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

Wei et al. SpringerPlus (06) 5:70 DOI 0.86/s40064-06-3358-z RESEARCH Jacobi spectral collocation method for the approximate solution of multidimensional nonlinear Volterra integral equation Open Access Yunxia Wei *, Yanping Chen, Xiulian Shi 3 and Yuanyuan Zhang 4 *Correspondence: yunxiawei@6.com College of Mathematic and Information Science, Shandong Institute of Business and Technology, Yantai 64005, China Full list of author information is available at the end of the article Abstract 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 ω(x) = d i= ( x i) α ( + x i ) β, <α, β< d (d denotes the space dimensions) as the collocation points. The error analysis in L -norm and L ω-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. Keywords: Multidimensional nonlinear Volterra integral equation, Jacobi collocation discretization, Multidimensional Gauss quadrature formula, Error estimates Mathematics Subject Classification: 65R0, 45J05, 65N Background We observe that there are many numerical approaches for solving one-dimensional Volterra integral equation, such as Runge Kutta method (Brunner 984; Yuan and Tang 990), polynomial collocation method (Brunner 986; Brunner et al. 00; Brunner and Tang 989), multistep method (Mckee 979; Houwen and Riele 985), hp-discontinuous Galerkin method (Brunner and Schötzau 006) and Taylor series method (Goldfine 977). The spectral collocation method is the most popular form of the spectral methods among practitioners. It is convenient to implement for one-dimensional problems and generally leads to satisfactory results an long as the problems possess sufficient smoothness. In the literature (Tang et al. 008), the authors proposed a Legendre spectral collocation method for Volterra integral equation with a regular kernel in one-dimensional space. Subsequently, Chen and Tang (009, 00), Chen et al. (03), developed the spectral collocation method for one-dimensional weakly singular Volterra integral equation. The proofs of the convergence properties of spectral collocation method for 06 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Wei et al. SpringerPlus (06) 5:70 Page of 6 Volterra integro-differential equation with a single spatial variable are given in Wei and Chen (0a, b, 03, 04). 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 977; Beesack 985; Pachpatte 0; Suryanarayana 97), even for the case with smooth kernel. We shall extend to several space dimensions the approximation results in Tang et al. (008) 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 y(t, t,..., t d ) + t t 0 0 td 0 K (t, s, t, s,..., t d, s d, y(s, s,..., s d )) ds d...ds ds = g(t, t,..., t d ), t i [0, T i ], i =,,..., d, () by the Jacobi spectral collocation method. Here, g :[0, T ] [0, T ] [0, T d ] R and K : D R R (where D := {(t, s, t, s,..., t d, s d ) : 0 s i t i T i, i =,,..., 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 004). This fact will be the standing point of this paper. Discretization scheme We consider now the domain = (, ) d and we denote an element of R d by x = (x, x,..., x d ). Let <α, β< d, if ω = ω(x) = d i= ( x i) α ( + x i ) β denotes a d-dimensional Jacobi weight function on, we denote by L ω ( ) the space of the measurable functions u : R such that u(x) ω(x)dx < +. It is a Banach space for the norm The space L ω ( ) is a Hilbert space for the inner product (u, v) ω = u(x)v(x)ω(x)dx. 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, α,..., α d ) of nonnegative integers, we set and u L ω ( ) = ( u(x) ω(x)dx). u L ( ) = ess sup x u(x). α =α + α + +α d

Wei et al. SpringerPlus (06) 5:70 Page 3 of 6 D α α v v = α x α x α. d x d We define Hω m( )= {v L ω ( ): for each nonnegative multi-index α with α m, the distributional derivative D α v belongs to L ω ( )}. This is a Hilbert space for the inner product (u, v) m,ω = α m D α u(x)d α v(x)ω(x)dx, which induces the norm v H m ω ( ) = D α v L ω ( ). α m Let { x j,0 j N} denote the Jacobi Gauss points on the one-dimensional interval (, ) (see Canuto et al. 006; Shen and Tang 006). 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 =,,..., d. Let us introduce the Jacobi Gauss points in : x j = ( x j, x j,..., x jd ) for j = (j, j,..., j d ) N d, j = max i d j i N, and denote by I N the interpolation operator at these points, i.e., for each continuous function u, I N u P N satisfies (I N u)( x j ) = u( x j ) for all j N d, j N. We can represent I N u as follows: I N u(x) = u( x j )F j (x), j N where F j (x) = F j (x )F j (x )...F jd (x d ), {F j } N j=0 is the Lagrange interpolation basis function associated with the Jacobi collocation points { x j } N j=0. The multidimensional Jacobi Gauss quadrature formula is f (x)dx j N f ( x j, x j,..., x jd )ω j ω j...ω jd. () We use the variable transformations t i = T i ( + x i), x i [, ] and s i = T i ( + τ i), τ i [, x i ], i =,,..., d to rewrite () as follows u(x, x,..., x d ) + x x xd ˆK (x, τ, x, τ,..., x d, τ d, u(τ, τ,..., τ d ))dτ d...dτ dτ = f (x, x,..., x d ). (3) Here,

