On the rate of convergence of the Legendre spectral collocation method for multi-dimensional nonline
本站小编 Free考研考试/2022-01-02
Nermeen A Elkot1, Mahmoud A Zaky,2, Eid H Doha1, Ibrahem G Ameen31Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt 2Department of Applied Mathematics, National Research Centre, Dokki, Cairo 12622, Egypt 3Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo, Egypt
First author contact:*Author to whom any correspondence should be addressed. Received:2020-07-22Revised:2020-11-12Accepted:2020-11-18Online:2021-01-25
Abstract While the approximate solutions of one-dimensional nonlinear Volterra-Fredholm integral equations with smooth kernels are now well understood, no systematic studies of the numerical solutions of their multi-dimensional counterparts exist. In this paper, we provide an efficient numerical approach for the multi-dimensional nonlinear Volterra-Fredholm integral equations based on the multi-variate Legendre-collocation approach. Spectral collocation methods for multi-dimensional nonlinear integral equations are known to cause major difficulties from a convergence analysis point of view. Consequently, rigorous error estimates are provided in the weighted Sobolev space showing the exponential decay of the numerical errors. The existence and uniqueness of the numerical solution are established. Numerical experiments are provided to support the theoretical convergence analysis. The results indicate that our spectral collocation method is more flexible with better accuracy than the existing ones. Keywords：spectral collocation method;convergence analysis;multi-dimensional integral equations
PDF (692KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Nermeen A Elkot, Mahmoud A Zaky, Eid H Doha, Ibrahem G Ameen. On the rate of convergence of the Legendre spectral collocation method for multi-dimensional nonlinear Volterra-Fredholm integral equations. Communications in Theoretical Physics, 2021, 73(2): 025002- doi:10.1088/1572-9494/abcfb3
1. Introduction
In practical applications, one frequently encounters the multi-dimensional nonlinear Volterra-Fredholm integral equation of the form$\begin{eqnarray}\begin{array}{rcl}\phi ({t}_{1},\ldots ,{t}_{d}) & = & {\displaystyle \int }_{0}^{1}\cdots {\displaystyle \int }_{0}^{1}{k}_{1}({t}_{1},{r}_{1},\ldots ,{t}_{d},{r}_{d})\\ & & \times f\left(\phi ({r}_{1},\ldots ,{r}_{d})\right){\rm{d}}{r}_{d}\cdots {\rm{d}}{r}_{1}\\ & & +{\displaystyle \int }_{0}^{{t}_{1}}\cdots {\displaystyle \int }_{0}^{{t}_{d}}{k}_{2}({t}_{1},{s}_{1},\ldots ,{t}_{d},{s}_{d})g\\ & & \times \left(\phi ({s}_{1},\ldots ,{s}_{d})\right){\rm{d}}{s}_{d}\cdots {\rm{d}}{s}_{1}\\ & & +y({t}_{1},\ldots ,{t}_{d}),\end{array}\end{eqnarray}$where φ(t1, t2,...,td) is the unknown function, $f,g:D\to {\mathbb{R}}$ where $D:= \{\left({s}_{1},\ldots {s}_{d}\right):0\leqslant {s}_{i}\leqslant 1,i=1,\ldots ,d\}$ satisfy the Lipschitz condition with respect to φ; y(t1, t2,…,td), k1(t1, r1,…,td, rd) and k2(t1, s1,…,td, sd) are given continuous functions. The multi-dimensional Volterra-Fredholm nonlinear integral equations arise in many physics, chemistry, biology and engineering applications, and they provide a vital tool for modeling many problems. Particular cases of such nonlinear integral equations arise in the mathematical design of the temporal-spatio development of an epidemic [1-4]. They also appear in the theory of porous filtering, antenna problems in electromagnetic theory, fracture mechanics, aerodynamics, in the quantum effects of electromagnetic fields in the black body whose interior is filled by Kerr nonlinear crystal.
During the last decade, many numerical schemes have been developed for the one- and two-dimensional version of the nonlinear Volterra-Fredholm integral equations [5-16]. However, the studies on analysis and derivation of numerical schemes for the multi-dimensional nonlinear integral equations are still limited. Mirzaee and Hadadiyan [17] solved the second kind three-dimensional linear Volterra-Fredholm integral equations using the modified block-pulse functions. Wei et al [18] studied the convergence analysis of the Chebyshev collocation method for approximating the solution of the second kind multi-dimensional nonlinear Volterra integral equation with a weakly singular kernel. Pan et al [19] developed a quadrature method based on multi-variate Bernstein polynomials for approximating the solution of multi-dimensional Volterra integral equations. Sadri et al [20] constructed an operational approach for linear and nonlinear three-dimensional Fredholm, Volterra, and mixed Volterra-Fredholm integral equations. Liu et al [21] proposed two interpolation collocation methods for solving the second kind nonlinear multi-dimensional Fredholm integral equations utilizing the modified weighted Lagrange and the rational basis functions. Assari et al [22] presented a discrete radial basis functions collocation scheme based on scattered points for solving the second kind two-dimensional nonlinear Fredholm integral equations on a non-rectangular domain. Wei et al [23] provided a spectral collocation scheme for multi-dimensional linear Volterra integral equation with a smooth kernel. Wei et al [24, 25] studied the convergence analysis of the Jacobi spectral collocation schemes for the numerical solution of multi-dimensional nonlinear Volterra integral equations. Doha et al [26] proposed Jacobi-Gauss-collocation scheme for approximating the solution of Fredholm, Volterra, and systems of Volterra-Fredholm integral equations with initial and nonlocal boundary conditions. Zaky and Hendy [27] developed and analyzed a spectral collocation scheme for a class of the second kind nonlinear Fredholm integral equations in multi-dimensions. Zaky and Ameen [28]developed Jacobi spectral collocation scheme for solving the second kind multi-dimensional integral equations with non-smooth solutions and weakly singular kernels.
