aainstitutetext: DAMTP, University of Cambridge, Wilberforce Road, Cambridge,
CB3 0WA, United Kingdom

# Fits to measurements of rare heavy flavour decays

Ben Allanach<EMAIL_ADDRESS>
###### Abstract

This write-up is intended to form part of the proceedings for Lepton-Photon
2023. We review the decays $b\rightarrow c\ell\bar{\nu}_{\ell}$ (where
$\ell\in\\{e,\mu,\tau\\}$) as well as $b\rightarrow s\mu^{+}\mu^{-}$ and
$b\rightarrow se^{+}e^{-}$, giving the current state-of-the-art in terms of
measurements. We review fits to such data of new physics weak effective field
theory operators, before closing with interpretations in terms of simplified
TeV-scale field theories.

###### Keywords:

Weak interactions, $B-$anomalies, $B$ hadron decays

##  1 $b\rightarrow c\tau\nu$ decays

Measurements of ratios of the branching ratios

$R(D{(\ast)})=\frac{B\rightarrow D^{(\ast)}\tau\bar{\nu}}{B\rightarrow
D{(\ast)}\ell\bar{\nu}},$ (1)

where $\ell\in\\{e,\mu\\}$ are of interest because they provide a test of
lepton flavour universality (LFU) of the couplings of $W^{\pm}$ bosons in the
Standard Model (SM). A recent estimate of the world average by the HFLAV
collaboration put the joint determination of $R(D)$ and $R(D\ast)$ at a
tension with SM predictions at the 3.2$\sigma$ level HFLAV . This tension is
called the $b\rightarrow c\tau\nu$ anomaly111We note that each observable
implicitly includes an average over different charge decays. and has led
various authors to suggest that quantum fields associated with TeV-scale
leptoquarks, charged Higgs’ or $W^{\prime}$ particles may be contributing to
the process. We show a preliminary 2023 HFLAV combination in Fig. 1, where
some measurements have been updated as compared to the preceding official
HFLAV combination HFLAV2 .

Figure 1: Preliminary HFLAV 2023 world average of $R(D)$ and $R(D\ast)$, along
with the HFLAV SM predictions. Figure taken from Ref. HFLAV . The horizontal
bands are 68$\%$ confidence level bands, whereas the filled ellipses are
39$\%$ bands. The dashed ellipse shows the locus of $p-$value where a
univariate Gaussian distribution would be at $3\sigma$ from the maximum. The
39$\%$ confidence level (CL) world average is shown as the filled red ellipse.
Near the top of the figure, an example leading order Feynman diagram has been
sketched in order to show a leading SM contribution to $B\rightarrow
D\tau\bar{\nu}$ decay.

Other measurements have recently come to our attention: a 2022 simultaneous
determination of $R(D)$ and $R(D\ast)$ based on semi-leptonic decays222This is
_not_ an official BaBar result lusiani ; that is currently in progress but not
yet complete. of the tagged $B$ meson in BaBar data thesis . The 2012 BaBar
determination featured in Fig. 1 was based instead on hadronic decays of the
tagged $B$ meson. The 2022 measurements are reported as

$R(D)=0.316\pm 0.062\pm 0.019,\qquad R(D\ast)=0.226\pm 0.022\pm
0.012,\qquad\rho=-0.82.$ (2)

It is our purpose here to augment the world average by these measurements. We
shall perform a more approximate job than has HFLAV. In particular, we have
not correlated any systematic errors; otherwise we have added all errors in
quadrature. Correlations between $R(D)$ and $R(D\ast)$ were taken into
account, though. We now check the level of agreement we obtain with HFLAV’s
more accurate calculation. We present the results of our calculation, based on
the same data, in Fig. 2. We display 68$\%$ confidence level contours
consistently throughout (except for the 3$\sigma$ one).

Figure 2: Our determination of the world average of $R(D)$ and $R(D\ast)$ and
the SM prediction (left panel) excluding and (right panel) including the 2022
semi-leptonic tag (SL) preliminary measurements from BaBar data. The
horizontal bands and error ellipses are 68$\%$ confidence level bands, except
for the dashed ellipse; this shows the locus of iso-$p-$value where a
univariate Gaussian distribution would be at $3\sigma$ from the maximum.

The results are in fact similar (although one should note the differences in
appearance coming from choosing somewhat different confidence levels and
aspect ratio), showing small differences due to the further approximations of
the latter combination. The central values differ by a percent relative
deviation between the two determinations. HFLAV´s determination holds that the
tension between data and the SM is at an equivalent univariate Gaussian
distribution value (EUGDV) of 3.2$\sigma$, whereas our calculation yields
3.4$\sigma$. We obtain a $p-$value of consistency of the measurements is .26
rather than .25. We judge that these results are close enough to validate our
approximations.

Our determination of the world average _now including the 2022 measurements of
the semi-leptonic tagged BaBar data_ is

$R(D)=0.339\pm 0.025,\qquad R(D\ast)=0.263\pm 0.01,\qquad\rho=-0.53,.$ (3)

which we calculate to be in only 1.5$\sigma$ EUGDV tension with the HFLAV 2023
SM prediction. We note here that the 2022 semi-leptonic tag result is in some
tension with the other measurements, reducing the $p-$value of the
measurements to an EUGDV of 2.7$\sigma$. The calculations used to produce Fig.
2 are included in the ancillary information of the arXiv version of this
manuscript.

Since the combined world average no longer calls strongly for new physics
effects in $b\rightarrow c\tau\nu$ decays (although we note the remaining
tensions between measurements and that there is still some room for new
physics effects), we move on to $b\rightarrow s\ell^{+}\ell^{-}$ decays.

##  2 $b\rightarrow s\ell^{+}\ell^{-}$ processes

Figure 3: An example of a penguin diagram contributing to the
$\bar{B}\rightarrow\bar{K}^{\ast 0}\ell^{+}\ell^{-}$ process.

In the SM, the dominant diagrams contributing to $B\rightarrow
K^{\ast}\ell^{+}\ell^{-}$ are one-loop electroweak box or penguin
contributions such as the penguin contribution shown in Fig. 3. These decays
are also useful for testing LFU. The amplitude is suppressed by the loop, by
the fact that it is an electroweak process rather than a strong process, and
by small CKM entries. This has the result that the branching ratio for
$BR(B\rightarrow K^{(\ast)}\ell^{+}\ell^{-})$ is of order $10^{-6}$ or less.
Since the electron- and muon- masses are much smaller than the $B$ meson mass,
they can be approximated as massless. This approximation leads to the
prediction $BR(B\rightarrow K^{(\ast)}\mu^{+}\mu^{-})=BR(B\rightarrow
K^{(\ast)}e^{+}e^{-})$, which holds to high precision except at low
$q^{2}:=m_{ll}^{2}$.

In practice, Feynman diagrams such as the one in Fig. 3 receive QCD
corrections. These can be parameterised by form factors (scalar functions of
$q^{2}$). The predictions for the rare decay $B\rightarrow M\ell^{+}\ell^{-}$
then are written in terms of form factors multiplied by kinematic variables
and pre-factors. The form factors come in two categories: _local_ and _non-
local_ form factors. For the local form factors, one can interpolate lattice
results which are valid at high $q^{2}$ (and therefore low $M$ recoil) and
light cone sum rule at low $q^{2}$. The non-local form factors currently have
no precise lattice estimates. Several of the SM predictions that are used to
produce fits that we shall show use QCD factorisation plus an ad-hoc
parameterisation of the long distance contribution which is fit to data (the
dominant contribution comes from charm loops). We will also show several SM
predictions from the EOS Gubernari:2022hxn ; Gubernari:2023puw  approach.
Here, the light cone operator product expansion at $q^{2}<0$ is
interpolated/extrapolated to various measurements of branching ratios and
angular distributions in decays, which are made at a $q^{2}$ value of the
$J/\psi$ resonance mass squared, $M_{J/\psi}^{2}$. Dispersion relations bound
the coefficients of a polynomial expansion in terms of a specific kinematic
variable, allowing truncations to finite order and a resulting fit of them to
a finite number of measurements. In the EOS approach Gubernari:2022hxn ;
Gubernari:2023puw , the dominant remaining uncertainty on the relevant
predictions typically comes from uncertainties in the local form factors.

The form factors for the SM prediction of $BR(B_{s}\rightarrow\mu^{+}\mu^{-})$
are known accurately by lattice computations. We display this branching ratio,
jointly measured with $BR(B_{s}\rightarrow\mu^{+}\mu^{-})$ in Fig. 4, along
with some other $b\rightarrow s\mu^{+}\mu^{-}$ observables.

Figure 4:  Various SM predictions and measurements of $bs\mu\mu$ processes.
(top left) $BR(B_{s}\rightarrow\mu^{+}\mu^{-})$: the SM prediction is from
Ref. Feldmann:2022ixt  and the combination of data in red is from Ref.
Allanach:2022iod . Empty ellipses show the 95$\%$ CL constraint in each case.
(top right) shows a certain angular distribution variable, $P_{5}^{\prime}$,
in $B\rightarrow K^{\ast}\mu^{+}\mu^{-}$ decays. (bottom left) shows
$BR(B_{s}\rightarrow\phi\mu^{+}\mu^{-})$ as a function of $q^{2}$ and (bottom
right) shows $BR(B\rightarrow K\mu^{+}\mu^{-})$ decay along with a hand-drawn
leading order SM contribution. In the latter three sub-plots, SM predictions
are made by EOS Gubernari:2022hxn ; Gubernari:2023puw .

Fig. 4 shows that several of the observables involving the
$b\bar{s}\mu^{+}\mu^{-}$ vertex display tensions between SM predictions and
measurements in several bins of $q^{2}$. One observes varying degrees of
tension in the SM predictions and data in each observable, although in the
$CP-$untagged $\overline{BR}(B_{s}\rightarrow\mu^{+}\mu^{-})$ decay, the
tension is only mild at 1.6$\sigma$ of an EUGDV. Many authors have interpreted
the tensions, dubbed the ‘$b\rightarrow s\mu\mu$ anomalies’, as requiring
contributions from processes involving quantum fields beyond the SM.

###  2.1 Lepton flavour universality ratios

LHCb has published a reanalysis LHCb:2022qnv  of the measurements of $R_{K}$
and $R_{K^{\ast}}$, where

$R_{X}(q^{2})=\frac{\int_{q^{2}_{min}}^{q^{2}_{max}}BR(B\rightarrow
X\mu^{+}\mu^{-}(q^{2}))}{\int_{q^{2}_{min}}^{q^{2}_{max}}BR(B\rightarrow
Xe^{+}e^{-}(q^{2}))}.$ (4)

$q^{2}_{min}$ and $q^{2}_{max}$ are the two extreme values of $q^{2}$ in the
particular $q^{2}$ bin under consideration. ‘low-$q^{2}$’ corresponds to
$q^{2}\in(0.1,\ 1,1)$ GeV2 whereas ‘central-$q^{2}$’ corresponds to
$q^{2}\in(1.1,\ 6.0)$ GeV2. These measurements show no significant deviation
from the SM.