Wei et al. SpringerPlus (06) 5:70 Page 4 of 6 ( T f (x, x,..., x d ) = g ( + x ), T ( + x ),..., T d ( T Td K T ˆK (x, τ, x, τ,..., x d, τ d, u) = T T ( + τ ),..., T d ( + x d), T d ( + τ d), u and u(x, x,..., x d ) = y( T ( + x ), T ( + x ),..., T d ) ( + x d), ) ( + x d ) is the smooth solution of problem (3). Firstly, Eq. (3) holds at the collocation points x j = ( x j, x j,..., x jd ) on, i.e., ), ( + x ), T ( + τ ), T ( + x ), xj xj xjd u( x j, x j,..., x jd ) + dτ d dτ dτ = f ( x j, x j,..., x jd ). ˆK ( x j, τ, x j, τ,..., x jd, τ d, u(τ, τ,..., τ d )) (4) In order to obtain high order accuracy for the problem (4), we transfer the integral domain [, x j ] [, x j ] [, x jd ] to a fixed interval u( x j, x j,..., x jd ) + K ( x j, τ ( x j, θ ), x j, τ ( x j, θ ),..., x jd, τ d ( x jd, θ d ), u(τ ( x j, θ ), τ ( x j, θ ),..., τ d ( x jd, θ d )))dθ d dθ dθ = f ( x j, x j,..., x jd ), by using the following transformation (5) τ i = τ i ( x ji, θ i ) = + x j i θ i + x j i, i =,,..., d, (6) where K ( x j, τ, x j, τ,..., x jd, τ d, u) = + x j + x j + x j d ˆK ( x j, τ, x j, τ,..., x jd, τ d, u). Next, let u j j j d be the approximation of the function value u( x j ) and use Legendre Gauss quadrature formula, (5) becomes u j j j d + k N K ( x j, τ ( x j, θ k ), x j, τ ( x j, θ k ),..., x jd, τ d ( x jd, θ kd ), u(τ ( x j, θ k ), τ ( x j, θ k ),..., τ d ( x jd, θ kd )))ω k ω k...ω kd = f ( x j, x j,..., x jd ). (7) Here, {θ k, k N} denotes the Legendre Gauss points on the multidimensional space and {ω k, k N} denotes the corresponding weights. Let u N (x, x,..., x d ) = i N u i i...i d F i (x )F i (x )...F id (x d ). 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 i i d satisfy the following collocation equation:

Wei et al. SpringerPlus (06) 5:70 Page 5 of 6 u j j j d + K ( x j, τ ( x j, θ k ), x j, τ ( x j, θ k ),..., x jd, τ d ( x jd, θ kd ), i N k N u i i...i d F i (τ ( x j, θ k ))F i (τ ( x j, θ k ))...F id (τ d ( x jd, θ kd )))ω k ω k...ω kd = f ( x j, x j,..., x jd ). (8) We can get the values of u i i 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 () can be written as y(t) = u(x) (t i = T i ( + x i), t i [0, T i ], x i [, ]), we can define its approximate solution y N (t) = u N (x). Then the corresponding error function satisfy ( ) ε y (t) := y(t) y N (t) = e u (x) = e u t, t,..., t d. T T T d Remark In our work, we let the multidimensional Jacobi weight function ω(x) = d i= ( x i) α ( + x i ) β, <α, β< d. So ω(x) = ( x)α ( + x) β, <α, β< for d =. In Tang et al. (008), the authors choose α = β = 0. Some lemmas The following result can be found in Canuto et al. (006). Lemma Assume that Gauss quadrature formula is used to integrate the product uφ, where u H m ( ) for some m > d and φ P N ( ). Then there exists a constant C independent of N such that (u, φ) (u, φ) N CN m u H ( ) φ L ( ), (9) where (, ) represents the continuous inner product in L ( ) space and (u, φ) N = u(θ j, θ j,..., θ jd )φ(θ j, θ j,..., θ jd )ω j ω j...ω jd. j N The seminorm is defined as m u H ( ) = k=min(m,n+) i= d k u x k i L ( ). Note that only pure derivatives in each spatial direction appear in this expression. From Fedotov (004), we have the following result on the Lebesgue constant for the Lagrange interpolation polynomials associated with the Jacobi-Gauss points. Lemma Let I N := max x k N F k (x )F k (x ) F kd (x d ), we have