Solving multi-dimensional Volterra-Fredholm integral equations is much more challenging than low-dimensional problems due to dimension effect, especially those are nonlinear. The main purpose of this paper is to propose and analyze a Legendre spectral collocation method for multi-dimensional nonlinear Volterra-Fredholm integral equations. Compared to Galerkin spectral methods, spectral collocation methods are more flexible to deal with complicated problems, especially those are nonlinear. We get the discrete scheme by using multi-variate Gauss quadrature formula for the integral term. We provide theoretical estimates of exponential decay for the errors of the approximate solutions. Moreover, we establish the existence and uniqueness of the numerical solution.
The paper is organized as follows. In the following section, we investigate the existence and uniqueness of the solution to equation (1.1). In section 3, we state some preliminaries and notation. The spectral collocation discretization of (1.1) is given in section 4. In section 5, the rate of convergence of the proposed scheme is derived. In section 6, we investigate the existence of the numerical solution. In section 7, we investigate the uniqueness of the numerical solution. The numerical experiments are performed in section 7 illustrating the performance of our scheme. We conclude the paper with some discussions in section 8.
2. Existence and uniqueness of the solution
In this section, we prove the existence and uniqueness of solution to (1.1). The L∞- norm is defined as$\begin{eqnarray*}\begin{array}{l}\parallel \phi {\parallel }_{\infty }=\ \mathrm{ess}\ \ \sup \ \{| \phi ({t}_{1},\ldots ,{t}_{d})| ,\\ ({t}_{1},\ldots ,{t}_{d})\in {I}^{d}:= [0,1]\times [0,1]...[0,1]\}.\end{array}\end{eqnarray*}$For sake of simplicity, we rewrite equation (1.1) as φ = Tφ, where$\begin{eqnarray*}\begin{array}{l}(T\phi )({t}_{1},\ldots ,{t}_{d})=y({t}_{1},\ldots ,{t}_{d})\\ +\,{\displaystyle \int }_{0}^{1}\cdots {\displaystyle \int }_{0}^{1}{k}_{1}({t}_{1},{r}_{1},\ldots ,{t}_{d},{r}_{d})f\left(\phi ({r}_{1},\ldots ,{r}_{d})\right){\rm{d}}{r}_{d}\cdots {\rm{d}}{r}_{1}\\ +\,{\displaystyle \int }_{0}^{{t}_{1}}\cdots {\displaystyle \int }_{0}^{{t}_{d}}{k}_{2}({t}_{1},{s}_{1},\ldots ,{t}_{d},{s}_{d})g\left(\phi ({s}_{1},\ldots ,{s}_{d})\right){\rm{d}}{s}_{d}\cdots {\rm{d}}{s}_{1},\end{array}\end{eqnarray*}$and consider the following assumptions:(T1) y ∈ L∞(Id). (T2) ${k}_{i}\in {L}^{\infty }({I}^{d}\times {I}^{d}),{M}_{i}={\parallel {k}_{i}\parallel }_{\infty },i=1,2.$ (T3)${d}_{1}={\sup }_{({t}_{1},\ldots ,{t}_{d})\in {I}^{d}}| f(0)| \lt \infty $, ${d}_{2}={\sup }_{({t}_{1},\ldots ,{t}_{d})\in {I}^{d}}| g(0)| \lt \infty ,$ and the nonlinear functions f, g satisfy the Lipschitz conditions $\begin{eqnarray*}\begin{array}{l}| f({\phi }_{1})-f({\phi }_{2})| \leqslant {L}_{1}| {\phi }_{1}-{\phi }_{2}| ,\quad | g({\phi }_{1})-g({\phi }_{2})| \\ \quad \leqslant \,{L}_{2}| {\phi }_{1}-{\phi }_{2}| ,\quad ({t}_{1},\ldots ,{t}_{d})\in {I}^{d},{\phi }_{1},{\phi }_{2}\in {\mathbb{R}},\end{array}\end{eqnarray*}$where L1 and L2 are non-negative real constants. (T4) ${\sum }_{i=1}^{2}{M}_{i}\,{L}_{i}\lt 1.$
If the assumptions (T1)-(T4) are satisfied, then equation (1.1) admits a unique solution in ${L}^{\infty }({I}^{d})$.
