPOMM), and we compared the results with the LWOMM. The numerical computations for *u*(*t*) and *v*(*t*) when *α* = 0.9 are revealed in Tables 1 and 2.

**Table 1.** Numerical solutions of *u*(*t*) when *α* = 0.9 attained by the introduced method and the LPOMM for Example 1.

**Table 2.** Numerical solutions of *v*(*t*) when *α* = 0.9 attained by the introduced method and the LPOMM for Example 1.

**Figure 1.** *Cont.*

**(b)**

**Figure 1.** Comparison of our solutions and the exact solution when *α* = 0.9, 0.7, 0.5 in Example 1: (**a**) Our solution *u*(*t*); and (**b**) Our solution *v*(*t*).

**Example 2.** *We considered the following nonlinear system of fractional differential equations [13]:*

$$\begin{aligned} D^{\frac{3}{7}}u(t) &= -8u(t) + v^2(t) - 4t^6 + 4t^3 + \frac{8t^{\frac{3}{2}}}{\sqrt{\pi}} - 1 \\ D^{\frac{1}{7}}v(t) &= t^2Du(t) + v(t) - 3t^4 - 2t^3 + \frac{32t^{\frac{3}{2}}}{5\sqrt{\pi}} - 1 \end{aligned}$$

$$u(0) = 0, \ v(0) = 1, \ u(1) = 1, \ v(1) = 3, \ u'(0) = 0, \ u'(1) = 3.$$

The exact solution of this system is known to be

$$u(t) = t^3, \ v(t) = 2t^3 + 1$$

Using the parameters *M* = 3 and *k* = 0, we applied both the proposed method and the LPOMM to solve this problem and show that our approach is more efficient and useful. Our numerical results supported the idea that our solution approaches the exact solution more than the approximate solution LPOMM. Comparisons of the approximate and exact solutions are presented in Tables 3 and 4.

**Table 3.** The numerical results attained by using the introduced method in comparison to the approximate solution LPOMM and the exact solution *u*(*t*) in Example 2.

**Table 4.** The numerical results attained by using the introduced method in comparison with the approximate solution LPOMM and exact solution *v*(*t*) in Example 2.

**Example 3.** *We considered the following nonlinear system of fractional differential equations with the initial conditions [8]*

$$\begin{array}{l}D^{a}u(t) = \frac{u(t)}{2} \\ D^{a}v(t) = u^{2}(t) + v(t) \end{array} \text{ \textquotedblleft} 0 \rightleftharpoons \begin{array}{l} \frac{1}{2} \\ v(0) = 1, \ v(0) = 0. \end{array}$$

The exact solution of this system when *α* = 1 is known to be

$$u(t) = e^{\left(\frac{\hbar}{2}\right)}, \ v(t) = te^t.$$

The parameters *M* = 2, *k* = 0, and *α* = 0.5, 0.7, 0.9 were utilized. A comparison of our results and the exact solution when *α* = 1 is displayed in Figure 2. The figures support that when *α* approximated 1, our results approximated the exact solution. We also solved this problem by using the LPOMM, and compared the results to the LWOMM. Finally, we present the numerical computations for *u*(*t*) and *v*(*t*) when *α* = 0.9 in Tables 5 and 6.

**Table 5.** Our solutions *u*(*t*) when *α* = 0.9 attained by the presented method and the LPOMM for Example 3.

**Figure 2.** Comparison of our solutions to the exact solution when *α* = 0.9, 0.7, 0.5 for Example 3: (**a**) Our solution *u*(*t*); and (**b**) Our solution *v*(*t*).

**Table 6.** Our solutions *v*(*t*) when *α* = 0.9 attained by the presented method and the LPOMM for Example 3.

**Example 4.** *We considered the following nonlinear system of FDEs with initial conditions [13]*

$$\begin{aligned} D^{\alpha}u(t) &= -1002u(t) + 1000v^2(t) \\ D^{\alpha}v(t) &= u(t) - v(t) - v^2(t) \end{aligned}$$

$$u(0) = 1, \ v(0) = 1$$

The exact solution of this system when *α* = 1 is known to be

$$u(t) = e^{-2t}, \ v(t) = e^{-t}$$

This example was analyzed for *M* = 3, *k* = 0, and *α* = 0.9, 0.7, 0.5. When the obtained results were matched against the exact solution when *α* = 1, as demonstrated in Figure 3, we can clearly observe that when *α* approached 1, our results approached the exact solution. We also solved this problem by using the LPOMM, and compared the results with the LWOMM. The numerical computations for *u*(*t*) and *v*(*t*) when *α* = 0.99 are also revealed in Tables 7 and 8.

**Table 7.** Numerical solutions of *u*(*t*) when *α* = 0.99 for Example 4.

**Table 8.** Numerical solutions of *v*(*t*) when *α* = 0.99 for Example 4.

**Figure 3.** Comparison of our solutions to the exact solution when *α* = 0.9, 0.7, 0.5 for Example 4: (**a**) Our solution *u*(*t*); and (**b**) Our solution *v*(*t*).