Wei et al. SpringerPlus (06) 5:70 Page 6 of 6 O ( (log N) d), if <α, β (, O (N max(α,β)+ ) d), if <α, β< d I N = (, O (N α+ ) d), if <β, <α< d (, O (N β+ ) d), if <α, <β< d. (0) Lemma 3 Assume that u(x) Hω m( ) for m > d 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 u I N u L ω ( ) CN m u H ω ( ), () u I N u L ( ) CN d+ m u H ω ( ). () Proof The inequality () can be found in Canuto et al. (006). We now prove (). From Canuto et al. (006), we have u I N u H l ω ( ) CN l m u H ω ( ), 0 l m. We know that H l ω ( ) is embedded in C( ) for l > d, namely, u I N u L ( ) C u I N u H l ω ( ) CN l m u H ω ( ) { CN d+ m u H, when d is an even number, ω ( ) CN d+ m u H, when d is an odd number. ω ( ) CN d+ m u H ω ( ). The following Gronwall Lemma, whose proof can be found in Headley (974), will be essential for establishing our main results. Lemma 4 Suppose M 0, a nonnegative integrable function E(x) satisfies E(x, x,..., x d ) M x x xd + G(x, x,..., x d ), (x, x,..., x d ), where G(x) is also an integrable function, we have E(τ, τ,..., τ d )dτ d...dτ dτ E L ω ( ) C G L ω ( ), E L ( ) C G L ( ). (3) (4) From Theorem in Nevai (984), we have the following mean convergence result of Lagrange interpolation based at the multidimensional Jacobi-Gauss points.

Wei et al. SpringerPlus (06) 5:70 Page 7 of 6 Lemma 5 For every bounded function v(x), there exists a constant C independent of v such that sup v( x j )F j (x) C max v(x). (5) N x j N L ω ( ) For r 0 and κ (0, ), C r,κ ( ) will denote the space of functions whose r-th derivatives are Hölder continuous with exponent κ, endowed with the norm: v C r,κ ( ) = max max α v(x) α r x x α xα xα d d α v(x ) x α + max sup xα xα d α v(x ) d x α xα xα d d α r x =x ( (x x ) + (x x ) + +(x d x d )) κ. We shall make use of a result of Ragozin (970, (97) in the following lemma. Lemma 6 For nonnegative integer r and κ (0, ), 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 v T N v L ( ) C r,κ N (r+κ) v C r,κ ( ), (6) Actually, T N is a linear operator from C r,κ ( ) into P N. Lemma 7 Assume there are constants L 0, L, L,..., L d such that ˆK (x, τ, x, τ,..., x d, τ d, v ) ˆK (x, τ, x, τ,..., x d, τ d, v ) L 0 v v, ˆK xi (x, τ, x, τ,..., x d, τ d, v ) ˆK xi (x, τ, x, τ,..., x d, τ d, v ) L i v v, i =,,..., d. Let M v,v be defined by M v,v (x) = x x xd [ ˆK (x, τ, x, τ,..., x d, τ d, v (τ, τ,..., τ d )) ˆK (x, τ, x, τ,..., x d, τ d, v (τ, τ,..., τ d ))]dτ d...dτ dτ. (7) Then, for any κ (0, ) and v, v C( ), there exists a positive constant C L 0, L, L,..., L d such that M v,v (x ) M v,v (x ) ( (x x ) + (x x ) + +(x d x d )) κ C max v (x) v (x), x (8) for any x, x and x = x. This implies that