Let $\phi \in {L}^{\infty }({I}^{d})$. Then, we deduce from the assumptions $({T}_{1})-({T}_{3})$ that$\begin{eqnarray*}\begin{array}{l}| (T\phi )({t}_{1},\ldots ,{t}_{d})| \leqslant | y({t}_{1},\ldots ,{t}_{d})| \\ +\,{\displaystyle \int }_{0}^{1}\cdots {\displaystyle \int }_{0}^{1}| {k}_{1}({t}_{1},{r}_{1},\ldots ,{t}_{d},{r}_{d})| | f\left(\phi ({r}_{1},\ldots ,{r}_{d})\right)| \,{\rm{d}}{r}_{d}\cdots {\rm{d}}{r}_{1}\\ +\,{\displaystyle \int }_{0}^{{t}_{1}}\cdots {\displaystyle \int }_{0}^{{t}_{d}}| {k}_{2}({t}_{1},{s}_{1},\ldots ,{t}_{d},{s}_{d})| | g\left(\phi ({s}_{1},\ldots ,{s}_{d})\right)| \,{\rm{d}}{s}_{d}\cdots {\rm{d}}{s}_{1}\\ \leqslant \,| y({t}_{1},\ldots ,{t}_{d})| +{\displaystyle \int }_{0}^{1}\cdots {\displaystyle \int }_{0}^{1}| {k}_{1}({t}_{1},{r}_{1},\ldots ,{t}_{d},{r}_{d})\\ \times \,| | f\left(\phi ({r}_{1},\ldots ,{r}_{d})\right)-f\left(0)\right)| | \,{\rm{d}}{r}_{d}\cdots {\rm{d}}{r}_{1}\\ +\,{\displaystyle \int }_{0}^{1}\cdots {\displaystyle \int }_{0}^{1}| {k}_{1}({t}_{1},{r}_{1},\ldots ,{t}_{d},{r}_{d})| | f\left(0)\right)| \,{\rm{d}}{r}_{d}\cdots {\rm{d}}{r}_{1}\\ +\,{\displaystyle \int }_{0}^{{t}_{1}}\cdots {\displaystyle \int }_{0}^{{t}_{d}}| {k}_{2}({t}_{1},{s}_{1},\ldots ,{t}_{d},{s}_{d})| | g\left(\phi ({s}_{1},\ldots ,{s}_{d})\right)\\ -\,g\left(0)\right)| | \,{\rm{d}}{s}_{d}\cdots {\rm{d}}{s}_{1}\\ +\,{\displaystyle \int }_{0}^{{t}_{1}}\cdots {\displaystyle \int }_{0}^{{t}_{d}}| {k}_{2}({t}_{1},{s}_{1},\ldots ,{t}_{d},{s}_{d})| | g\left(0)\right)| \,{\rm{d}}{s}_{d}\cdots {\rm{d}}{s}_{1}\\ \leqslant \,{\parallel y\parallel }_{\infty }+\displaystyle \sum _{n=1}^{2}{M}_{n}\left({L}_{n}{\parallel \phi \parallel }_{\infty }+{d}_{n}\right)\lt \infty ,\quad ({t}_{1},\ldots ,{t}_{d})\in {I}^{d}.\end{array}\end{eqnarray*}$Hence, ${\parallel T\phi \parallel }_{\infty }\lt \infty $ and the operator T map ${L}^{\infty }({I}^{d})$ to ${L}^{\infty }({I}^{d})$. Consequently, it follows directly from the assumptions $(T2)-(T4)$ that the operator T is a contraction$\begin{eqnarray*}\parallel T\phi -{Tv}{\parallel }_{\infty }\leqslant \sum _{n=1}^{2}{M}_{n}{L}_{n}\parallel \phi -v{\parallel }_{\infty }.\end{eqnarray*}$Thus, the proof follows directly from the Banach fixed point theorem. □
3. Multi-variate Legendre-Gauss interpolation
In this section, we provide some properties of the Legendre polynomials.Let ${{ \mathcal P }}_{N}$ be the space of polynomials of degree at most N in Ω, where Ω $:= $ ( − 1, 1) and Ωd $:= $ ( − 1, 1)d. Let ωμ,ν(y) = (1 − y)μ(1 + y)ν be a non-negative weight function defined in Ω and corresponding to the Jacobi parameters μ, ν > − 1. Let ${\mathbb{R}}$ be the set of all real numbers, ${\mathbb{N}}$ be the set of all non-negative integers, and ${{\mathbb{N}}}_{0}={\mathbb{N}}\cup 0$. The lowercase boldface letters denotes d-dimensional vectors and multi-indexes, e.g. ${\boldsymbol{j}}=({j}_{1},\ldots ,{j}_{d})\in {{\mathbb{N}}}_{0}^{d}$ and ${\boldsymbol{b}}=({b}_{1},\ldots ,{b}_{d})\in {{\mathbb{R}}}^{d}$. We denote by ek = (0,…,1,…,0) the kth unit vector in ${{\mathbb{R}}}^{d}$ and ${\bf{1}}=(1,1,\ldots ,1)\in {{\mathbb{N}}}^{d}$. For a constant $c\in {\mathbb{R}}$, we introduce the following operations: $\begin{eqnarray*}\begin{array}{l}{\boldsymbol{a}}+{\boldsymbol{b}}=({a}_{1}+{b}_{1},\ldots ,{a}_{d}+{b}_{d}),\\ {\boldsymbol{a}}+c:= {\boldsymbol{a}}+c{\bf{1}}=({a}_{1}+c,\ldots ,{a}_{d}+c),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{\boldsymbol{a}}\geqslant {\boldsymbol{b}}\iff {\forall }_{1\leqslant i\leqslant d}\ {a}_{i}\geqslant {b}_{i};\\ {\boldsymbol{a}}\geqslant c\iff {\boldsymbol{a}}\geqslant c{\bf{1}}\iff {\forall }_{1\leqslant i\leqslant d}\ {a}_{i}\geqslant c.