**Example 5.** *We considered the following fractional order Brusselator system [16,17]:*

$$\begin{aligned} D^\mathfrak{u}u(t) &= -2\mathfrak{u}(t) + \mathfrak{u}^2(t)v(t) \\ D^\mathfrak{v}v(t) &= \mathfrak{u}(t) - \mathfrak{u}^2(t)v(t) \end{aligned}$$

$$\mathfrak{u}(0) = 1, \ v(0) = 1.$$

The approximate solutions of this system when *α* = 1 and *α* = 0.98 were presented by Chang and Isah using the LWPT [17] and by Bota and Caruntu using the PLSM [16]. These solutions when *α* = 98 are given by

```
uLWPT(t) = 1 − 1.0791t + 0.2711t
                                 2 − 0.0638t
                                             3, vLWPT(t) = 1 + 0.0151t + 0.4185t
                                                                                2 − 0.2624t
                                                                                3
uPLSM(t) = 1 − 1.08655t + 0.311138t
                                     2 + 0.0243682t
                                                    3, vPLSM(t) = 1 + 0.0349127t + 0.333424t
                                                                                2 − 0.184414t
                                                                                3 .
```
The parameters *M* = 2, *k* = 0, and *α* = 0.98 were used. A comparison of our results to the approximate solutions introduced by Bota and Caruntu [16] and Chang and Isah [17] when *α* = 0.98 is displayed in Figure 4. Finally, we also present the numerical computations for *u*(*t*) and *v*(*t*) when *α* = 0.98 in Tables 9 and 10.

**Table 9.** Numerical solutions of *u*(*t*) when *α* = 0.98 obtained by the introduced method, the LWPT, and the PLSM for Example 5.

**Table 10.** Numerical solutions of *v*(*t*) when *α* = 0.98 obtained by the introduced method, the LWPT, and the PLSM for Example 5.

**Figure 4.** Comparison of our solutions to the approximate solution LWPT and the approximate solution PLSM when *α* = 0.98 for Example 5: (**a**) Our solution *u*(*t*); and (**b**) Our solution *v*(*t*).


**4. Conclusions**

The study successfully illustrated the nanopollution of water media during the simulated use of NEPs by characterising product-released ENMs from a wide range of products. The product-released nTiO2 were elongated (7–9 × 66–70 nm) or angular (21–80 × 25–79 nm) in shape; product-released nAg were near-spherical (12–49 nm) or angular (21–76 × 29–77 nm) and product-released nZnO were angular (32–36 × 32–40 nm)

in shape. The ENMs release rate was determined to be *ca* 0.4–95% relative to the initial amount of ENMs added to NEPs. The extent and characteristics of product-released ENMs were influenced by receiving water quality, ENMs *loci* in the product, and the formulation of the product matrix, while illumination variation essentially did not exert influence. Predominantly, the product-released ENMs were released in association with coating agents (Si and Al) and ionic forms. Considering the influential role the surface coating exerts on the behaviour and toxicity of ENMs in water resources, we highly recommend the reporting of the presence and characteristics of coating agents on product-released ENMs since it is currently not standard practice.

SUN1, CA1 and SK1 released binary ENMs. Typically, there is currently limited information on the environmental implications of ENMs mixtures, more so for productreleased ENMs; hence, we encourage more studies to unravel the exposure and hazard dynamics of product-released ENMs mixtures.

Nanopollution is an emerging environmental health issue that is yet to be clearly quantified. Nevertheless, proactive mitigation measures can reduce environmental exposure, for instance, the reduction in ENMs quantity in NEPs (safety-by-design principle), since this study demonstrated that the NEPs sample caused nanopollution. In low- and middle-income countries, such as South Africa, where the current study was carried out, there must be accelerated efforts to estimate the size of the NEPs market to refine the extent of nanopollution, as developed regions have advanced in that aspect.