Figure 5: (top) Recent LHCb measurements of $R_{K}$ and $R_{K^{\ast}}$
LHCb:2022qnv . (bottom-left) LHCb measurement of $R_{pK}$. (bottom-right)
Belle measurements of $R_{K}$ and $R_{K^{\ast}}$ and LHCb measurements of
other LFU ratios.

This is a relatively new development, and indicates that, if there is new
physics in $b\rightarrow s\mu\mu$ transitions, as indicated above, there may
also be associated new physics in $b\rightarrow se^{+}e^{-}$ in order that
$R_{K}$ and $R_{K^{\ast}}$ may be close to the lepton flavour universality
limit.

##  3 Weak effective theory fits

We now interpret the new physics effects in terms of the Weak Effective Theory
(WET), which parameterises the effects of integrated out heavy new physics
fields in terms of new effectively non-renormalisable operators at the weak
scale or below. Two such operators that have been shown to substantially
ameliorate the fit to the $b\rightarrow s\mu^{+}\mu^{-}$ anomalies are

${\mathcal{L}}=\ldots+N\left(C_{9\mu}^{NP}(\bar{b}\gamma^{\alpha}P_{L}s)(\bar{\mu}\gamma_{\alpha}\mu)+C_{10\mu}^{NP}(\bar{b}\gamma^{\alpha}P_{L}s)(\bar{\mu}\gamma_{\alpha}\gamma_{5}\mu)+H.c.\right),$
(5)

$N:=4G_{F}e^{2}|V_{ts}|/(16\pi^{2}\sqrt{2})$ is a normalising constant, where
$G_{F}$ is the Fermi decay constant, $e$ the electromagnetic gauge coupling
and $V_{ij}$ the entries of the CKM matrix.

Figure 6:  Fits of two non-zero WET operators to $b\rightarrow s\ell\ell$
anomaly data. The inner (outer) coloured bands show the 68$\%$ ($95\%$ CL)
regions. (left) flavio fits Greljo:2022jac . ‘$b\rightarrow s\mu\mu$’ contains
observables such as $B\rightarrow K^{(\ast)}\mu^{+}\mu^{-}$ branching ratios
and angular distributions. The global constraint using all three sets of
observables is shown in black at the 95$\%$ CL. (right-panel) shows the global
fit regions of flavio Greljo:2022jac  and other fitting groups Alguero:2023jeh
; Ciuchini:2022wbq  such as superIso Mahmoudi:2008tp  which use different sets
of experimental data and different treatments of the SM predictions and
theoretical uncertainties BERNAT .

We see the fit by the flavio program in the left-hand panel of Fig. 6. Since
the three coloured regions overlap at 95$\%$ Cl, we conclude that there is
still some compatibility with the measured values of $R_{K}$ and
$R_{K^{\ast}}$. The figure also shows a comparison of the fits obtained by
different fitting groups in order to show the spread in predictions. While
there is broad agreement that the fits disagree with SM predictions333Some
estimates in Ref. Ciuchini:2022wbq  fit an unidentified non-perturbative SM
contribution (that mimics a $q^{2}-$dependent lepton-family universal $C_{9}$)
in tandem with the new physics WET operators. As argued in Ref.
Isidori:2023unk , a similar non-perturbative effect cannot explain the
2$\sigma$ deficit in the $BR(B\rightarrow X_{s}\mu^{+}\mu^{-})$ high
$q^{2}-$bin, which is compatible with the low $q^{2}-$deficits., there are
some quantitative differences visible between the fits.

Now, we consider also turning on a new physics operator involving di-electron
pairs

${\mathcal{L}}=\ldots+NC_{9e}^{NP}(\bar{b}\gamma^{\alpha}P_{L}s)(\bar{e}\gamma_{\alpha}e)+H.c.$
(6)

We display a joint WET fit to non-zero $C_{9\mu}^{NP}$ and $C_{9e}^{NP}$ in
Fig. 7, where we see the preference for $C_{9e}^{NP}\neq 0$. Overall, the fit
has a pull away from the SM limit (the origin), equivalent to a UEDGV of
5.2$\sigma$, a significant change for only two fitted parameters.

Figure 7:  Fits of $C_{9\mu}^{NP}$ and $C_{9e}^{NP}$ to $b\rightarrow
s\ell\ell$ anomaly data. The inner (outer) coloured bands show the 68$\%$
($95\%$ CL) regions, respectively. Figure taken from Ref. Alguero:2023jeh .

We now turn to some simple bottom-up models which can generate the non-zero
WET operators which we mention above and which were found to significantly
ameliorate SM predictions.

##  4 Simple models

We examine simple $Z^{\prime}$ models with a spontaneously broken $U(1)_{X}$
gauge extension of the SM Allanach:2023uxz , where

$3B_{3}-\left(X_{e}L_{e}+X_{\mu}L_{\mu}+[3-X_{e}-X_{\mu}]L_{\tau}\right),$ (7)

i.e. where $B_{3}$ is third family baryon number, $L_{e}$ is electron number,
$L_{\mu}$ is muon number and $X_{e}$ and $X_{\tau}$ are arbitrary integer
parameters. Such an assignment is anomaly free if three right-handed neutrinos
augment the SM chiral fermion content. Once the $Z^{\prime}$ is coupled to di-
electron pairs as would be implied by $X_{e}\neq 0$, one should apply bounds
from LEP which come from differential measurements of scattering to di-lepton
pairs. These do not show significant tensions with SM predictions
Falkowski:2015krw  and so they bound the contributions coming from Feynman
diagrams such as those shown in Fig. 8.

Figure 8:  Leading Feynman diagrams of $Z^{\prime}$ contributions to di-lepton
production at the LEP collider.

(7) allows one to interpolate between $Z^{\prime}$ models which couple to
charged leptons via muons only, to those which couple equally to electrons and
muons. We display a set of global fits to different values of $X_{e}$, for
$X_{\mu}=10$, in Fig. 9.

Figure 9:  Two parameter fits of a set of $Z^{\prime}$ models to $b\rightarrow
s\ell\ell$ anomaly data. Figure taken from Ref. Allanach:2023uxz . Regions
where the ‘global’ curve is below the dashed line are compatible with data
globally, to 95$\%$ CL, assuming the hypothesis of the line of models.

The figure shows that a significant improvement on $\chi^{2}$ is obtained as
compared to the SM for $X_{e}/X_{\mu}\approx 1/2$, although any value in the
range $(-0.4,1.3)$ is within the 95$\%$ CL. We also see that the limits of
zero coupling of the $Z^{\prime}$ boson to di-electron pairs and equal
coupling to di-electron and di-muon pairs are fit more-or-less equally well,
with only an insignificant difference in best-fit $\chi^{2}$ value of 0.7
between them.

We anticipate that requiring a single leptoquark (LQ) to couple to electrons
and muons with a similar strength in order to fit $R_{K}$ and $R_{K^{\ast}}$
will generically lead to contravention of the $BR(\mu\rightarrow
e\gamma)<4.2\times 10^{-13}$ (90$\%$ CL) bound from the MEG collaboration
MEG:2016leq  through processes such as the one shown in Fig.
10444$\mu\rightarrow e$ conversion also provides very strong constraints upon
such a LQ..

Figure 10:  A scalar LQ’s (that has been fit to $b\rightarrow
s\ell^{+}\ell^{-}$ anomalies and therefore has a coupling to the strange
quark) contribution to $\mu\rightarrow e\gamma$. There is another contribution
with $s\rightarrow b$.

However, Ref. Allanach:2022iod  found that LQs that do _not_ couple to di-
electron pairs, can provide an improvement on SM predictions similar to that
of the ‘$B_{3}-L_{2}$’ model with $X_{\mu}=3,X_{e}=0$.

## Acknowledgements

This work has been partially supported by STFC HEP Theory Consolidated grant
ST/000694. We thank other members of the Cambridge Pheno Working group for
helpful discussions and Lepton-Photon 2023 for facilitating the talk.

## References

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# Richardson extrapolation for the iterated Galerkin solution of Urysohn
integral equations with Green’s kernels

Gobinda RAKSHIT111School of Mathematical Sciences, UM-DAE Centre for
Excellence in Basic Sciences, University of Mumbai Campus, Kalina, Mumbai 400
098, India, email<EMAIL_ADDRESS>Akshay S. RANE222Department of
Mathematics, Institute of Chemical Technology, Nathalal Parekh Marg, Matunga,
Mumbai 400 019, India, email<EMAIL_ADDRESS>and Kshitij
PATIL333Department of Mathematics, Institute of Chemical Technology, Nathalal
Parekh Marg, Matunga, Mumbai 400 019, India, email :
<EMAIL_ADDRESS>
###### Abstract

We consider a Urysohn integral operator $\mathcal{K}$ with kernel of the type
of Green’s function. For $r\geq 1$, a space of piecewise polynomials of degree
$\leq r-1$ with respect to a uniform partition is chosen to be the
approximating space and the projection is chosen to be the orthogonal
projection. Iterated Galerkin method is applied to the integral equation
$x-\mathcal{K}(x)=f$. It is known that the order of convergence of the
iterated Galerkin solution is $r+2$ and, at the above partition points it is
$2r$. We obtain an asymptotic expansion of the iterated Galerkin solution at
the partition points of the above Urysohn integral equation. Richardson
extrapolation is used to improve the order of convergence. A numerical example
is considered to illustrate our theoretical results.

Key Words : Urysohn integral operator, Green’s kernel, Galerkin method,
Richardson extrapolation.

AMS subject classification : 45G10, 65B05, 65J15, 65R20

##  1 Introduction

Let $\mathcal{X}=L^{\infty}[0,1]$ and consider the following Urysohn integral
operator

$\displaystyle\mathcal{K}(x)(s)=\int_{0}^{1}\kappa\left(s,t,x(t)\right)dt,\;\;\;s\in[0,1],\;x\in\mathcal{X},$
(1.1)

where $\kappa(s,t,u)$ is a real valued continuous function defined on
$\Omega=[0,1]\times[0,1]\times{\mathbb{R}}.$ Then $\mathcal{K}$ is a compact
operator from $L^{\infty}[0,1]$ to $C[0,1].$ Consider the Urysohn integral
equation

$\displaystyle
x(s)-\int_{0}^{1}\kappa\left(s,t,x(t)\right)dt=f(s),\;\;\;s\in[0,1],$ (1.2)

where $f\in\mathcal{X}$ is given and $x$ is the unknown to be determined. We
assume that $\varphi$ is an isolated solution of the above equation and
consider its numerical approximations.

We are interested in approximate solution which converges to $\varphi$
uniformly. We consider some projection methods associated with a sequence of
orthogonal projections converging to the Identity operator point-wise.