Wei et al. SpringerPlus (06) 5:70 Page 8 of 6 M v,v C 0,κ ( ) C max v (x) v (x). x (9) Proof For ease of exposition, and without essential loss of generality, we will proof this lemma for d = and assume x < x, x < x, Here, M v,v (x ) M v,v (x ) x = [ ˆK (x, τ, x, τ, v (τ, τ )) ˆK (x, τ, x, τ, v (τ, τ ))]dτ dτ x [ ˆK (x, τ, x, τ, v (τ, τ )) ˆK (x, τ, x, τ, v (τ, τ ))]dτ dτ E + E. E = x [ ˆK (x, τ, x, τ, v (τ, τ )) ˆK (x, τ, x, τ, v (τ, τ ))]dτ dτ [ ˆK (x, τ, x, τ, v (τ, τ )) ˆK (x, τ, x, τ, v (τ, τ ))]dτ dτ (0) x x ˆK (x, τ, x, τ, v (τ, τ )) ˆK (x, τ, x, τ, v (τ, τ )) dτ dτ CL 0 v v L ( )(x x ) CL 0 v v L ( )(x x )+κ (x x ) κ C v v L ( )[(x x ) + (x x ) ] κ, () E = where P = = [ ˆK (x, τ, x, τ, v (τ, τ )) ˆK (x, τ, x, τ, v (τ, τ ))]dτ dτ [ ˆK (x, τ, x, τ, v (τ, τ )) ˆK (x, τ, x (P + P + P 3 )dτ C max(p + P + P 3 ), x [ ˆK (x, τ, x, τ, v (τ, τ )) ˆK (x, τ, x, τ, v (τ, τ ))]dτ [ ˆK (x, τ, x, τ, v (τ, τ )) ˆK (x, τ, x, τ, v (τ, τ ))]dτ dτ, τ, v (τ, τ ))]dτ [ ˆK x (x, τ, ξ, τ, v (τ, τ )) ˆK x (x, τ, ξ, τ, v (τ, τ ))](x x )dτ CL v v L ( )(x x ) C v v L ( )(x x )+κ (x x ) κ C v v L ( )[(x x ) + (x x ) ] κ, ξ (x, x ). () (3)

Wei et al. SpringerPlus (06) 5:70 Page 9 of 6 similarly, P = = [ ˆK (x, τ, x, τ, v (τ, τ )) ˆK (x, τ, x, τ, v (τ, τ ))]dτ [ ˆK (x, τ, x, τ, v (τ, τ )) ˆK (x, τ, x, τ, v (τ, τ ))]dτ [ ˆK x (η, τ, x, τ, v (τ, τ )) ˆK x (η, τ, x, τ, v (τ, τ ))](x x )dτ CL v v L ( )(x x ) C v v L ( )[(x x ) + (x x ) ] κ, η (x, x ). (4) P 3 = [ ˆK (x, τ, x, τ, v (τ, τ )) ˆK (x, τ, x, τ, v (τ, τ ))]dτ [ ˆK (x, τ, x, τ, v (τ, τ )) ˆK (x, τ, x, τ, v (τ, τ ))]dτ x L 0 v v dτ C v v L ( )(x x )+κ (x x ) κ C v v L ( )[(x x ) + (x x ) ] κ. The estimate (8) for d = is obtained by combining (0) (4). (5) Error estimates Theorem Let u(x) be the exact solution of the multidimensional nonlinear Volterra integral equation (3), which is smooth. u N (x) is the approximate solution, i.e., u(x) u N (x). Assume that k θ k i K (x, θ, x, θ,..., x d, θ d, v ) k Then there is a constant C such that the errors satisfy for m > d +, θ k i L ik v v, i =,,..., d; k =,,..., m, L = max L ik. i d, k m K (x, θ, x, θ,..., x d, θ d, v ) u u N L ( ) CN m (log N) d K + N d+ u H ω ( ), if <α, β ( ), dk N max(α,β)+ + N d+ u H ω ( ), if <α, β< d ( ), dk N α+ + N d+ u H ω ( ), if <β, <α< d ( ), dk N β+ + N d+ u H ω ( ), if <α, <β< d. (6)