\end{array}\end{eqnarray*}$ We define $\begin{eqnarray*}\begin{array}{l}{\left|{\boldsymbol{q}}\right|}_{1}=\displaystyle \sum _{i=1}^{d}{q}_{i},{\left|{\boldsymbol{q}}\right|}_{\infty }=\mathop{\max }\limits_{1\leqslant i\leqslant d}{q}_{i},\prod {\boldsymbol{x}}{{\boldsymbol{y}}}^{{\boldsymbol{z}}}=\prod _{i=1}^{d}{x}_{i}{y}_{i}^{{z}_{i}},\\ {\displaystyle \int }_{{\boldsymbol{p}}}^{{\boldsymbol{q}}}g({\boldsymbol{x}}){\rm{d}}{\boldsymbol{x}}={\displaystyle \int }_{{p}_{1}}^{{q}_{1}}\cdots {\displaystyle \int }_{{p}_{d}}^{{q}_{d}}g({x}_{1},\cdots ,{x}_{d})\ {\rm{d}}{x}_{d}\cdots {\rm{d}}{x}_{1}.\end{array}\end{eqnarray*}$ Given a multi-variate function φ(z), we define the qth partial (mixed) derivative as $\begin{eqnarray*}{{\boldsymbol{\partial }}}_{{\boldsymbol{z}}}^{{\boldsymbol{q}}}\phi =\displaystyle \frac{{\partial }^{{\left|{\boldsymbol{q}}\right|}_{1}}\phi }{{\partial }_{{z}_{1}}^{{q}_{1}}\cdots {\partial }_{{z}_{d}}^{{q}_{d}}}={\partial }_{{z}_{1}}^{{q}_{1}}\cdots {\partial }_{{z}_{d}}^{{q}_{d}}\phi .\end{eqnarray*}$ The set of Legendre polynomials $\left\{{{L}}_{n}(y)\right\}$ forms a complete orthogonal system in L2(Ω), i.e.$\begin{eqnarray*}{\int }_{{\rm{\Omega }}}{{L}}_{n}(y){{L}}_{m}(y){\rm{d}}y={\alpha }_{n}{\delta }_{n,m},\end{eqnarray*}$where δm,n is the Kronecker symbol and$\begin{eqnarray*}{\alpha }_{n}=\displaystyle \frac{2}{2n+1}.\end{eqnarray*}$The d-dimensional Legendre polynomial is given by$\begin{eqnarray*}{{P}}_{{\boldsymbol{n}}}({\boldsymbol{y}})=\prod _{i=1}^{d}{{L}}_{{n}_{i}}({y}_{i}),\quad {\boldsymbol{y}}\in {{\rm{\Omega }}}^{d}.\end{eqnarray*}$Hence,$\begin{eqnarray*}{\int }_{{{\rm{\Omega }}}^{d}}{{P}}_{{\boldsymbol{n}}}({\boldsymbol{y}}){{P}}_{{\boldsymbol{m}}}({\boldsymbol{y}})={{\boldsymbol{\alpha }}}_{{\boldsymbol{n}}}{{\boldsymbol{\delta }}}_{{\boldsymbol{n}},{\boldsymbol{m}}}=\displaystyle \prod _{i=1}^{d}{\alpha }_{{n}_{i}}{\delta }_{{n}_{i},{m}_{i}},\quad {\boldsymbol{n}},{\boldsymbol{m}}\geqslant 0.\end{eqnarray*}$Let ${\left\{{\bar{\omega }}_{{j}_{k}},{\xi }_{{j}_{k}}\right\}}_{{j}_{k}=0}^{N}$ be the Gauss-Legendre weights and nodes in Ω, and let ${I}_{{z}_{k},N}$ be the corresponding interpolation operator in zk direction. Then, the d-dimensional weights and nodes ${\left\{{\bar{{\boldsymbol{\omega }}}}_{{\boldsymbol{r}}},{{\boldsymbol{\xi }}}_{{\boldsymbol{r}}}\right\}}_{{\left|{\boldsymbol{r}}\right|}_{\infty }\leqslant N}$ in Ωd are given by$\begin{eqnarray*}{\bar{{\boldsymbol{\omega }}}}_{{\boldsymbol{r}}}=\left({\bar{\omega }}_{{r}_{1}},\ldots ,{\bar{\omega }}_{{r}_{d}}\right),\qquad {{\boldsymbol{\xi }}}_{{\boldsymbol{r}}}=\left({\xi }_{{r}_{1}},\ldots ,{\xi }_{{r}_{d}}\right).\end{eqnarray*}$The multi-dimensional Gauss-Legendre quadrature formula is given by$\begin{eqnarray}{\int }_{{{\rm{\Omega }}}^{d}}f({\boldsymbol{z}}){\rm{d}}{\boldsymbol{z}}=\displaystyle \sum _{{\left|{\boldsymbol{r}}\right|}_{\infty }\leqslant N}f\left({{\boldsymbol{\xi }}}_{{\boldsymbol{r}}}\right){\bar{{\boldsymbol{\omega }}}}_{{\boldsymbol{r}}},\ \forall f({\boldsymbol{z}})\in {{ \mathcal P }}_{2N+1}^{d}.