In this paper, we consider the case when the kernel $\kappa$ of the integral
operator $\mathcal{K}$, is of the type of Green’s function in its domain. We
allow the partial derivatives of the kernel $\kappa$ to have jump
discontinuities along the diagonal $s=t$. For $r\geq 1,$ let $\mathscr{X}_{n}$
be a space of piecewise polynomials of degree $\leq r-1$ with respect to a
uniform partition of $[0,1]$ with $n$ subintervals each of length
${h=\frac{1}{n}}.$ Let $\pi_{n}$ be the restriction to $L^{\infty}[0,1]$ of
the orthogonal projection from $L^{2}[0,1]$ onto $\mathscr{X}_{n}.$ Galerkin
method is a classical projection method for the approximate solution of an
integral equation. In this method, (1.2) is approximated by

$x_{n}^{G}-\pi_{n}\mathcal{K}(x_{n}^{G})=\pi_{n}f.$ (1.3)

The above projection method has been studied extensively in the research
literature. See Krasnoselsii [6], Krasnoselskii, Vainikko et al [7] and
Krasnoselskii and Zabreiko [8] for details.

The iterated Galerkin solution is defined by

$x_{n}^{S}=\mathcal{K}(x_{n}^{G})+f.$

Note that

$x_{n}^{G}=\pi_{n}x_{n}^{S}$

and then the iterated Galerkin solution satisfies the following equation:

$x_{n}^{S}-\mathcal{K}(\pi_{n}x_{n}^{S})=f.$ (1.4)

From Atkinson-Potra [3], we quote the following orders of convergence:

If $r=1,$ then

$\|x_{n}^{G}-\varphi\|_{\infty}=O(h),\;\;\;\|x_{n}^{S}-\varphi\|_{\infty}=O(h^{2}),$
(1.5)

whereas if $r\geq 2,$ then

$\left\lVert
x_{n}^{G}-\varphi\right\rVert_{\infty}=O\left(h^{r}\right)~{},\quad\left\lVert
x_{n}^{S}-\varphi\right\rVert_{\infty}=O\left(h^{r+2}\right).$ (1.6)

Asymptotic error analysis and extrapolation methods are classical topics in
numerical analysis. Richardson extrapolation is the popular one. In Ford et
al. [5], Hammerstein integral equation with Green’s function type of kernel is
considered. Composite trapezoidal quadrature method is used to approximate the
integral operator, and then an asymptotic error expansion is obtained for the
approximate solution at the node points. Richardson extrapolation is applied
to improve the orders of convergence. In Kulkarni-Rane [12], the authors have
defined the Nyström operator based on the composite midpoint and the composite
modified Simpson rules to approximate the integral operator of a Hammerstein
integral equation with Green’s function type kernel. Asymptotic expansions for
the approximate solution at the node points as well as at the partition
points, are obtained and Richardson extrapolation is used to obtain
approximate solutions with higher orders of convergence. Hammerstein integral
equation is a special case of Urysohn integral equation. The case when the
kernel of the Urysohn integral equation is sufficiently smooth, asymptotic
error analysis are investigated for various projection methods in Kulkarni-
Nidhin [10]. In the case of a linear integral equation of the second kind with
smooth kernel, asymptotic series expansion for the iterated Galerkin solution
is proved by McLean [14]. The case of asymptotic expansion for approximate
solution of integral equations with Green’s kernel, at the partition points in
the case of Nyström method with midpoint rule, modified Simpsons rule and
iterated collocation method is treated in Kulkarni-Rane [11].

In Rakshit-Rane [15], we considered a Fredholm integral equation with kernel
of the type of Green’s function and then asymptotic error analysis is
investigated for the iterated Galerkin solution at the partition points.
Richardson extrapolation is applied to obtain an approximate solution with
higher rate of convergence.

In this paper we shall analyze asymptotic expansion for the iterated Galerkin
solution of Urysohn integral equation with Green’s function type kernel. We
will use Richardson extrapolation to improve the order of convergence.

The paper is organized as follows. In Section 2, notation is set and some
preliminary results are proved for later use. In Section 3, asymptotic error
analysis for the iterated Galerkin solution at the partition points are
investigated. Numerical illustration is given in Section 4.

##  2 Preliminaries

In this section we describe the Urysohn integral operator with Green’s
function type kernel, its Fréchet derivatives and related preliminary results.
We introduce the following notations.

For an integer $\alpha\geq 0$, let $C^{\alpha}[0,1]$ denotes the space of all
real valued $\alpha$-times continuously differentiable functions on $[0,1]$
with the following norm.

$\left\lVert x\right\rVert_{\alpha,\infty}=\max_{0\leq j\leq\alpha}\left\lVert
x^{(j)}\right\rVert_{\infty},$

where $x^{(j)}$ is the $j^{\text{th}}$ derivative of the function $x$ and

$\left\lVert x^{(j)}\right\rVert_{\infty}=\sup_{0\leq t\leq
1}\left|x^{(j)}(t)\right|.$

Define

$\left\lVert\kappa\right\rVert_{\alpha,\infty}=\max_{0\leq
i+j+k\leq\alpha}\left\lVert D^{(i,j,k)}\kappa(s,t,u)\right\rVert_{\infty},$

where

$D^{(i,j,k)}\kappa(s,t,u)=\frac{\partial^{i+j+k}\kappa}{\partial s^{i}\partial
t^{j}\partial u^{k}}(s,t,u).$

###  2.1 Properties of the kernel (Green’s function type)

Let $r\geq 1$ be an integer and assume that the kernel $\kappa$ has the
following properties.

  1. 1.

For $i=1,2,3,4$, the functions
$\kappa,\displaystyle{\frac{\partial^{i}\kappa}{\partial u^{i}}\in
C(\Omega),}$ where $C(\Omega)$ denotes the space of all real valued continuous
function on $\Omega=[0,1]\times[0,1]\times{\mathbb{R}}$.

  2. 2.

Let

$\Omega_{1}=\\{(s,t,u):0\leq t\leq s\leq
1,\;u\in{\mathbb{R}}\\},\;\;\;\Omega_{2}=\\{(s,t,u):0\leq s\leq t\leq
1,\;u\in{\mathbb{R}}\\}.$

There are two functions $\kappa_{j}\in C^{r}(\Omega_{j}),j=1,2,$ such that

$\kappa(s,t,u)=\left\\{{\begin{array}[]{ll}\kappa_{1}(s,t,u),\;\;\;(s,t,u)\in\Omega_{1},\\\
\kappa_{2}(s,t,u),\;\;\;(s,t,u)\in\Omega_{2}.\end{array}}\right.$

  3. 3.

Denote $\displaystyle{\ell(s,t,u)=\frac{\partial\kappa}{\partial u}(s,t,u)}$
and $\displaystyle{q(s,t,u)=\frac{\partial^{2}\kappa}{\partial
u^{2}}(s,t,u)},$ for all $(s,t,u)\in\Omega.$ The partial derivatives of
$\ell(s,t,u)$ and $q(s,t,u)$ with respect to $s$ and $t$ have jump
discontinuities on $s=t$.

  4. 4.

There are functions $\ell_{j},q_{j}\in C^{r}(\Omega_{j}),j=1,2,$ with

$\ell(s,t,u)=\left\\{{\begin{array}[]{ll}\ell_{1}(s,t,u),\;\;\;(s,t,u)\in\Omega_{1},\\\
\ell_{2}(s,t,u),\;\;\;(s,t,u)\in\Omega_{2},\end{array}}\right.$
$q(s,t,u)=\left\\{{\begin{array}[]{ll}q_{1}(s,t,u),\;\;\;(s,t,u)\in\Omega_{1},\\\
q_{2}(s,t,u),\;\;\;(s,t,u)\in\Omega_{2}.\end{array}}\right.$

Following Atkinson-Potra [3], if the kernel $\kappa$ satisfies the above
conditions, then we say that $\kappa$ is of class $\mathscr{G}_{4}(r,0)$.

Under the above assumptions, the operator $\mathcal{K}$ is four times Fréchet
differentiable and its Fréchet derivatives at $x\in\mathcal{X}$ are given by

$\mathcal{K}^{\prime}(x)v_{1}(s)=\int_{0}^{1}\frac{\partial\kappa}{\partial
u}\left(s,t,x(t)\right)\>v_{1}(t)\>dt$

and

$\mathcal{K}^{(i)}(x)(v_{1},\ldots,v_{i})(s)=\int_{0}^{1}\frac{\partial^{i}\kappa}{\partial
u^{i}}\left(s,t,x(t)\right)\>v_{1}(t)\cdots v_{i}(t)\>dt,\qquad i=2,3,4,$
(2.1)

where

$\displaystyle\frac{\partial^{i}\kappa}{\partial
u^{i}}\left(s,t,x(t)\right)=\frac{\partial^{i}\kappa}{\partial
u^{i}}\left(s,t,u\right)|_{u=x(t)},\quad i=1,2,3,4$

and $v_{1},v_{2},v_{3},v_{4}\in\mathcal{X}$. Note that
$\mathcal{K}^{\prime}(x):\mathcal{X}\rightarrow\mathcal{X}$ is linear and
$\mathcal{K}^{(i)}(x):\mathcal{X}^{i}\rightarrow\mathcal{X}$ are multi-linear
operators, where $\mathcal{X}^{i}$ is the cartesian product of $i$ copies of
$\mathcal{X}$. See Rall [16]. We define

$\left\lVert\mathcal{K}^{(i)}(x)\right\rVert\>=\>\sup_{\stackrel{{\scriptstyle\left\lVert
v_{j}\right\rVert_{\infty}\leq
1}}{{j=1,\ldots,i}}}\left\lVert\mathcal{K}^{(i)}(x)(v_{1},\ldots,v_{i})\right\rVert_{\infty},\qquad
i=1,2,3,4.$

It follows that

$\displaystyle\left\lVert\mathcal{K}^{(i)}(x)\right\rVert$ $\displaystyle\leq$
$\displaystyle\sup_{0\leq s,t\leq 1}\left|\frac{\partial^{i}\kappa}{\partial
u^{i}}\left(s,t,x(t)\right)\right|,\qquad i=1,2,3,4.$

We rewrite the equation (1.2) as

$\displaystyle x-\mathcal{K}(x)=f,\quad x\in\mathcal{X}.$

Let

$\mathcal{T}(x)=\mathcal{K}(x)+f,\quad x\in\mathcal{X}.$ (2.2)

Assume that $\varphi$ is a fixed point of $\mathcal{T}$. Since $\mathcal{K}$
is compact, $\mathcal{K}^{\prime}(\varphi)$ is a compact linear operator. See
Krasnoselskii [6]. Assume that $1$ is not an eigenvalue of
$\mathcal{K}^{\prime}(\varphi).$ Then, $\varphi$ is an isolated solution of
(1.2). Let $f\in C^{\alpha}[0,1]$, then by the Corollary 3.2 of Atkinson-Potra
[3], it follows that $\varphi\in C^{\alpha}[0,1].$

###  2.2 Approximating Space and Projection operator

Let $n\in\mathbb{N}$ and consider the following uniform partition of $[0,1]:$

$\Delta:0<\frac{1}{n}<\cdots<\frac{n-1}{n}<1.$ (2.3)