Wei et al. SpringerPlus (06) 5:70 Page 0 of 6 where K = max j N K ( x j, θ, x j, θ,..., x jd, θ d, u(θ, θ,..., θ d )) H ( ), C d, L, L 0, L, L,..., L d. Proof We subtract (8) from (5) to get the error equation u( x j, x j,..., x jd ) u j j...j d + where Using the variable transformation (6), we have [ K ( x j, τ ( x j, θ ), x j, τ ( x j, θ ),..., x jd, τ d ( x jd, θ d ), u(τ ( x j, θ ), τ ( x j, θ ),..., τ d ( x jd, θ d ))) K ( x j, τ ( x j, θ ), x j, τ ( x j, θ ),..., x jd, τ d ( x jd, θ d ), u N (τ ( x j, θ ), τ ( x j, θ ),..., τ d ( x jd, θ d )))]dθ d dθ dθ = I( x j, x j,..., x jd ), I( x j, x j,..., x jd ) = k N K ( x j, τ ( x j, θ k ), x j, τ ( x j, θ k ),..., x jd, τ d ( x jd, θ kd ), u N (τ ( x j, θ k ), τ ( x j, θ k ),..., τ d ( x jd, θ kd )))ω k ω k...ω kd K ( x j, τ ( x j, θ ), x j, τ ( x j, θ ),..., x jd, τ d ( x jd, θ d ), u N (τ ( x j, θ ), τ ( x j, θ ),..., τ d ( x jd, θ d )))dθ d...dθ dθ. u( x j, x j,..., x jd ) u j j...j d + xj xj... xjd [ ˆK ( x j, τ, x j, τ,..., x jd, τ d, u(τ, τ,..., τ d )) ˆK ( x j, τ, x j, τ,..., x jd, τ d, u N (τ, τ,..., τ d ))]dτ d...dτ dτ = I( x j, x j,..., x jd ). (7) Multiplying F j (x )F j (x )...F jd (x d ) on both sides of Eq. (7) and summing up j N yield x x e u (x, x,..., x d ) + Consequently, xd [ ˆK (x, τ, x, τ,..., x d, τ d, u(τ, τ,..., τ d )) ˆK (x, τ, x, τ,..., x d, τ d, u N (τ, τ,..., τ d ))]dτ d...dτ dτ = J (x, x,..., x d ) + J (x, x,..., x d ) + J 3 (x, x,..., x d ). (8) x x xd e u (x, x,..., x d ) L 0... e u (τ, τ,..., τ d ) dτ d...dτ dτ + J (x, x,..., x d ) + J (x, x,..., x d ) + J 3 (x, x,..., x d ), (9) where

Wei et al. SpringerPlus (06) 5:70 Page of 6 J (x) = I( x j, x j,..., x jd )F j (x )F j (x )...F jd (x d ), j N J (x) = u(x, x,..., x d ) (I N u)(x, x,..., x d ), J 3 (x) = x x xd [ ˆK (x, τ, x, τ,..., x d, τ d, u(τ, τ,..., τ d )) ˆK (x, τ, x, τ,..., x d, τ d, u N (τ, τ,..., τ d ))]dτ d...dτ dτ I N x x xd [ ˆK (x, τ, x, τ,..., x d, τ d, u(τ, τ,..., τ d )) ˆK (x, τ, x, τ,..., x d, τ d, u N (τ, τ,..., τ d ))]dτ d...dτ dτ. It follows from the Gronwall inequality in Lemma 4 that e u L ( ) C ( J L ( ) + J L ( ) + J 3 L ( )). Using (9) and (0), we have ( ) J L ( ) C I N max I( x j, x j,..., x jd ) j N C I N N m K ( x j, θ, x j, θ,..., x jd, θ d, u N (θ, θ,..., θ d )) H ( ), C I N N m (K + K ( x j, θ, x j, θ,..., x jd, θ d, u N (θ, θ,..., θ d )) K ( x j, θ, x j, θ,..., x jd, θ d, u(θ, θ,..., θ d )) H ( ) ). (30) (3) A straightforward computation shows that K ( x j, θ, x j, θ,..., x jd, θ d, u N (θ, θ,..., θ d )) K ( x j, θ, x j, θ,..., x jd, θ d, u(θ, θ,..., θ d )) H ( ) ( m d k K θi k ( x j, θ, x j, θ,..., x jd, θ d, u N (θ, θ,..., θ d )) k= i= k θ k i K ( x j, θ, x j, θ,..., x jd, θ d, u(θ, θ,..., θ d )) L ( ) ) ( m k= i= d L ik u N u L ( ) ) L m d u N u L ( ) C e u L ( ). (3) Due to Lemma 3, J L ( ) CN d+ m u H ω ( ). (33) By virtue of Lemmas 6 and 7,