\end{eqnarray}$Hence$\begin{eqnarray}\begin{array}{l}\displaystyle \sum _{{\left|{\boldsymbol{k}}\right|}_{\infty }\leqslant N}{{P}}_{{\boldsymbol{n}}}\left({{\boldsymbol{\xi }}}_{{\boldsymbol{k}}}\right){{P}}_{{\boldsymbol{m}}}\left({{\boldsymbol{\xi }}}_{{\boldsymbol{k}}}\right){\bar{{\boldsymbol{\omega }}}}_{{\boldsymbol{k}}}={{\boldsymbol{\alpha }}}_{{\boldsymbol{n}}}{{\boldsymbol{\delta }}}_{{\boldsymbol{n}},{\boldsymbol{m}}},\\ \qquad \forall \ 0\leqslant {\boldsymbol{m}}+{\boldsymbol{n}}\leqslant 1+2N.\end{array}\end{eqnarray}$For any φ ∈ C(Ωd), then the Gauss-Legendre interpolation operator ${{\boldsymbol{I}}}_{{\boldsymbol{\xi }},N}:C({{\rm{\Omega }}}^{d})\longrightarrow {{ \mathcal P }}_{N}^{d}$ is computed uniquely by$\begin{eqnarray*}\left({{\boldsymbol{I}}}_{{\boldsymbol{\xi }},N}\phi \right)({{\boldsymbol{\xi }}}_{{\boldsymbol{r}}})=\phi ({{\boldsymbol{\xi }}}_{{\boldsymbol{r}}})\quad \forall \ {\boldsymbol{r}}\in {{\mathbb{N}}}^{d},{\left|{\boldsymbol{r}}\right|}_{\infty }\leqslant N.\end{eqnarray*}$For each direction, we assume that the number of Gauss quadrature points is N + 1 points. We define$\begin{eqnarray}{{\boldsymbol{I}}}_{{\boldsymbol{y}},N}={I}_{{y}_{1},N}\circ \ldots \circ {I}_{{y}_{d},N}.\end{eqnarray}$Since ${{\boldsymbol{I}}}_{{\boldsymbol{y}},N}\phi \in {{ \mathcal P }}_{N}^{d}$, then we obtain that$\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{I}}}_{{\boldsymbol{y}},N}\phi \left({\boldsymbol{y}}\right) & = & \displaystyle \sum _{{\left|{\boldsymbol{r}}\right|}_{\infty }\leqslant N}{\hat{\phi }}_{{\boldsymbol{r}}}{{P}}_{{\boldsymbol{r}}}\left({\boldsymbol{y}}\right),\mathrm{where}\ {\hat{\phi }}_{{\boldsymbol{r}}}\\ & = & \displaystyle \frac{1}{{{\boldsymbol{\alpha }}}_{{\boldsymbol{r}}}}\displaystyle \sum _{{\left|{\boldsymbol{j}}\right|}_{\infty }\leqslant N}\phi \left({{\boldsymbol{\xi }}}_{{\boldsymbol{j}}}\right){{P}}_{{\boldsymbol{r}}}({\xi }_{{\boldsymbol{j}}}){\bar{{\boldsymbol{\omega }}}}_{{\boldsymbol{j}}}.\end{array}\end{eqnarray}$We define the space ${\tilde{B}}^{q}\left({{\rm{\Omega }}}^{d}\right)$ for q ≥ d equipped with the semi-norm and norm:$\begin{eqnarray}\begin{array}{rcl}{\left|\mu \right|}_{{\tilde{B}}^{q}\left({{\rm{\Omega }}}^{d}\right)} & = & {\left(\displaystyle \sum _{j=1}^{d}\displaystyle \sum _{{\boldsymbol{q}}\in {\mho }_{j}}{\parallel {{\boldsymbol{\partial }}}_{{\boldsymbol{x}}}^{{\boldsymbol{q}}}\mu \parallel }_{{{\boldsymbol{\omega }}}^{{q}_{j}{{\boldsymbol{e}}}_{j},{q}_{j}{{\boldsymbol{e}}}_{j}}}^{2}\right)}^{\tfrac{1}{2}},\\ {\parallel \mu \parallel }_{{\tilde{B}}^{q}\left({{\rm{\Omega }}}^{d}\right)} & = & {\left({\parallel \mu \parallel }^{2}+{\left|\mu \right|}_{{\tilde{B}}^{q}\left({{\rm{\Omega }}}^{d}\right)}^{2}\right)}^{\tfrac{1}{2}},\end{array}\end{eqnarray}$respectively. For 1 ≤ j ≤ d, the index sets ℧j are defined as$\begin{eqnarray*}{\mho }_{j}=\left\{{\boldsymbol{q}}\in {{\mathbb{N}}}_{0}^{d}:\ d\leqslant {q}_{j}\leqslant q;{q}_{i}\in \left\{0,1\right\},i\ne j;\displaystyle \sum _{k=1}^{d}{q}_{k}=q\right\}.\end{eqnarray*}$
([29]).For $\mu \in {\tilde{B}}^{q}\left({{\rm{\Omega }}}^{d}\right)$ with $d\leqslant q\leqslant M+1$,$\begin{eqnarray}\parallel {{\boldsymbol{I}}}_{{\boldsymbol{x}},M}\mu -\mu \parallel \leqslant c\sqrt{\displaystyle \frac{(M-q+1)!}{M!}}{\left(M+q\right)}^{-(q+1)/2}{\left|\mu \right|}_{{\tilde{B}}^{q}\left({{\rm{\Omega }}}^{d}\right)},\end{eqnarray}$where c is a non-negative constant independent of $q,M$ and μ.
