Jacobi spectral collocation method for the approximate solution of multidimensional nonlinear Volterra integral equation

We present in this paper the convergence properties of Jacobi spectral collocation method when used to approximate the solution of multidimensional nonlinear Volterra integral equation. The solution is sufficiently smooth while the source function and the kernel function are smooth. We choose the Jacobi–Gauss points associated with the multidimensional Jacobi weight function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega ({\mathbf{x}})=\Pi _{i=1}^d(1-x_i)^\alpha (1+x_i)^\beta ,\; -1<\alpha , \beta <\frac{1}{d}-\frac{1}{2}$$\end{document}ω(x)=Πi=1d(1-xi)α(1+xi)β,-1<α,β<1d-12 (d denotes the space dimensions) as the collocation points. The error analysis in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty$$\end{document}L∞-norm and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\omega ^2$$\end{document}Lω2-norm theoretically justifies the exponential convergence of spectral collocation method in multidimensional space. We give two numerical examples in order to illustrate the validity of the proposed Jacobi spectral collocation method.

Volterra integro-differential equation with a single spatial variable are given in Wei and Chen (2012a, b, 2013, 2014. Nevertheless, to the best of our knowledge, there have been no works regarding the theoretical analysis of the spectral approximation for multidimensional Volterra integral equation (Atdev and Ashirov 1977;Beesack 1985;Pachpatte 2011;Suryanarayana 1972), even for the case with smooth kernel.
We shall extend to several space dimensions the approximation results in Tang et al. (2008) for a single spatial variable. The expansion of Jacobi will be considered. We will be concerned with Sobolev-type norms that are most frequently applied to the convergence analysis of spectral methods. We get the discrete scheme by using multidimensional Gauss quadrature formula for the integral term. We will provide a rigorous verification of the exponential decay of the errors for approximate solution.
We study the multidimensional nonlinear Volterra integral equation of the form by the Jacobi spectral collocation method. Here, g : [0, T 1 ] × [0, T 2 ] × · · · × [0, T d ] → R and K : D × R → R (where D := {(t 1 , s 1 , t 2 , s 2 , . . . , t d , s d ) : 0 ≤ s i ≤ t i ≤ T i , i = 1, 2, . . . , d}) are given smooth functions. If the given functions are smooth on their respective domains, the solution y is also the smooth function (see Brunner 2004). This fact will be the standing point of this paper.

Discretization scheme
We consider now the domain � = (−1, 1) d and we denote an element of R d by x = (x 1 , x 2 , . . . , x d ). Let −1 < α, β < 1 (1 − x i ) α (1 + x i ) β denotes a d-dimensional Jacobi weight function on , we denote by L 2 ω (�) the space of the measurable functions u : → R such that � |u(x)| 2 ω(x)dx < +∞. It is a Banach space for the norm The space L 2 ω (�) is a Hilbert space for the inner product L ∞ (�) is the Banach space of the measurable functions u : → R that are bounded outside a set of measure zero, equipped with the norm Given a multi-index α = (α 1 , α 2 , . . . , α d ) of nonnegative integers, we set and (1) This is a Hilbert space for the inner product which induces the norm Let {x j , 0 ≤ j ≤ N } denote the Jacobi Gauss points on the one-dimensional interval (−1, 1) (see Canuto et al. 2006;Shen and Tang 2006). We now consider multidimensional Jacobi interpolation. Let P N (�) be the space of all algebraic polynomials of degree up to N in each variable x i for i = 1, 2, . . . , d. Let us introduce the Jacobi Gauss points in : and denote by I N the interpolation operator at these points, i.e., for each continuous function u, I N u ∈ P N satisfies We can represent I N u as follows: is the Lagrange interpolation basis function associated with the Jacobi collocation points {x j } N j=0 . The multidimensional Jacobi Gauss quadrature formula is Here, u(x 1 , x 2 , . . . , Firstly, Eq. (3) holds at the collocation points x j = (x j 1 ,x j 2 , . . . ,x j d ) on , i.e., In order to obtain high order accuracy for the problem (4), we transfer the integral domain [−1,x j 1 ] × [−1,x j 2 ] · · · × [−1,x j d ] to a fixed interval ¯ by using the following transformation where Next, let u j 1 j 2 ···j d be the approximation of the function value u(x j ) and use Legendre Gauss quadrature formula, (5) becomes Here, {θ k , �k� ≤ N } denotes the Legendre Gauss points on the multidimensional space and {ω k , �k� ≤ N } denotes the corresponding weights. Let . Now, we use u N to approximate the solution u. Then, the Jacobi spectral collocation method is to seek u N such that u i 1 i 2 ···i d satisfy the following collocation equation: We can get the values of u i 1 i 2 ···i d by solving (8) and obtain the expressions of u N (x) accordingly.
Let the error function of the solution be written as e u (x) := u(x) − u N (x) . Since the exact solution of the problem (1) can be written as Then the corresponding error function satisfy Remark In our work, we let the multidimensional Jacobi weight function Tang et al. (2008), the authors choose α = β = 0.