Define

$t_{j}=\frac{j}{n},\;\;\;j=0,\ldots,n.$ (2.4)

Let

$\Delta_{j}=[t_{j-1},t_{j}]\;\;\;\mbox{and}\;\;\;h=t_{j}-t_{j-1}=\frac{1}{n},\;\;\;j=1,\ldots,n.$

Consider a finite dimensional approximating space as

$\mathscr{X}_{n}=\left\\{g\in L^{\infty}[0,1]:g\text{ is a polynomial of
degree }\leq r-1\text{ on }\Delta_{j},~{}j=1,2,\dots,n\right\\}.$

As no continuity conditions are imposed at the partition points, the dimension
of $\mathscr{X}_{n}$ is $nr$ and $\displaystyle{\mathscr{X}_{n}\subset
L^{\infty}[0,1].}$

Let $\pi_{n}$ be the restriction to $L^{\infty}[0,1]$ of the orthogonal
projection from $L^{2}[0,1]$ onto $\mathscr{X}_{n}$, which converges to the
Identity operator pointwise. Then

$\sup_{n\geq 1}\left\lVert\pi_{n}\right\rVert_{L^{\infty}[0,1]\rightarrow
L^{\infty}[0,1]}<\infty.$ (2.5)

If $x\in C^{\alpha}[0,1]$, it is well-known that

$\displaystyle\left\lVert(I-\pi_{n})x\right\rVert_{\infty}$
$\displaystyle\leq$ $\displaystyle C_{1}\|x^{(\beta)}\|_{\infty}h^{\beta},$
(2.6)

where $\beta=\min\left\\{\alpha,r\right\\}$ and $C_{1}$ is a constant
independent of $h$. See Atkinson [1], Chatelin-Lebbar [4]. Denote

$\pi_{n,j}x=\pi_{n}x|_{\Delta_{j}},\quad j=1,2,\ldots,n.$

For $x\in C^{\alpha}(\Delta_{j}),$ we have

$\displaystyle\|(I-\pi_{n,j})x\|_{\Delta_{j},\infty}$ $\displaystyle\leq$
$\displaystyle C_{2}\|x^{(\beta)}\|_{\Delta_{j},\infty}h^{\beta},$ (2.7)

where $\beta=\min\left\\{\alpha,r\right\\}$ and $C_{2}$ is a constant
independent of $h$. See Atkinson-Potra [3, Corollary 4.3].

###  2.3 Asymptotic Expansions and the higher order terms

Let $\varphi\in C^{2r+2}[0,1]$. For $\delta>0,$ let

$\mathcal{B}(\varphi,\delta)=\left\\{x\in\mathcal{X}:\left\lVert
x-\varphi\right\rVert_{\infty}\leq\delta\right\\}$

denote the closed $\delta$-neighbourhood of $\varphi$. Without loss of
generality, we assume that the Galerkin solution $x_{n}^{G}$ and the iterated
Galerkin solution $x_{n}^{S}$ belong to the above neighbourhood.

Denote

$\ell_{*}(s,t)=\frac{\partial\kappa}{\partial
u}\left(s,t,\varphi(t)\right),\quad s,t\in[0,1],$
$q_{*}(s,t)=\frac{\partial^{2}\kappa}{\partial
u^{2}}\left(s,t,\varphi(t)\right),\quad s,t\in[0,1].$

It follows that

$\mathcal{K}^{\prime}(\varphi)v(s)=\int_{0}^{1}\ell_{*}(s,t)\>v(t)\>dt,\qquad
v\in\mathcal{X},\>s\in[0,1],$ (2.8)
$\mathcal{K}^{\prime\prime}(\varphi)(v_{1},v_{2})(s)=\int_{0}^{1}q_{*}(s,t)~{}v_{1}(t)\>v_{2}(t)\>dt,\qquad
v_{1},v_{2}\in\mathcal{X},\>s\in[0,1],$

where the kernels $\ell_{*}(\cdot,\cdot),\;q_{*}(\cdot,\cdot)\in C[0,1]\times
C[0,1]$ are of the type of Green’s function as mentioned in the section 2.1.
Then

$\left\lVert\mathcal{K}^{\prime}(\varphi)\right\rVert\leq\sup_{0\leq t\leq
1}\int_{0}^{1}\left|\ell_{*}(s,t)\right|\>ds,$
$\left\lVert\mathcal{K}^{\prime\prime}(\varphi)\right\rVert\leq\sup_{0\leq
t\leq 1}\int_{0}^{1}\left|q_{*}(s,t)\right|\>ds.$ (2.9)

See Atkinson [1].

By assumption, $I-\mathcal{K}^{\prime}(\varphi)$ is invertible. Let

$\mathcal{M}=\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\mathcal{K}^{\prime}(\varphi),$
$\mathcal{M}_{2}=\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\mathcal{K}^{\prime\prime}(\varphi),$
$\mathcal{M}_{3}=\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\mathcal{K}^{(3)}(\varphi).$

Then $\mathcal{M}$, $\mathcal{M}_{2}$ and $\mathcal{M}_{3}$ are respectively
compact linear, bi-linear and tri-linear integral operators. See Riesz-Nagy
[17]. For $v\in\mathcal{X}$, let

$\mathcal{M}v(s)=\int_{0}^{1}m(s,t)\>v(t)\>dt,\quad s\in[0,1],$

Note that the kernels of $\mathcal{M}$, $\mathcal{M}_{2}$ and
$\mathcal{M}_{3}$ inherit the same smoothness properties as the kernels of
$\mathcal{K}^{\prime}(\varphi)$, $\mathcal{K}^{\prime\prime}(\varphi)$ and
$\mathcal{K}^{(3)}(\varphi)$ respectively. See Atkinson-Potra [Lemma 5.1][3].
Hence, the kernels of the above three operators are of the type of Green’s
function as mentioned in section 2.1.

We quote the following result from Rakshit-Rane [15].

$\mathcal{M}\varphi(t_{i})=\mathcal{M}\pi_{n}\varphi(t_{i})+\left(\mathcal{A}_{2r}\varphi\right)(t_{i})h^{2r}+O\left(h^{2r+2}\right),\quad
i=0,1,\ldots,n,$ (2.10)

where

$(\mathcal{A}_{2r}\varphi)(t_{i})=\bar{b}_{2r,2r}\int_{0}^{1}m(t_{i},t)(t)~{}\varphi^{(2r)}(t)~{}dt\\\
~{}~{}~{}~{}~{}+\sum_{p=1}^{2r-1}\bar{b}_{2r,p}\Bigg{\\{}\left[\left(\frac{\partial}{\partial
t}\right)^{2r-p-1}\left(m(t_{i},t)\varphi^{(p)}(t)\right)\right]_{t=0}^{t=1}\\\
-\left[\left(\frac{\partial}{\partial
t}\right)^{2r-p-1}\left(m(t_{i},t)\varphi^{(p)}(t)\right)\right]_{t=t_{i}-}^{t=t_{i}+}\Bigg{\\}}$

with

$\bar{b}_{2r,p}=\int_{0}^{1}\int_{0}^{1}\Lambda_{r}(\sigma,\tau)\frac{(\sigma-\tau)^{p}}{p!}\frac{B_{2r-p}(\tau)}{(2r-p)!}\>d\sigma\>d\tau,$

$\displaystyle\Lambda_{r}(\sigma,\tau)=\sum_{q=0}^{r-1}e_{q}(\sigma)e_{q}(\tau)$,
$\left\\{e_{0},e_{1},e_{2},\ldots\right\\}$ is the sequence of orthonormal
polynomials in $L^{2}[0,1]$ and $B_{k}$ is the Bernoulli polynomial of degree
$k$.

As in Kulkarni-Nidhin [10, Lemma 2.4], it can be shown that for all
$s\in[0,1]$,

$\mathcal{M}_{2}(\pi_{n}\varphi-\varphi)^{2}(s)=\mathcal{V}_{1}(\varphi)(s)h^{2r}+O(h^{2r+2})$
(2.11)

and

$\mathcal{M}_{3}(\pi_{n}\varphi-\varphi)^{3}(s)=\mathcal{V}_{2}(\varphi)(s)h^{3r}+O(h^{3r+1}),$
(2.12)

where

$\mathcal{V}_{1}(\varphi)=\left(\int_{0}^{1}[\chi_{r}(t)]^{2}dt\right)\mathcal{M}_{2}\left(\varphi^{(r)}\right)^{2},$
$\mathcal{V}_{2}(\varphi)=\left(\int_{0}^{1}[\chi_{r}(t)]^{3}dt\right)\mathcal{M}_{3}\left(\varphi^{(r)}\right)^{3}\text{
and }~{}\mathcal{V}_{2}(\varphi)=0\text{ for }~{}r=1$

with

$\chi_{r}(t)=\int_{0}^{1}\Lambda_{r}(\sigma,t)\frac{(\sigma-t)^{r}}{r!}d\sigma,$

are independent of $h$. In the proof of Lemma 2.4 in Kulkarni-Nidhin [10], the
authors used Euler-McLaurin expansion for smooth kernel. Since, the kernels of
$\mathcal{M}_{2}$ and $\mathcal{M}_{3}$ are of the type of Green’s function,
we use the extended Euler-McLaurin summation formula from Kulkarni-Rane [11].
It follows that

$\left\lVert\mathcal{M}_{3}(\pi_{n}\varphi-\varphi)^{3}\right\rVert_{\infty}=O\left(h^{4}\right)\quad\text{
for }~{}r=1.$ (2.13)

Let

$C_{3}=\max\left\\{\sup_{\stackrel{{\scriptstyle 0\leq t\leq s\leq
1}}{{|u|\leq\|\varphi\|_{\infty}+\delta}}}\left|D^{(1,0,0)}q_{1}(s,t,u)\right|,\sup_{\stackrel{{\scriptstyle
0\leq s\leq t\leq
1}}{{|u|\leq\|\varphi\|_{\infty}+\delta}}}\left|D^{(1,0,0)}q_{2}(s,t,u)\right|\right\\},$
$C_{4}=\max\left\\{\sup_{0\leq t\leq s\leq
1}\left|D^{(1,0)}\ell_{*,1}(s,t)\right|,\sup_{0\leq s\leq t\leq
1}\left|D^{(1,0)}\ell_{*,2}(s,t)\right|\right\\}.$

We first prove the following preliminary result which is needed later on.