4. Legendre collocation discretization
In this section, we describe the procedure of solving problem (1.1) in the domain Ωd $:= $ ( − 1, 1)d. Using the change of variables:$\begin{eqnarray*}{t}_{i}\to \,{\psi }_{i}({x}_{i})=\displaystyle \frac{(1+{x}_{i})}{2},{x}_{i}\in [-1,1],\end{eqnarray*}$and employing the linear transformations:$\begin{eqnarray*}{r}_{i}={\psi }_{i}({\sigma }_{i}),{\sigma }_{i}\in [-1,1],\quad {s}_{i}={\psi }_{i}({\tau }_{i}),{\tau }_{i}\in [-1,{x}_{i}],\end{eqnarray*}$then equation (1.1) can be expressed as follows:$\begin{eqnarray}\begin{array}{l}\phi \left({\psi }_{1}({x}_{1}),\ldots ,{\psi }_{d}({x}_{d})\right)=y\left({\psi }_{1}({x}_{1}),\ldots ,{\psi }_{d}({x}_{d})\right)\\ +\,\displaystyle \frac{1}{{2}^{d}}{\displaystyle \int }_{-1}^{1}\cdots {\displaystyle \int }_{-1}^{1}{k}_{1}\left({\psi }_{1}({x}_{1}),{\psi }_{1}({\sigma }_{1}),\ldots ,{\psi }_{d}({x}_{d}),{\psi }_{d}({\sigma }_{d})\right)\\ \times \,f\left(\phi \left({\psi }_{1}({\sigma }_{1}),\ldots ,{\psi }_{d}({\sigma }_{d})\right)\right){\rm{d}}{\sigma }_{d}\cdots {\rm{d}}{\sigma }_{1}\\ +\,\displaystyle \frac{1}{{2}^{d}}{\displaystyle \int }_{-1}^{{x}_{1}}\cdots {\displaystyle \int }_{-1}^{{x}_{d}}{k}_{2}\left({\psi }_{1}({x}_{1}),{\psi }_{1}({\tau }_{1}),\ldots ,{\psi }_{d}({\tau }_{d}),{\psi }_{d}({x}_{d})\right)\\ \times \,g\left(\phi \left({\psi }_{1}({\tau }_{1}),\ldots ,{\psi }_{d}({\tau }_{d})\right)\right){\rm{d}}{\tau }_{d}\cdots {\rm{d}}{\tau }_{1},\end{array}\end{eqnarray}$which may be written in the compact form$\begin{eqnarray}\begin{array}{rcl}{\rm{\Phi }}({\boldsymbol{x}}) & = & Y({\boldsymbol{x}})+\displaystyle \frac{1}{{2}^{d}}\,{\int }_{-{\bf{1}}}^{{\bf{1}}}{K}_{1}({\boldsymbol{x}},{\boldsymbol{\sigma }})F({\rm{\Phi }}({\boldsymbol{\sigma }})){\rm{d}}{\boldsymbol{\sigma }}\\ & & +\displaystyle \frac{1}{{2}^{d}}\,{\int }_{-{\bf{1}}}^{{\boldsymbol{x}}}{K}_{2}({\boldsymbol{x}},{\boldsymbol{\tau }})G({\rm{\Phi }}({\boldsymbol{\tau }})){\rm{d}}{\boldsymbol{\tau }},\end{array}\end{eqnarray}$where$\begin{eqnarray*}\begin{array}{rcl}{\rm{\Phi }}({\boldsymbol{x}}) & = & {\rm{\Phi }}({x}_{1},\ldots ,{x}_{d})=\phi ({\psi }_{1}({x}_{1}),\ldots {\psi }_{d}({x}_{d})),\\ Y({\boldsymbol{x}}) & = & Y({x}_{1},\ldots ,{x}_{d})=y({\psi }_{1}({x}_{1}),\ldots ,{\psi }_{d}({x}_{d})),\\ {K}_{1}({\boldsymbol{x}},{\boldsymbol{\sigma }}) & = & {K}_{1}({x}_{1},{\sigma }_{1},\ldots ,{x}_{d},{\sigma }_{d})\\ & = & {k}_{1}({\psi }_{1}({x}_{1}),{\psi }_{1}({\sigma }_{1}),\ldots ,{\psi }_{d}({x}_{d}),{\psi }_{d}({\sigma }_{d})),\\ {K}_{2}({\boldsymbol{x}},{\boldsymbol{\tau }}) & = & {K}_{2}({x}_{1},{\tau }_{1},\ldots ,{x}_{d},{\tau }_{d})\\ & = & {k}_{2}({\psi }_{1}({x}_{1}),{\psi }_{1}({\tau }_{1}),\ldots ,{\psi }_{d}({x}_{d}),{\psi }_{d}({\tau }_{d})),\\ F({\rm{\Phi }}({\boldsymbol{\sigma }})) & = & f(\phi ({\psi }_{1}({\sigma }_{1}),\ldots ,{\psi }_{d}({\sigma }_{d})),\\ G({\rm{\Phi }}({\boldsymbol{\tau }})) & = & g(\phi ({\psi }_{1}({\tau }_{1}),\ldots ,{\psi }_{d}({\tau }_{d}))).