Wei et al. SpringerPlus (06) 5:70 Page of 6 J 3 L ( ) = (I I N )M u,un L ( ) = (I I N )(M u,un T N M u,un ) L ( ) ( + I N ) M u,un T N M u,un L ( ) C I N N κ M u,un C 0,κ ( ) C I N N κ e u L ( ) C(log N) d N κ e u L ( ), if <α, β, C(N max(α,β)+ ) d N κ e u L ( ), if <α, β< d, C(N α+ ) d N κ e u L ( ), if <β, <α< d, C(N β+ ) d N κ e u L ( ), if <α, <β< d. (34) We now obtain the estimate for e u L ( ) by using (30) (34), e u L ( ) CN m( ) I N K + N d+ u H, ω ( ) where in last step we have used the following assumption, 0 <κ<, if <α, β, (max(α, β) + )d <κ<, if <α, β< d, (α + )d <κ<, if <β, <α< d, (β + )d <κ<, if <α, <β< d. This completes the proof of the theorem. (35) Theorem If the hypotheses given in Theorem hold and κ satisfies (35), then Proof m u u N L ω ( ) CN ( + N κ (log N) d )K + ( + N d+ κ ) u H ω ( ), ( if <α, β, + N κ( ) ) N max(α,β)+ d K + ( + N d+ κ ) u H ω ( ), if <α, β< d, ( + N κ( N α+ ) d) K + ( + N d+ κ ) u H ω ( ), if <β, <α< d, ( + N κ( N β+ ) d ) K + ( + N d+ κ ) u H ω ( ), if <α, <β< d. By using (8) and Gronwall inequality in Lemma 4, we obtain that ) e u L ω ( ) ( J C L ω ( ) + J L ω ( ) + J 3 L ω ( ). (36) (37) Using Lemmas, 5 and (3) we have for J L ω ( ) C max I(x) CN m (K + e u L x ω ( ) ). (38) Due to Lemma 3,

Wei et al. SpringerPlus (06) 5:70 Page 3 of 6 J L ω ( ) CN m u H ω ( ). (39) By virtue of Lemmas 6 and 7, J 3 L ω ( ) = (I I N )M u,un L ω ( ) = (I I N )(M u,un T N M u,un ) L ω ( ) M u,un T N M u,un L ω ( ) + I N (M u,un T N M u,un ) L ω ( ) C M u,un T N M u,un L ( ) CN κ e u L ( ) CN m κ( ) I N K + N d+ u H. ω ( ) (40) The desired estimate (36) is obtained by combining (37) (40) and using the same technique as in the proof of Theorem. Numerical results We give two numerical examples to confirm our analysis. To examine the accuracy of the results, L ω and L errors are employed to assess the efficiency of the method. All the calculations are supported by the software Matlab. Example We consider the following two-dimensional Volterra integral equation x y u(x, y) + cos(x + y)e ξη u(ξ, η)dηdξ = e xy + cos(x + y)(x + )(y + ). (4) The corresponding exact solution is given by u(x, y) = e xy. We select α = 3, β =. Table shows the errors u u N L ω ( ) and u u N L ( ) obtained by using the spectral collocation method described above. Furthermore, the numerical results are plotted for N in Fig.. It is observed that the desired exponential rate of convergence is obtained. Example Consider the equation with Table The errors u u N L ω ( ) and u u N L ( ) N 4 6 L -error 9.373e 003 3.3409e 005 5.698e 008 L ω-error.85e 003.454e 006.0899e 009 N 8 0 L -error 4.5534e 0 6.58e 04 6.7390e 04 L ω-error 6.0859e 03.30e 03.34e 03