######  Lemma 2.1.

Let $x\in\mathcal{B}(\varphi,\delta)$. Then, for $v_{1},v_{2}\in\mathcal{X}$,

$\left\lVert\left(\mathcal{K}^{\prime\prime}(x)(v_{1},v_{2})\right)^{{}^{\prime}}\right\rVert_{\infty}\leq
C_{3}\left\lVert v_{1}\right\rVert_{\infty}\left\lVert
v_{2}\right\rVert_{\infty}.$

###### Proof.

Let $s\in[0,1]$. Then

$\displaystyle\mathcal{K}^{\prime\prime}(x)(v_{1},v_{2})(s)$
$\displaystyle=\int_{0}^{1}q(s,t,x(t))\>v_{1}(t)\>v_{2}(t)\>dt$
$\displaystyle=\int_{0}^{s}q_{1}(s,t,x(t))\>v_{1}(t)\>v_{2}(t)\>dt+\int_{s}^{1}q_{2}(s,t,x(t))\>v_{1}(t)\>v_{2}(t)\>dt.$

It follows that

$\displaystyle\left(\mathcal{K}^{\prime\prime}(x)(v_{1},v_{2})\right)^{{}^{\prime}}(s)$
$\displaystyle=\int_{0}^{s}\frac{\partial q_{1}}{\partial
s}\left(s,t,x(t)\right)v_{1}(t)\>v_{2}(t)\>dt+q_{1}(s,s,x(s))\>v_{1}(s)\>v_{2}(s)$
$\displaystyle~{}~{}+\int_{s}^{1}\frac{\partial q_{2}}{\partial
s}\left(s,t,x(t)\right)v_{1}(t)\>v_{2}(t)\>dt-
q_{2}(s,s,x(s))\>v_{1}(s)\>v_{2}(s).$

Since $q$ is continuous on $\Omega$,

$\displaystyle\left(\mathcal{K}^{\prime\prime}(x)(v_{1},v_{2})\right)^{{}^{\prime}}(s)=\int_{0}^{s}\frac{\partial
q_{1}}{\partial
s}\left(s,t,x(t)\right)v_{1}(t)v_{2}(t)\>dt+\int_{s}^{1}\frac{\partial
q_{2}}{\partial s}\left(s,t,x(t)\right)v_{1}(t)v_{2}(t)\>dt.$

Hence,

$\left\lVert\left(\mathcal{K}^{\prime\prime}(x)(v_{1},v_{2})\right)^{{}^{\prime}}\right\rVert_{\infty}\leq
C_{3}\left\lVert v_{1}\right\rVert_{\infty}\left\lVert
v_{2}\right\rVert_{\infty}.$

This completes the proof. ∎

Let $x\in\mathcal{B}(\varphi,\delta)$. Then by the above lemma, we obtain

$\left\lVert(I-\pi_{n})\left(\mathcal{K}^{\prime\prime}(x)(v_{1},v_{2})\right)\right\rVert_{\infty}\leq
C_{1}C_{3}\left\lVert v_{1}\right\rVert_{\infty}\left\lVert
v_{2}\right\rVert_{\infty}h.$ (2.14)

Similarly, for any $v\in\mathcal{X}$, it can be shown that the function
$\mathcal{K}^{\prime}(\varphi)v$ is differentiable on $[0,1]$ and

$\left\lVert\left(\mathcal{K}^{\prime}(\varphi)v\right)^{{}^{\prime}}\right\rVert_{\infty}\leq
C_{4}\left\lVert v\right\rVert_{\infty}.$ (2.15)

It follows that

$\left\lVert(I-\pi_{n})\left(\mathcal{K}^{\prime}(\varphi)v\right)\right\rVert_{\infty}\leq
C_{1}C_{4}\left\lVert v\right\rVert_{\infty}h.$ (2.16)

The following crucial estimate

$\displaystyle\left\lVert\mathcal{K}^{\prime}(\varphi)(I-\pi_{n})\varphi\right\rVert_{\infty}=\left\\{{\begin{array}[]{ll}O\left(h^{2}\right),~{}~{}~{}~{}~{}~{}r=1,\\\
O\left(h^{r+2}\right),~{}~{}~{}\>r\geq 2.\end{array}}\right.$ (2.19)

follows from Lemma 9 of Chatelin-Lebbar [4]. From Theorem 3.1 of Kulkarni [9],
we have

$\displaystyle\left\lVert(I-\pi_{n})\mathcal{K}^{\prime}(\varphi)(I-\pi_{n})\varphi\right\rVert_{\infty}=O\left(h^{r+2}\right),\quad
r\geq 1.$ (2.20)

In order to prove our main result, we need to establish the following lemmas
and propositions. Note that

$x_{n}^{G}-\varphi=\pi_{n}(x_{n}^{S}-\varphi)-(I-\pi_{n})\varphi.$ (2.21)

We use the above relation between Galerkin and iterated Galerkin solution
several times in the following lemmas and propositions.

######  Lemma 2.2.

Let $x_{n}^{G}$ be the Galerkin solution defined by the equation (1.3). Then
for $r\geq 1$,

$\displaystyle\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\mathcal{K}^{\prime\prime}(\varphi)(x_{n}^{G}-\varphi)^{2}(s)=(\mathcal{V}_{1}(\varphi))(s)\>h^{2r}+O\left(h^{2r+2}\right),\quad
s\in[0,1],$ (2.22)

where $\mathcal{V}_{1}$ is defined by (2.11).

###### Proof.

Using (2.21), we write

$\displaystyle\mathcal{K}^{\prime\prime}(\varphi)(x_{n}^{G}-\varphi)^{2}$
$\displaystyle=~{}\mathcal{K}^{\prime\prime}(\varphi)\left[\pi_{n}(x_{n}^{S}-\varphi)-(I-\pi_{n})\varphi\right]^{2}$
$\displaystyle=~{}\mathcal{K}^{\prime\prime}(\varphi)\left(\pi_{n}(x_{n}^{S}-\varphi)\right)^{2}$
$\displaystyle~{}~{}~{}-2~{}\mathcal{K}^{\prime\prime}(\varphi)\left(\pi_{n}(x_{n}^{S}-\varphi),(I-\pi_{n})\varphi\right)$
$\displaystyle~{}~{}~{}+~{}\mathcal{K}^{\prime\prime}(\varphi)\left((I-\pi_{n})\varphi\right)^{2}.$

It follows that

$\displaystyle\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\mathcal{K}^{\prime\prime}(\varphi)(x_{n}^{G}-\varphi)^{2}$
$\displaystyle=~{}\mathcal{M}_{2}\left(\pi_{n}(x_{n}^{S}-\varphi)\right)^{2}$
$\displaystyle~{}~{}~{}-2~{}\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\mathcal{K}^{\prime\prime}(\varphi)\left(\pi_{n}(x_{n}^{S}-\varphi),(I-\pi_{n})\varphi\right)$
$\displaystyle~{}~{}~{}+~{}\mathcal{M}_{2}\left((I-\pi_{n})\varphi\right)^{2}.$
(2.23)

Since
$\displaystyle{\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\mathcal{K}^{(3)}(\varphi)}$
is bounded, from (1.6), (2.5), (2.6) and (2.9), it is easy to see that

$\left\lVert\mathcal{M}_{2}\left(\pi_{n}(x_{n}^{S}-\varphi)\right)^{2}\right\rVert_{\infty}=\left\\{{\begin{array}[]{ll}O\left(h^{4}\right),~{}~{}~{}~{}r=1,\\\
O\left(h^{2r+4}\right),~{}~{}~{}\>r\geq 2.\end{array}}\right.$ (2.24)

and

$\left\lVert\mathcal{K}^{\prime\prime}(\varphi)\left(\pi_{n}(x_{n}^{S}-\varphi),(I-\pi_{n})\varphi\right)\right\rVert_{\infty}=\left\\{{\begin{array}[]{ll}O\left(h^{3}\right),~{}~{}~{}~{}r=1,\\\
O\left(h^{2r+2}\right),~{}~{}~{}\>r\geq 2.\end{array}}\right.$

When $r=1$, that is, when $\mathscr{X}_{n}$ is the space of piecewise constant
functions, the order of the term
$\left\lVert\mathcal{K}^{\prime\prime}(\varphi)\left(\pi_{n}(x_{n}^{S}-\varphi),(I-\pi_{n})\varphi\right)\right\rVert_{\infty}$
can be improved to $h^{4}$ in the following way. Note that

$\displaystyle\mathcal{K}^{\prime\prime}(\varphi)\left(\pi_{n}(x_{n}^{S}-\varphi),(I-\pi_{n})\varphi\right)(s)$
$\displaystyle=\int_{0}^{1}q_{*}(s,t)\>(\pi_{n}(x_{n}^{S}-\varphi))(t)\>(I-\pi_{n})\varphi(t)\>dt$
$\displaystyle=\sum_{j=1}^{n}\int_{t_{j-1}}^{t_{j}}q_{*}(s,t)(\pi_{n,j}(x_{n}^{S}-\varphi))(t)\>(I-\pi_{n,j})\varphi(t)\>dt.$

Since, the range of $\pi_{n}$ is $\mathscr{X}_{n}$,
$\pi_{n,j}(x_{n}^{S}-\varphi)$ is a constant on $[t_{j-1},t_{j}]$. It follows
that

$\mathcal{K}^{\prime\prime}(\varphi)\left(\pi_{n}(x_{n}^{S}-\varphi),(I-\pi_{n})\varphi\right)(s)=\\\
\sum_{j=1}^{n}(\pi_{n,j}(x_{n}^{S}-\varphi))\left(\frac{t_{j-1}+t_{j}}{2}\right)\int_{t_{j-1}}^{t_{j}}q_{*}(s,t)\>(I-\pi_{n,j})\varphi(t)\>dt.$

As in the proof of Lemma 9 of Chatelin-Lebbar [4], it can be shown that

$\left|\int_{t_{j-1}}^{t_{j}}q_{*}(s,t)\>(I-\pi_{n,j})\varphi(t)\>dt\right|=O\left(h^{3}\right).$

Since $\left\\{\pi_{n,j}\right\\}$ is uniformly bounded and $\left\lVert
x_{n}^{S}-\varphi\right\rVert_{\infty}=O\left(h^{2}\right)$, from the above
estimate it follows that

$\left\lVert\mathcal{K}^{\prime\prime}(\varphi)\left(\pi_{n}(x_{n}^{S}-\varphi),(I-\pi_{n})\varphi\right)\right\rVert_{\infty}=O\left(h^{4}\right),\quad
r=1.$

Hence,

$\left\lVert\mathcal{K}^{\prime\prime}(\varphi)\left(\pi_{n}(x_{n}^{S}-\varphi),(I-\pi_{n})\varphi\right)\right\rVert_{\infty}=\left\\{{\begin{array}[]{ll}O\left(h^{4}\right),~{}~{}~{}~{}~{}~{}~{}r=1,\\\
O\left(h^{2r+2}\right),~{}~{}~{}\>r\geq 2.\end{array}}\right.$ (2.25)

Hence, (2.22) follows from (2.11), (2.3) and (2.25), the proof of the
proposition is complete. ∎

######  Lemma 2.3.

Let $s\in[0,1]$ and $r\geq 1$. Then

$\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\mathcal{K}^{(3)}(\varphi)(x_{n}^{G}-\varphi)^{3}(s)=\left\\{{\begin{array}[]{ll}O\left(h^{4}\right),\qquad\qquad\qquad\qquad~{}r=1,\\\
(\mathcal{V}_{2}(\varphi))(s)\>h^{3r}+O\left(h^{3r+1}\right),~{}~{}\>r\geq
2.\end{array}}\right.$ (2.26)

where $\mathcal{V}_{2}$ is defined by (2.12).