\end{array}\end{eqnarray*}$For computing the second integral term in (4.2), we use the linear transformation:$\begin{eqnarray*}\begin{array}{l}{\tau }_{i}({x}_{i},{\beta }_{i})=\displaystyle \frac{{x}_{i}+1}{2}\,{\beta }_{i}+\displaystyle \frac{{x}_{i}-1}{2},\\ {\beta }_{i}\in [-1,1],i=1,\ldots ,d,\end{array}\end{eqnarray*}$to transform the integral interval [ − 1, xi] to [ − 1, 1]. Therefore, equation (4.2) becomes$\begin{eqnarray}\begin{array}{rcl}{\rm{\Phi }}({\boldsymbol{x}}) & = & Y({\boldsymbol{x}})+\displaystyle \frac{1}{{2}^{d}}\,{\displaystyle \int }_{-{\bf{1}}}^{{\bf{1}}}{K}_{1}({\boldsymbol{x}},{\boldsymbol{\sigma }})F({\rm{\Phi }}({\boldsymbol{\sigma }})){\rm{d}}{\boldsymbol{\sigma }}\\ & & +\displaystyle \frac{\prod (1+{\boldsymbol{x}})}{{4}^{d}}\,{\displaystyle \int }_{-{\bf{1}}}^{{\bf{1}}}{K}_{2}({\boldsymbol{x}},{{\boldsymbol{\tau }}}_{{\boldsymbol{x}},{\boldsymbol{\beta }}})G({\rm{\Phi }}({{\boldsymbol{\tau }}}_{{\boldsymbol{x}},{\boldsymbol{\beta }}})){\rm{d}}{\boldsymbol{\beta }},\end{array}\end{eqnarray}$where$\begin{eqnarray*}{{\boldsymbol{\tau }}}_{{\boldsymbol{x}},{\boldsymbol{\beta }}}=\left({\tau }_{1}({x}_{1},{\beta }_{1}),\ldots ,{\tau }_{d}({x}_{d},{\beta }_{d})\right).\end{eqnarray*}$Setting$\begin{eqnarray}{{\rm{\Phi }}}_{N}({\bf{x}})=\displaystyle \sum _{| {\boldsymbol{i}}{| }_{\infty }\leqslant N}{\hat{\phi }}_{{\boldsymbol{i}}}\,{{P}}_{{\boldsymbol{i}}}({\boldsymbol{x}})\in {{ \mathcal P }}_{N}^{d},\end{eqnarray}$and inserting (4.4) into (4.3) lead to the following scheme$\begin{eqnarray}\begin{array}{l}{{\rm{\Phi }}}_{N}({\boldsymbol{x}})={{\boldsymbol{I}}}_{{\boldsymbol{x}},N}\,Y({\boldsymbol{x}})\\ +\,\displaystyle \frac{1}{{2}^{d}}\,{\displaystyle \int }_{-{\bf{1}}}^{{\bf{1}}}\,{{\boldsymbol{I}}}_{{\boldsymbol{x}},N}{{\boldsymbol{I}}}_{{\boldsymbol{\sigma }},N}\left[{K}_{1}({\boldsymbol{x}},{\boldsymbol{\sigma }})F({{\rm{\Phi }}}_{N}({\boldsymbol{\sigma }}))\right]\,{\rm{d}}{\boldsymbol{\sigma }}\\ +\,\displaystyle \frac{1}{{4}^{d}}\,{{\boldsymbol{I}}}_{{\boldsymbol{x}},N}{\displaystyle \int }_{-{\bf{1}}}^{{\bf{1}}}{{\boldsymbol{I}}}_{{\boldsymbol{\beta }},N}\left[\prod (1+{\boldsymbol{x}}){K}_{2}({\boldsymbol{x}},{{\boldsymbol{\tau }}}_{{\boldsymbol{x}},{\boldsymbol{\beta }}})G({{\rm{\Phi }}}_{N}({{\boldsymbol{\tau }}}_{{\boldsymbol{x}},{\boldsymbol{\beta }}}))\right]\,{\rm{d}}{\boldsymbol{\beta }}.\end{array}\end{eqnarray}$The implementation of the spectral collocation scheme is performed as follows:
Let Φ and ΦN be the solutions of (4.3) and (4.5), respectively. Let ${\rm{\Phi }}\in {\tilde{B}}^{s}({{\rm{\Omega }}}^{d})$, $d\leqslant s\leqslant N+1,$ and the nonlinear functions F and G satisfy the Lipschitz conditions$\begin{eqnarray*}\begin{array}{l}\left|F({{\rm{\Phi }}}_{1})-F({{\rm{\Phi }}}_{2})\right|\leqslant {L}_{1}\left|{{\rm{\Phi }}}_{1}-{{\rm{\Phi }}}_{2}\right|,\\ \left|G({{\rm{\Phi }}}_{1})-G({{\rm{\Phi }}}_{2})\right|\leqslant {L}_{2}\left|{{\rm{\Phi }}}_{1}-{{\rm{\Phi }}}_{2}\right|,\end{array}\end{eqnarray*}$where L1 and L2 are real non-negative constants satisfy ${M}_{1}{L}_{1}+{M}_{2}{L}_{2}\lt 1$ with ${M}_{i}=\parallel {K}_{i}{\parallel }_{\infty },i=1,2.$ Then, we have$\begin{eqnarray*}\begin{array}{rcl}\parallel {E}_{N}\parallel & \leqslant & c\sqrt{\displaystyle \frac{(1+N-s)!}{N!}}{\left(s+N\right)}^{\tfrac{-(1+s)}{2}}\\ & & \times \left(| {\rm{\Phi }}{| }_{{\tilde{B}}^{s}({{\rm{\Omega }}}^{d})}+| F(.,{\rm{\Phi }}(.)){| }_{{\tilde{B}}^{s}({{\rm{\Omega }}}^{d})}+| G(.,{\rm{\Phi }}(.)){| }_{{\tilde{B}}^{s}({{\rm{\Omega }}}^{d})}\right).\end{array}\end{eqnarray*}$