**Supplementary Materials:** The following are available online at https://www.mdpi.com/2079-499 1/11/10/2537/s1, S1.1 Properties of the release media of SUN1–3, Table S1. Average physicochemical properties of release media before and after ENMs release, Figure S1. The EDX spectra of productreleased ENMs obtained under light conditions for SUN1 detected in milli-Q water (A), freshwater (B), swimming pool water (C), seawater (D); SUN2 detected in milli-Q water (E), freshwater (F), swimming pool water (G), seawater (H) and SUN3 detected in milli-Q water (I), freshwater (J), swimming pool water (K), seawater (L), Figure S2. TEM images of product-released ENMs obtained under dark conditions for SUN1 detected in milli-Q water (A), freshwater (B), swimming pool water (C), seawater (D); SUN2 detected in milli-Q water (E), freshwater (F), swimming pool water (G), seawater (H) and SUN3 detected in milli-Q water (I), freshwater (J), swimming pool water (K), seawater (L), Figure S3. Corresponding EDX images of product-released ENMs obtained under dark conditions for SUN1 detected in milli-Q water (A), freshwater (B), swimming pool water (C), seawater (D); SUN2 detected in milli-Q water (E), freshwater (F), swimming pool water (G), seawater (H) and SUN3 detected in milli-Q water (I), freshwater (J), swimming pool water (K), seawater (L), Figure S4. EDX elemental mapping showing adsorption and desorption of ENMs coating agents (Si and Al) on SUN1-released ENMs, Figure S5. EDX elemental mapping showing adsorption and desorption of ENMs coating agents (Si) on SUN2 (A)- and SUN3 (B)-released ENMs, Figure S6A. Violin plot showing particle distribution of SUN1(A)-, SUN2 (B)-, and SUN3 (C)-released ENMs obtained under light conditions. The upper and lower quartiles are highlighted by a solid line, while the dotted line indicates the median. The denser the violin shape, the higher the number of particle size in that region, Figure S6B. Violin plot showing particle distribution of SUN1(A)-, SUN2 (B)-, and SUN3 (C)-released ENMs obtained under dark conditions. The upper and lower quartiles are highlighted by a solid line, while the dotted line indicates the median. The denser the violin shape, the higher the number of particle size in that region, Figure S7. Zeta potential of SUN1–3-released ENMs obtained under dark conditions in different release media of milli-Q water (MQ), freshwater (FW), swimming pool water (SPW), and seawater (SS), Figure S8. Violin plot showing particle distribution of CA1- released ENMs (obtained under light and dark conditions) and SAN1-released ENMs. The upper and lower quartiles are highlighted by a solid line, while the dotted line indicates the median. The denser the violin shape, the higher the number of particle size in that region, Figure S9. TEM-EDX image showing of CA1 product-released nAg and product-released nTiO2 obtained under dark conditions, Figure S10. Violin plot showing particle distribution SK1-released ENMs. The upper and lower quartiles are highlighted by a solid line, while the dotted line indicates the median. The denser the violin shape, the higher the number of particle size in that region, Figure S11. Elemental mapping of binary SK1-released ENMs identified as product-released nTiO2 (yellow) and product-released nAg (red). The images further show evidence of SK1-released nTiO2 particles partially still coated

with Si and Al, Figure S12. TEM images showing the thick layer introduced by washing SK1 with sodium dodecyl sulfate release media, Figure S13. EDX elemental mapping illustrating Si desorbed from CA1-released ENMs.

**Author Contributions:** Conceptualisation, M.T. and R.F.L.; methodology, R.F.L. and M.T.; software, R.F.L.; validation, M.T.; formal analysis, R.F.L.; investigation, R.F.L.; resources, M.T.; data curation, R.F.L.; writing—original draft preparation, R.F.L.; writing—review and editing, R.F.L. and M.T.; visualisation, R.F.L. and M.T.; supervision, M.T.; project administration, M.T.; funding acquisition, M.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the South African Department of Science and Technology (DST) under the Nanotechnology Health, Safety and Environment Risk Research Platform (grant number: 0085/2015).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on a reasonable request from the corresponding author.

**Acknowledgments:** This work was supported by the South African Department of Science and Technology (DST) under the Nanotechnology Health, Safety and Environment Risk Research Platform (grant number: 0085/2015). We further wish to thank Jérôme Rose, Melanie Auffan, and Danielle Slomberg funded by Gov4Nano H2020 European Commission project (grant number: 814401), the SERENADE project (Labex Serenade programme (grant number. ANR-11-LABX-0064), and the "Investissements d'Avenir" programme of the French National Research Agency (ANR) through the A\*MIDEX project (grant number: ANR-11-IDEX-0001-02) for hosting and mentoring L.R.F.

**Conflicts of Interest:** The authors declare no conflict of interest.


*Supporting open minds since 2005*

Use of Gamma Radiation Techniques in Peaceful Applications http://dx.doi.org/10.5772/intechopen.78481 Edited by Basim A. Almayahi

#### Contributors

Wanchai De-Eknamkul, Thongchai Koobkokkruad, Praderm Wanichananan, Chalermpol Kirdmanee, Bertil Rolf Ragnar Persson, Hai Lin, Seyed Mahyar Mirmajlessi, Hossein Ahari Mostafavi, Evelin Loit, Neda Najdabbasi, Marika Mänd, Christopher Busby, Hassane Erramli, Jaouad El Asri, Nicolas Thierry Fourches, Magdalena Zielinska, Gabriel Charles, Lubna Mehdi Alasadi, Hasna Albander, Vladimir Krasilnikov, Nikolay Kamyshanchenko, Alexander Parkhomenko, Victor Robuk, Boris Loncar, Luka Rubinjoni, Katarina Karadžić, Kinga Polaczek-Grelik, Aneta Kawa-Iwanicka, Marek Rygielski, Łukasz Michalecki, Alexander Barzilov, Monia Kazemeini, John Vargas, Woosoon Yim, Elizabeth Carvalho Leite Cardoso, Duclerc Fernandes Parra, Sandra Scagliusi, Ademar Benevolo Lugão

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