###### Proof.

We write

$\displaystyle\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\mathcal{K}^{(3)}(\varphi)(x_{n}^{G}-\varphi)^{3}$
$\displaystyle=~{}\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\mathcal{K}^{(3)}(\varphi)\left[\pi_{n}(x_{n}^{S}-\varphi)-(I-\pi_{n})\varphi\right]^{3}$
$\displaystyle=~{}\mathcal{M}_{3}\left(\pi_{n}(x_{n}^{S}-\varphi)\right)^{3}$
$\displaystyle~{}~{}~{}-3~{}\mathcal{M}_{3}\left(\pi_{n}(x_{n}^{S}-\varphi),\pi_{n}(x_{n}^{S}-\varphi),(I-\pi_{n})\varphi\right)$
$\displaystyle~{}~{}~{}+3~{}\mathcal{M}_{3}\left(\pi_{n}(x_{n}^{S}-\varphi),(I-\pi_{n})\varphi,(I-\pi_{n})\varphi\right)$
$\displaystyle~{}~{}~{}-~{}\mathcal{M}_{3}\left((I-\pi_{n})\varphi\right)^{3}$
(2.27)

Since
$\displaystyle{\mathcal{M}_{3}=\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\mathcal{K}^{(3)}(\varphi)}$
is bounded, (1.6), (2.5), (2.6) we have

$\left\lVert\mathcal{M}_{3}\left(\pi_{n}(x_{n}^{S}-\varphi)\right)^{3}\right\rVert_{\infty}=\left\\{{\begin{array}[]{ll}O\left(h^{6}\right),\qquad~{}r=1,\\\
O\left(h^{3r+6}\right),\quad r\geq 2.\end{array}}\right.$ (2.28)
$\left\lVert\mathcal{M}_{3}\left(\pi_{n}(x_{n}^{S}-\varphi),\pi_{n}(x_{n}^{S}-\varphi),(I-\pi_{n})\varphi\right)\right\rVert_{\infty}=\left\\{{\begin{array}[]{ll}O\left(h^{5}\right),\qquad~{}r=1,\\\
O\left(h^{3r+4}\right),\quad r\geq 2.\end{array}}\right.$ (2.29)
$\left\lVert\mathcal{M}_{3}\left(\pi_{n}(x_{n}^{S}-\varphi),(I-\pi_{n})\varphi,(I-\pi_{n})\varphi\right)\right\rVert_{\infty}=\left\\{{\begin{array}[]{ll}O\left(h^{4}\right),\qquad~{}r=1,\\\
O\left(h^{3r+2}\right),\quad r\geq 2.\end{array}}\right.$ (2.30)

On the other hand, from (2.12) we have

$\mathcal{M}_{3}\left((I-\pi_{n})\varphi\right)^{3}(s)=\left\\{{\begin{array}[]{ll}O\left(h^{4}\right),\qquad\qquad\qquad\qquad~{}r=1,\\\
(\mathcal{V}_{2}(\varphi))(s)\>h^{3r}+O\left(h^{3r+1}\right),~{}~{}\>r\geq
2.\end{array}}\right.$

Hence, (2.26) follows from (2.3), (2.28), (2.29), (2.30) and the above
equation. ∎

##  3 The Main Result

Recall that the Iterated Galerkin solution is defined by

$\displaystyle x_{n}^{S}-\mathcal{K}(\pi_{n}x_{n}^{S})=f$

and the exact solution as

$\varphi-\mathcal{K}(\varphi)=f.$

In this section, we prove our main result about the asymptotic series
expansion for the iterated Galerkin solution $x_{n}^{S}$ at the partition
points $t_{i},\>i=0,1,\dots,n$. That is, we will show the following.

$\displaystyle\varphi(t_{i})-x_{n}^{S}(t_{i})=\mathcal{A}_{2r}(t_{i})h^{2r}+O\left(h^{2r+2}\right),$
(3.1)

where $\mathcal{A}_{2r}$ is a function independent of $n$.

Then, we can apply Richardson extrapolation to obtain an approximation of
$\varphi$ with higher order at the partition points. From Ford et al [5,
Section 5], it can be shown that a continuous function can be reconstructed
from the extrapolated discrete values at the partition points and it
approximates the exact solution $\varphi$ to higher order in the uniform norm.
We will not discuss this thing here. Our main aim is to prove (3.1).

Recall that

$\mathcal{M}=\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\mathcal{K}^{\prime}(\varphi)$

with

$\displaystyle(\mathcal{M}x)(s)=\int_{0}^{1}m(s,t)~{}x(t)~{}dt,\;\;\;s\in[0,1],\;x\in\mathcal{X},$
(3.2)

where the kernel $m$ is of the type of Green’s function.

We quote the following expression for the error in the iterated Galerkin
solution from Atkinson et al [2, equation (2.28)]:

$\displaystyle x_{n}^{S}-\varphi$
$\displaystyle~{}=~{}\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\left\\{\left[\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right]\right\\}$
$\displaystyle~{}~{}~{}-\mathcal{M}(I-\pi_{n})\left[\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right]$
$\displaystyle~{}~{}~{}-\mathcal{M}(I-\pi_{n})\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)$
$\displaystyle~{}~{}~{}-\mathcal{M}(I-\pi_{n})\varphi.$ (3.3)

By the following propositions, we will prove that the second and the third
terms on the right hand side of the above equation are of the order $2r+2$ or
higher, and the first and the last term has an asymptotic expansion at the
partition points.

Let

$\displaystyle C_{5}$ $\displaystyle=$ $\displaystyle\max_{0\leq i\leq
4}\left(\sup_{\stackrel{{\scriptstyle
s,t\in[0,1]}}{{|u|\leq\|\varphi\|_{\infty}+\delta}}}\left|\frac{\partial^{i}\kappa}{\partial
u^{i}}(s,t,u)\right|\right).$

Let us investigate the first term on the right hand side of the equation
(2.28) for an asymptotic expansion.

######  Proposition 3.1.

Let $x_{n}^{G}$ be the Galerkin solution defined by the equation (1.3) and
$s\in[0,1]$. Then for $r\geq 1,$

$\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\left[\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right](s)=\mathcal{V}_{1}(\varphi)(s)\>h^{2r}+O\left(h^{2r+2}\right),$

where $\mathcal{V}_{1}$ is defined by (2.11).

###### Proof.

Using the generalized Taylor’s series expansion (see Linz [13]) in the
neighbourhood $\mathcal{B}(\varphi,\delta)$, we obtain

$\displaystyle\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-$
$\displaystyle\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)$
$\displaystyle=\frac{1}{2}\mathcal{K}^{\prime\prime}(\varphi)(x_{n}^{G}-\varphi)^{2}+\frac{1}{6}\mathcal{K}^{(3)}(\varphi)(x_{n}^{G}-\varphi)^{3}+\mathcal{R}_{4}\left(x_{n}^{G},\varphi\right),$

where

$\mathcal{R}_{4}\left(x_{n}^{G},\varphi\right)=\frac{1}{6}\int_{0}^{1}\mathcal{K}^{(4)}\left(\varphi+\theta(x_{n}^{G}-\varphi)\right)(x_{n}^{G}-\varphi)^{4}(1-\theta)^{3}\;d\theta.$

By (2.1), we have

$\displaystyle\mathcal{K}^{(4)}\left(\varphi+\theta(x_{n}^{G}-\varphi)\right)(x_{n}^{G}-\varphi)^{4}(s)=\int_{0}^{1}\frac{\partial^{4}\kappa}{\partial
u^{4}}\left(s,t,\varphi(t)+\theta(x_{n}^{G}-\varphi)(t)\right)(x_{n}^{G}-\varphi)^{4}(t)\>dt.$

Since $\left\lVert x_{n}^{G}-\varphi\right\rVert_{\infty}\rightarrow 0$ as
$n\rightarrow\infty$ and $\theta\in(0,1)$,

$\varphi+\theta(x_{n}^{G}-\varphi)\in\mathcal{B}(\varphi,\delta).$

It follows that

$\displaystyle\left\lVert\mathcal{K}^{(4)}\left(\varphi+\theta(x_{n}^{G}-\varphi)\right)(x_{n}^{G}-\varphi)^{4}\right\rVert_{\infty}\leq
C_{5}\left\lVert x_{n}^{G}-\varphi\right\rVert_{\infty}^{4}$

Using (1.6) and the above estimate, we obtain

$\mathcal{R}_{4}\left(x_{n}^{G},\varphi\right)=O\left(h^{4r}\right).$ (3.4)

Note that

$\displaystyle\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}$
$\displaystyle\left[\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\>\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\mathcal{K}^{\prime\prime}(\varphi)(x_{n}^{G}-\varphi)^{2}+\frac{1}{6}\>\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\mathcal{K}^{(3)}(\varphi)(x_{n}^{G}-\varphi)^{3}$
$\displaystyle~{}+\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\mathcal{R}_{4}\left(x_{n}^{G},\varphi\right).$

Hence, by Lemma 2.2, Lemma 2.3, equation (3.4) and and the above estimate, we
obtain the followings.
For $r=1$,

$\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\left[\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right](s)=\mathcal{V}_{1}(\varphi)(s)\>h^{2r}+O\left(h^{4}\right),$

for $r\geq 2$,

$\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\left[\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right](s)\\\
=\mathcal{V}_{1}(\varphi)(s)\>h^{2r}+\mathcal{V}_{2}(\varphi)(s)\>h^{3r}+O\left(h^{2r+2}\right).$

Since $3r\geq 2r+2$ for $r\geq 2$,

$\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\left[\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right](s)=\mathcal{V}_{1}(\varphi)(s)\>h^{2r}+O\left(h^{2r+2}\right),\quad
r\geq 2.$

This completes the proof. ∎

Now we investigate the second term on the R.H.S. of the equation (3).