The exact solution of this equation is $\phi (\zeta )=\cos (\zeta ).$ Using the variables $\zeta =\tfrac{\pi }{2}t,s=\tfrac{\pi }{2}r$ and $y(t)=\phi (\tfrac{\pi }{2}t)$, we obtain the following equivalent integral equation$\begin{eqnarray}\begin{array}{rcl}y(t) & = & \displaystyle \frac{1}{36}\left(35\cos \left(\displaystyle \frac{\pi }{2}t\right)-1\right)+\displaystyle \frac{\pi }{4}{\displaystyle \int }_{0}^{t}\sin \left(\displaystyle \frac{\pi }{2}r\right){y}^{2}(r){\rm{d}}r\\ & & +\displaystyle \frac{\pi }{72}{\displaystyle \int }_{0}^{1}\left({\cos }^{3}\left(\displaystyle \frac{\pi }{2}t\right)\right.\\ & & \left.+\cos \left(\displaystyle \frac{\pi }{2}t\right)\right)y\left(\displaystyle \frac{\pi }{2}r\right){\rm{d}}r,t\in [0,1].\end{array}\end{eqnarray}$
In table 1, for various values of N, we report the numerical results of the presented method and the numerical schemes based on shifted piecewise cosine basis [6], Haar wavelets [30] and the Picard iteration method [31] to solve this example. As it is shown in this table, the present numerical results are more accurate than those reported in [6, 30, 31] to solve example 1.
Table 1. Table 1.The L2- errors for example 1 versus N.
Consider the following two-dimensional Volterra integral equation [32]:$\begin{eqnarray}\begin{array}{l}\phi (x,y)=f(x,y)+{\int }_{0}^{y}{\int }_{0}^{x}\displaystyle \frac{x+t-y-z}{4}\phi (t,z){\rm{d}}t{\rm{d}}z,\\ (x,y)\in {[0,1]}^{2},\end{array}\end{eqnarray}$where$\begin{eqnarray}\begin{array}{l}f(x,y)\\ =\,\displaystyle \frac{{\rm{e}}\,x\left(x(3-5x\gamma +3y\gamma )+{{\rm{e}}}^{-y\gamma }(\gamma x(-6y+5x)-3x+24{\gamma }^{2})\right)}{24\gamma },\\ \quad \gamma \in {\mathbb{R}}-\{0\}.\end{array}\end{eqnarray}$
The exact solution is given as$\begin{eqnarray}\phi (x,y)=\gamma x\,{{\rm{e}}}^{1-\gamma y}.\end{eqnarray}$The Lipschitz conditions are satisfied with M1 = 0, M2 = 0.5, d1 = d2 = 0, L1 = 0, L2 = 1. For different values for γ, the exact solution φ may have large total variation$\begin{eqnarray}{TV}(\phi ,{\rm{\Omega }})={\int }_{{\rm{\Omega }}}\parallel {\rm{\nabla }}\phi (x){\parallel }_{2}{\rm{d}}x.\end{eqnarray}$The L∞-errors for this example are given in table 2. These results indicate that, when γ decreases, TV(φ) increases and the method converges slower. The absolute error eN, for N = 9 and γ = 1, − 1, − 2, is displayed in figures 1-3.
The extension of existing numerical methods for one-dimensional integral equations to their corresponding high-dimensional integral equations is not trivial. We presented an efficient spectral collocation scheme for the numerical solution of the multi-dimensional nonlinear Volterra-Fredholm integral equations based on multi-variate Legendre-collocation method. We have studied the existence and uniqueness of the solution using the Banach fixed point theorem. Moreover, we provided rigorous error estimates showing that the numerical errors decay exponentially in the weighted Sobolev space. We have also established the existence and uniqueness of the numerical solution. Our numerical tests confirmed the theoretical findings, showing that an elevated rate of convergence in comparison with the numerical results reported in [6, 30-32]. Our spectral collocation method is more flexible with better accuracy than the existing ones. In our future extension, we will consider a unified spectral collocation method for multi-dimensional nonlinear systems of integral equations with convergence analysis.
The authors would like to thank the editor Bolin Wang, the associate editor, and the anonymous reviewers for their constructive comments and suggestions which improved the quality of this paper.
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,2, Eid H Doha1, Ibrahem G Ameen31Department of Mathematics, Faculty of Science, 
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