Some lemmas
The following result can be found in Canuto et al. (2006).

Lemma 1 Assume that Gauss quadrature formula is used to integrate the product uφ ,
where u ∈ H m (�) for some m > d 2 and φ ∈ P N (�). Then there exists a constant C independent of N such that where (·, ·) represents the continuous inner product in L 2 (�) space and The seminorm is defined as Note that only pure derivatives in each spatial direction appear in this expression.
From Fedotov (2004), we have the following result on the Lebesgue constant for the Lagrange interpolation polynomials associated with the Jacobi-Gauss points. Wei et al. SpringerPlus (2016) 5:1710 Lemma 3 Assume that u(x) ∈ H m ω (�) for m > d 2 and denote (I N u)(x) its interpolation polynomial associated with the multidimensional Jacobi Gauss points {x j , �j� ≤ N }. Then the following estimates hold Proof The inequality (11) can be found in Canuto et al. (2006). We now prove (12). From Canuto et al. (2006), we have The following Gronwall Lemma, whose proof can be found in Headley (1974), will be essential for establishing our main results.

also an integrable function, we have
From Theorem 1 in Nevai (1984), we have the following mean convergence result of Lagrange interpolation based at the multidimensional Jacobi-Gauss points.

Lemma 5 For every bounded function v(x), there exists a constant C independent of v such that
For r ≥ 0 and κ ∈ (0, 1), C r,κ (�) will denote the space of functions whose r-th derivatives are Hölder continuous with exponent κ, endowed with the norm: We shall make use of a result of Ragozin (1970Ragozin ( , (1971 in the following lemma. Lemma 6 For nonnegative integer r and κ ∈ (0, 1), there exists a constant C r,κ > 0 such that for any function v ∈ C r,κ (�), there exists a polynomial function T N v ∈ P N such that Actually, T N is a linear operator from C r,κ (�) into P N .
Lemma 7 Assume there are constants L 0 , L 1 , L 2 , . . . , L d such that Let M v 1 ,v 2 be defined by Then, for any κ ∈ (0, 1) and v 1 , v 2 ∈ C(�), there exists a positive constant C ∼ L 0 , L 1 , L 2 , . . . , L d such that for any x ′ , x ′′ ∈¯ and x ′ � = x ′′ . This implies that Proof For ease of exposition, and without essential loss of generality, we will proof this lemma for d = 2 and assume x ′′

By virtue of Lemmas 6 and 7,
The desired estimate (36) is obtained by combining (37)-(40) and using the same technique as in the proof of Theorem 1.

Numerical results
We give two numerical examples to confirm our analysis. To examine the accuracy of the results, L 2 ω and L ∞ errors are employed to assess the efficiency of the method. All the calculations are supported by the software Matlab.
Example 1 We consider the following two-dimensional Volterra integral equation The corresponding exact solution is given by u(x, y) = e − xy 2 . We select α = − 2 3 , β = − 1 2 . Table 1 shows the errors �u − u N � L 2 ω (�) and �u − u N � L ∞ (�) obtained by using the spectral collocation method described above. Furthermore, the numerical results are plotted for 2 ≤ N ≤ 12 in Fig. 1. It is observed that the desired exponential rate of convergence is obtained.

Conclusions
In this paper, we proposed a spectral collocation method based on Jacobi orthogonal polynomials to obtain approximate solution for multidimensional nonlinear Volterra integral equation. The most important contribution of this work is that we are able to (42) v(x, y) + cos(x + ξ)v(ξ , η)dηdξ = sin(x + y) − 1 4 sin(3x + y) + 1 4 sin(x + y − 2) − 1 2 (x + 1)cos(x − y) sin(x − 3).  demonstrate rigorously that the errors of spectral approximations decay exponentially in both L ∞ (�) norm and L 2 ω (�) norm on d-dimensional space, which is a desired feature for a spectral method.