######  Proposition 3.2.

Let $\displaystyle{\left\\{t_{i}:i=0,1,\dots,n\right\\}}$ be the set of all
partition points defined by (2.4). Then for $r\geq 1$,

$\mathcal{M}(I-\pi_{n})\left[\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right](t_{i})=O\left(h^{2r+2}\right).$

###### Proof.

By the generalized Taylor’s theorem we obtain

$\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)=\int_{0}^{1}\mathcal{K}^{\prime\prime}\left(\varphi+\theta(x_{n}^{G}-\varphi)\right)(x_{n}^{G}-\varphi)^{2}(1-\theta)\;d\theta.$

Let

$\mathcal{R}_{2}(x_{n}^{G},\varphi)=\int_{0}^{1}\mathcal{K}^{\prime\prime}\left(\varphi+\theta(x_{n}^{G}-\varphi)\right)(x_{n}^{G}-\varphi)^{2}(1-\theta)\;d\theta.$
(3.5)

Note that

$\displaystyle\mathcal{K}^{\prime\prime}\left(\varphi+\theta(x_{n}^{G}-\varphi)\right)(x_{n}^{G}-\varphi)^{2}(s)=\int_{0}^{1}\frac{\partial^{2}\kappa}{\partial
u^{2}}\left(s,t,\varphi(t)+\theta(x_{n}^{G}-\varphi)(t)\right)(x_{n}^{G}-\varphi)^{2}(t)\>dt.$

It follows that

$\displaystyle\left\lVert\mathcal{K}^{\prime\prime}\left(\varphi+\theta(x_{n}^{G}-\varphi)\right)(x_{n}^{G}-\varphi)^{2}\right\rVert_{\infty}\leq
C_{3}\left\lVert x_{n}^{G}-\varphi\right\rVert_{\infty}^{2}.$

Since $\displaystyle{\left\lVert
x_{n}^{G}-\varphi\right\rVert_{\infty}=O\left(h^{r}\right)}$,

$\displaystyle\left\lVert\mathcal{K}^{\prime\prime}\left(\varphi+\theta(x_{n}^{G}-\varphi)\right)(x_{n}^{G}-\varphi)^{2}\right\rVert_{\infty}=O\left(h^{2r}\right).$
(3.6)

Let $s\in[0,1]$ be fixed and $m_{s}(t)=m(s,t),~{}t\in[0,1]$, then

$\mathcal{M}(I-\pi_{n})\left[\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right](s)\\\
=\left<m_{s},\>(I-\pi_{n})\left[\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right]\right>,$

where $\left<\cdot,\cdot\right>$ is the usual inner product in $L^{2}[0,1]$,
i.e.,

$\left<x,y\right>=\int_{0}^{1}x(t)\>y(t)\>dt,\quad x,y\in L^{2}[0,1].$

Since $I-\pi_{n}$ is self-adjoint,

$\mathcal{M}(I-\pi_{n})\left[\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right](t_{i})\\\
=\left<(I-\pi_{n})m_{t_{i}},\>(I-\pi_{n})\left[\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right]\right>.$
(3.7)

Note that $m_{t_{i}}$ is continuous on $[t_{j-1},t_{j}]$ and $r$ times
continuously differentiable on $(t_{j-1},t_{j})$ for all $j=1,2,\dots,n$.
Therefore by (2.7)

$\displaystyle\left\lVert(I-\pi_{n,j})m_{t_{i}}\right\rVert_{\Delta_{j},\infty}=O\left(h^{r}\right).$
(3.8)

Using (2.5), (3.6), (3.7) and the above estimate, we obtain

$\mathcal{M}(I-\pi_{n})\left[\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right](t_{i})=O\left(h^{3r}\right),\quad
r\geq 2.$ (3.9)

Consider the case $\mathbf{r=1}$.

From (3.5), it is easy to see that if
$\mathcal{K}^{\prime\prime}\left(\varphi+\theta(x_{n}^{G}-\varphi)\right)(x_{n}^{G}-\varphi)^{2}$
is differentiable, then $\mathcal{R}_{2}(x_{n}^{G},\varphi)$ is
differentiable. Since
$\varphi+\theta(x_{n}^{G}-\varphi)\in\mathcal{B}(\varphi,\delta)$, from Lemma
2.1 we have $\mathcal{R}_{2}(x_{n}^{G},\varphi)$ is differentiable. Thus, from
(3.5)

$\left(\mathcal{R}_{2}(x_{n}^{G},\varphi)\right)^{{}^{\prime}}=\int_{0}^{1}\left(\mathcal{K}^{\prime\prime}\left(\varphi+\theta(x_{n}^{G}-\varphi)\right)(x_{n}^{G}-\varphi)^{2}\right)^{{}^{\prime}}(1-\theta)\;d\theta.$

This implies,

$\left\lVert\left(\mathcal{R}_{2}(x_{n}^{G},\varphi)\right)^{{}^{\prime}}\right\rVert_{\infty}\leq\frac{1}{2}\left\lVert\left(\mathcal{K}^{\prime\prime}\left(\varphi+\theta(x_{n}^{G}-\varphi)\right)(x_{n}^{G}-\varphi)^{2}\right)^{{}^{\prime}}\right\rVert_{\infty},\qquad
0<\theta<1.$

Using Lemma 2.1,

$\left\lVert\left(\mathcal{R}_{2}(x_{n}^{G},\varphi)\right)^{{}^{\prime}}\right\rVert_{\infty}\leq
C_{3}\left\lVert x_{n}^{G}-\varphi\right\rVert^{2}.$

From (2.6), it follows that

$\displaystyle\left\lVert(I-\pi_{n})\left[\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right]\right\rVert_{\infty}$
$\displaystyle~{}=~{}\left\lVert(I-\pi_{n})\mathcal{R}_{2}(x_{n}^{G},\varphi)\right\rVert_{\infty}$
$\displaystyle~{}\leq~{}C_{1}C_{3}\left\lVert
x_{n}^{G}-\varphi\right\rVert^{2}h.$

By (1.5), (3.7), (3.8) and the above estimate, we obtain

$\mathcal{M}(I-\pi_{n})\left[\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right](t_{i})=O\left(h^{4}\right),\quad
r=1.$ (3.10)

Hence, the required result follows from (3.9) and (3.10). ∎

Next we investigate the third term on the R.H.S. of the equation (3).

######  Proposition 3.3.

Let $\displaystyle{\left\\{t_{i}:i=0,1,\dots,n\right\\}}$ be the set of all
partition points defined by (2.4). Then

$\mathcal{M}(I-\pi_{n})\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)(t_{i})=O\left(h^{2r+2}\right),\quad\text{
for }r\geq 1.$

###### Proof.

From (3.2) we have

$\displaystyle\mathcal{M}(I-\pi_{n})\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)(t_{i})$
$\displaystyle=\int_{0}^{1}m(t_{i},t)~{}(I-\pi_{n})\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)(t)~{}dt$
$\displaystyle=\left<m_{t_{i}}~{},~{}(I-\pi_{n})\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right>$
$\displaystyle=\left<(I-\pi_{n})m_{t_{i}}~{},~{}(I-\pi_{n})\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right>.$
(3.11)

It is easy to see that, $m_{t_{i}}$ is continuous on $[t_{j-1},t_{j}]$ and $r$
times continuously differentiable on $(t_{j-1},t_{j})$ for all
$j=1,2,\dots,n$. Therefore by (2.7)

$\displaystyle\left\lVert(I-\pi_{n,j})m_{t_{i}}\right\rVert_{\Delta_{j},\infty}=O\left(h^{r}\right).$
(3.12)

Note that

$\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)=\mathcal{K}^{\prime}(\varphi)\left(\pi_{n}(x_{n}^{S}-\varphi)\right)-\mathcal{K}^{\prime}(\varphi)(I-\pi_{n})\varphi.$
(3.13)

Thus, by (1.6), (2.5) and (2.19) we obtain,

$\left\lVert\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right\rVert_{\infty}=O\left(h^{r+2}\right),\quad
r\geq 2.$

Hence, from (3), (3.12) and the above estimate, we obtain

$\mathcal{M}(I-\pi_{n})\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)(t_{i})=O\left(h^{2r+2}\right),\quad
r\geq 2.$ (3.14)

When $\mathbf{r=1}$, that is, when $\mathscr{X}_{n}$ is the space of piecewise
constant functions, it is easy to see from (1.5), (2.19) and (3.13) that

$\left\lVert\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right\rVert_{\infty}=O\left(h^{2}\right),$

which is not equal to $O\left(h^{2r+2}\right)$ with $r=1$. We consider this
case separately.

Note that,

$(I-\pi_{n})\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)=(I-\pi_{n})\mathcal{K}^{\prime}(\varphi)\left(\pi_{n}(x_{n}^{S}-\varphi)\right)-(I-\pi_{n})\mathcal{K}^{\prime}(\varphi)(I-\pi_{n})\varphi.$
(3.15)

From (2.16), we have

$\left\lVert(I-\pi_{n})\mathcal{K}^{\prime}(\varphi)\left(\pi_{n}(x_{n}^{S}-\varphi)\right)\right\rVert_{\infty}\leq
C_{1}C_{4}\left\lVert\pi_{n}\right\rVert\left\lVert
x_{n}^{S}-\varphi\right\rVert_{\infty}h.$

From (1.5) and (2.5), it follows that

$\left\lVert(I-\pi_{n})\mathcal{K}^{\prime}(\varphi)\left(\pi_{n}(x_{n}^{S}-\varphi)\right)\right\rVert_{\infty}=O\left(h^{3}\right).$

By (2.20), (3), (3.12), (3.15) and the above estimate, we obtain

$\mathcal{M}(I-\pi_{n})\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)(t_{i})=O\left(h^{4}\right),\quad
r=1.$ (3.16)

Hence, the required result follows from (3.14) and (3.16). ∎

Now, we prove our main theorem.

######  Theorem 3.4.

Let $f\in C^{2r}[0,1]$, and the kernel of the Urysohn integral operator (1.1)
be of class $\mathscr{G}_{4}(r,0)$. Let $\varphi$ be a fixed point of the
operator $\mathcal{T}$ defined by (2.2), with $1$ not an eigenvalue of
$\mathcal{K}^{\prime}(\varphi)$. For $r\geq 1$, let $\mathscr{X}_{n}$ be the
space of piecewise polynomials of degree $\leq r-1$ with respect to the
partition (2.3) and $\pi_{n}$ be the orthogonal projection defined by
(2.5)–(2.6). Let $x_{n}^{S}$ be the iterated Galerkin solution defined by
(1.4). Then, for $i=0,1,\ldots,n$,

$x_{n}^{S}(t_{i})-\varphi(t_{i})=-\zeta_{2r}(t_{i})\>h^{2r}+O\left(h^{2r+2}\right),$

where $\zeta_{2r}$ is a function bounded by a constant independent of $h$.

###### Proof.

From the equation (3)

$\displaystyle x_{n}^{S}(t_{i})-\varphi(t_{i})$
$\displaystyle~{}=~{}\left(I-\mathcal{K}^{\prime}(\varphi)\right)^{-1}\left[\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right](t_{i})$
$\displaystyle~{}~{}~{}-\mathcal{M}(I-\pi_{n})\left[\mathcal{K}(x_{n}^{G})-\mathcal{K}(\varphi)-\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)\right](t_{i})$
$\displaystyle~{}~{}~{}-\mathcal{M}(I-\pi_{n})\mathcal{K}^{\prime}(\varphi)(x_{n}^{G}-\varphi)(t_{i})$
$\displaystyle~{}~{}~{}-\mathcal{M}(I-\pi_{n})\varphi(t_{i}).$

Let

$\displaystyle\zeta_{2r}=\mathcal{A}_{2r}+\mathcal{V}_{1}(\varphi).$

Hence, the proof of this theorem follows from the equation (2.10), Proposition
3.1, Proposition 3.2 and Proposition 3.3.

∎

We can now apply one step of Richardson extrapolation and obtain an
approximations of $\varphi$ of order $h^{2r+2}$ at the partition points.

Define

$x_{n}^{EX}=\frac{2^{4r}x_{2n}^{S}-x_{n}^{S}}{2^{4r}-1}.$

Then under the assumptions of Theorem 3.4, we have the following result

$x_{n}^{EX}{(t_{i})}-\varphi{(t_{i})}=O\left(h^{2r+2}\right),\quad
i=0,1,2,\ldots,n.$ (3.17)

##  4 Numerical Illustration

For the sake of numerical illustration, we consider the following example of a
non-linear Hammerstein integral equation from Kulkarni-Rane [12]. Consider

$\varphi(s)-\int_{0}^{1}\kappa(s,t)\left[\psi\left(t,\varphi(t)\right)\right]\>dt=f(s),\;\;\;0\leq
s\leq 1,$ (4.1)

where

$\kappa(s,t)=\frac{1}{\gamma\sinh\gamma}\left\\{{\begin{array}[]{ll}\sinh\gamma
s\>\sinh\gamma(1-t),&~{}0\leq t\leq s\leq 1,\\\ \gamma(1-s)\sinh\gamma
t,&~{}0\leq s\leq t\leq 1,\end{array}}\right.$

with $\gamma=\sqrt{12},$ and

$\psi(t,\varphi(t))=\gamma^{2}\varphi(t)-2\left(\varphi(t)\right)^{3},\quad
t\in[0,1].$

We have
$f(s)=\frac{1}{\gamma\sinh\gamma}\left\\{2\sinh\gamma(1-s)+\frac{2}{3}\sinh\gamma
s\right\\}.$ The exact solution of (4.1) is given by

$\displaystyle\varphi(s)=\frac{2}{2s+1},\quad s\in[0,1].$

Let $\mathscr{X}_{n}$ be the space of piecewise constant polynomials with
respect to the uniform partition (2.3) of the interval $[0,1]$ considered
before. Let $\pi_{n}:L^{\infty}[0,1]\rightarrow\mathscr{X}_{n}$ be the
orthogonal projection defined by (2.5)–(2.6).
In this case, it is given by

$(\pi_{n}\varphi)(s)=\displaystyle\frac{1}{h}\int_{(i-1)h}^{ih}\varphi(t)\>dt,\;s\in[(i-1)h,ih],$

where $h=\frac{1}{n}$. In the definition of projection operator defined above,
if we replace the integral by the right hand rule, then
$(\pi_{n}\varphi)(ih-)=\varphi(ih-).$ Let $x_{n}^{S}$ be the Sloan solution
defined by (1.4). Then
$x_{n}^{G}(ih)=\displaystyle\frac{\pi_{n}x_{n}^{S}(ih-)+\pi_{n}x_{n}^{S}(ih+)}{2}$
at the partition points is obtained by solving the approximate system of non-
linear equations which gives the values of $\pi_{n}x_{n}^{S}(ih-)$ and
$\pi_{n}x_{n}^{S}(ih+).$ The system is as follows:

$\alpha_{j}=h\sum_{l=1}^{n}\kappa(s_{j},s_{l})\left[\gamma^{2}\alpha_{l}-2\frac{\alpha_{l}^{3}}{h}\right]+\frac{f(s_{j})}{\sqrt{h}},\quad
j=1,2,\ldots,n,$ (4.2)

where $\alpha_{l}=\pi_{n}x_{n}^{G}(s_{l})$ and $s_{l}=(l-\frac{1}{2})h$ for
$l=1,2,\ldots,n$. The above system is obtained by replacing all the integrals
by numerical integration formula.

We have used Picard’s iteration to solve the above system of non-linear
equations.

Let $t_{i}=(i-1)/20,i=1,2,\ldots,21$ be the partition points with step size
$h=\frac{1}{20}.$ It is easy to see that

$\displaystyle
E_{1}^{n}(t_{i})=|\varphi(t_{i})-x_{n}^{S}(t_{i})|=O\left(h^{2}\right).$

We define

$\displaystyle
x_{n}^{EX}(t_{i})=\frac{4x_{2n}^{S}(t_{i})-x_{n}^{S}(t_{i})}{3}.$

Then

$E_{2}^{n}(t_{i})=\left|\varphi(t_{i})-x_{n}^{EX}(t_{i})\right|=O\left(h^{4}\right).$

The orders of convergence are calculated using the formula :

$\displaystyle\begin{aligned}
\alpha_{1}=\frac{log(E_{1}^{n}(t_{i})/E_{1}^{2n}(t_{i}))}{log(2)},\\\
\beta=\frac{log(E_{2}^{n}(t_{i})/E_{2}^{2n}(t_{i}))}{log(2)},\end{aligned}~{}~{}~{}~{}~{}~{}~{}n=40$

$\displaystyle\alpha_{2}=\frac{log(E_{1}^{n}(t_{i})/E_{1}^{2n}(t_{i}))}{log(2)},\quad
n=20.$

We expect $\alpha_{1}=\alpha_{2}=2$ and $\beta=4.$

Table 4.1

$t_{i}$ | $E_{1}^{n}(t_{i}):n=20$ |  $E_{1}^{n}(t_{i}):n=40$ | $E_{1}^{n}(t_{i}):n=80$ |  $\alpha_{1}$ |  $\alpha_{2}$
---|---|---|---|---|---
$0.05$ | $8.6\times 10^{-3}$ | $2.15\times 10^{-3}$ | $5.37\times 10^{-4}$ | $2.00$ | $2.00$
$0.1$ | $7.56\times 10^{-3}$ | $1.89\times 10^{-3}$ | $4.72\times 10^{-4}$ | $2.00$ | $2.00$
$0.15$ | $6.79\times 10^{-3}$ | $1.7\times 10^{-3}$ | $4.24\times 10^{-4}$ | $2.00$ | $2.00$
$0.2$ | $6.22\times 10^{-3}$ | $1.55\times 10^{-3}$ | $3.89\times 10^{-4}$ | $2.00$ | $2.00$
$0.25$ | $5.78\times 10^{-3}$ | $1.44\times 10^{-3}$ | $3.61\times 10^{-4}$ | $2.00$ | $2.00$
$0.3$ | $5.45\times 10^{-3}$ | $1.36\times 10^{-3}$ | $3.4\times 10^{-4}$ | $2.00$ | $2.00$
$0.35$ | $5.19\times 10^{-3}$ | $1.3\times 10^{-3}$ | $3.24\times 10^{-4}$ | $2.00$ | $2.00$
$0.4$ | $4.98\times 10^{-3}$ | $1.25\times 10^{-3}$ | $3.11\times 10^{-4}$ | $2.00$ | $2.00$
$0.45$ | $4.82\times 10^{-3}$ | $1.2\times 10^{-3}$ | $3.01\times 10^{-4}$ | $2.00$ | $2.00$
$0.5$ | $4.68\times 10^{-3}$ | $1.17\times 10^{-3}$ | $2.92\times 10^{-4}$ | $2.00$ | $2.00$
$0.55$ | $4.55\times 10^{-3}$ | $1.14\times 10^{-3}$ | $2.84\times 10^{-4}$ | $2.00$ | $2.00$
$0.6$ | $4.44\times 10^{-3}$ | $1.11\times 10^{-3}$ | $2.77\times 10^{-4}$ | $2.00$ | $2.00$
$0.65$ | $4.33\times 10^{-3}$ | $1.08\times 10^{-3}$ | $2.7\times 10^{-4}$ | $2.00$ | $2.00$
$0.7$ | $4.22\times 10^{-3}$ | $1.05\times 10^{-3}$ | $2.64\times 10^{-4}$ | $2.00$ | $2.00$
$0.75$ | $4.10\times 10^{-3}$ | $1.02\times 10^{-3}$ | $2.56\times 10^{-4}$ | $2.00$ | $2.00$
$0.8$ | $3.98\times 10^{-3}$ | $9.94\times 10^{-4}$ | $2.48\times 10^{-4}$ | $2.00$ | $2.00$
$0.85$ | $3.84\times 10^{-3}$ | $9.6\times 10^{-4}$ | $2.4\times 10^{-4}$ | $2.00$ | $2.00$
$0.9$ | $3.69\times 10^{-3}$ | $9.22\times 10^{-4}$ | $2.3\times 10^{-4}$ | $2.00$ | $2.00$
$0.95$ | $3.52\times 10^{-3}$ | $8.8\times 10^{-4}$ | $2.2\times 10^{-4}$ | $2.00$ | $2.00$

Table 4.2

$t_{i}$ | $E_{2}^{n}(t_{i}):n=20$ |  $E_{2}^{n}(t_{i}):n=40$ |  $\beta$
---|---|---|---
$0.05$ | $2.98\times 10^{-6}$ | $1.87\times 10^{-7}$ | $3.99$
$0.1$ | $2.23\times 10^{-6}$ | $1.41\times 10^{-7}$ | $3.99$
$0.15$ | $1.59\times 10^{-6}$ | $1.01\times 10^{-7}$ | $3.99$
$0.2$ | $1.09\times 10^{-6}$ | $6.94\times 10^{-8}$ | $3.97$
$0.25$ | $7.13\times 10^{-7}$ | $4.58\times 10^{-8}$ | $3.96$
$0.3$ | $4.46\times 10^{-7}$ | $2.91\times 10^{-8}$ | $3.94$
$0.35$ | $2.7\times 10^{-7}$ | $4.58\times 10^{-8}$ | $3.91$
$0.4$ | $1.69\times 10^{-7}$ | $4.58\times 10^{-8}$ | $3.86$
$0.45$ | $1.3\times 10^{-7}$ | $4.58\times 10^{-8}$ | $3.83$
$0.5$ | $1.41\times 10^{-7}$ | $4.58\times 10^{-8}$ | $3.85$
$0.55$ | $1.91\times 10^{-7}$ | $4.58\times 10^{-8}$ | $3.89$
$0.6$ | $2.72\times 10^{-7}$ | $4.58\times 10^{-8}$ | $3.93$
$0.65$ | $3.75\times 10^{-7}$ | $4.58\times 10^{-8}$ | $3.95$
$0.7$ | $4.95\times 10^{-7}$ | $4.58\times 10^{-8}$ | $3.97$
$0.75$ | $6.26\times 10^{-7}$ | $4.58\times 10^{-8}$ | $3.98$
$0.8$ | $7.6\times 10^{-7}$ | $4.58\times 10^{-8}$ | $3.99$
$0.85$ | $8.94\times 10^{-7}$ | $4.58\times 10^{-8}$ | $3.99$
$0.9$ | $1.02\times 10^{-6}$ | $4.58\times 10^{-8}$ | $3.99$
$0.95$ | $1.14\times 10^{-6}$ | $4.58\times 10^{-8}$ | $4$

This verifies the result (3.17).

Acknowledgment

The author Akshay S. Rane would like to thank UGC faculty recharge program,
India for their support.

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