Wei et al. SpringerPlus (06) 5:70 Page 4 of 6 0 4 L error L ω error errors in log scale 0 6 0 8 0 0 0 4 6 8 0 N Fig. The errors u u N versus the number of collocation points in L and L ω norms x y v(x, y) + cos(x + ξ)v(ξ, η)dηdξ = sin(x + y) 4 sin(3x + y) + 4 sin(x + y ) (x + )cos(x y) + (x + )cos(x + ) + 4 sin(3x ) sin(x 3). 4 (4) The corresponding exact solution is given by v(x, y) = sin(x + y). We select α = 3, β = 3 4. Table shows the errors v v N L ω ( ) and v v N L ( ). The numerical results are plotted for N in Fig.. 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 Table The errors v v N L ω ( ) and v v N L ( ) N 4 6 L -error.0746e 00.307e 003 7.3430e 006 L ω-error 4.797e 00 3.5805e 004.395e 006 N 8 0 L -error.503e 008 5.699e 0.99e 03 L ω-error 3.6864e 009 6.785e 0 7.9865e 04

Wei et al. SpringerPlus (06) 5:70 Page 5 of 6 0 L error L ω error 0 4 errors in log scale 0 6 0 8 0 0 0 4 6 8 0 N Fig. The errors v v N versus the number of collocation points in L and L ω norms demonstrate rigorously that the errors of spectral approximations decay exponentially in both L ( ) norm and L ω ( ) norm on d-dimensional space, which is a desired feature for a spectral method. Authors contributions YW and YC carried out the spectral collocation method studies, performed the error analysis and drafted the manuscript. XS participated in the numerical experiments. YZ helped to draft the manuscript. All authors read and approved the final manuscript. Author details College of Mathematic and Information Science, Shandong Institute of Business and Technology, Yantai 64005, China. School of Mathematical Sciences, South China Normal University, Guangzhou 5063, China. 3 School of Mathematics and Statistics, Zhaoqing University, Zhaoqing 5606, China. 4 Department of Mathematics and Information Science, Yantai University, Yantai 64005, China. Acknowledgements This work is supported by National Natural Science Foundation of China (40347, 943004, 745, 64055, 4693). Competing interests The authors declare that they have no competing interests. Received: 6 June 06 Accepted: September 06 References Atdev S, Ashirov S (977) Solutions of multidimensional nonlinear Volterra operator equations. Ukr Math J 9:437 44 Beesack PR (985) Systems of multidimensional Volterra integral equations and inequalities. Nonlinear Anal Theory 9:45 486 Brunner H (984) Implicit Runge Kutta methods of optimal order for Volterra integro-differential equations. Math Comput 4:95 09 Brunner H (986) Polynomial spline collocation methods for Volterra integro-differential equations with weakly singular kernels. IMA J Numer Anal 6: 39 Brunner H (004) Collocation methods for Volterra integral and related functional equations. Cambridge University Press, Cambridge Brunner H, Tang T (989) Polynomial spline collocation methods for the nonlinear Basset equation. Comput Math Appl 8:449 457 Brunner H, Schötzau D (006) hp-discontinuous Galerkin time-stepping for Volterra integrodifferential equations. SIAM J Numer Anal 44:4 45 Brunner H, Pedas A, Vainikko G (00) Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels. SIAM J Numer Anal 39:957 98 Canuto C, Hussaini MY, Quarteroni A et al (006) Spectral methods fundamentals in single domains. Springer, Berlin

Wei et al. SpringerPlus (06) 5:70 Page 6 of 6 Chen Y, Tang T (009) Spectral methods for weakly singular Volterra integral equations with smooth solutions. J Comput Appl Math 33:938 950 Chen Y, Tang T (00) Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Math Comput 79:47 67 Chen Y, Li X, Tang T (03) A note on Jacobi spectral-collocation methods for weakly singular Volterra integral equations with smooth solutions. J Comput Math 3:47 56 Fedotov AI (004) Lebesgue constant estimation in multidimensional Sobolev space. J Math 4:5 3 Goldfine A (977) Taylor series methods for the solution of Volterra integral and integro-differential equations. Math Comput 3:69 707 Headley VB (974) A multidimensional nonlinear Gronwall inequality. J Math Anal Appl 47:50 55 Mckee S (979) Cyclic multistep methods for solving Volterra integro-differential equations. SIAM J Numer Anal 6:06 4 Nevai P (984) Mean convergence of Lagrange interpolation. Trans Am Math Soc 8:669 698 Pachpatte BG (0) Multidimensional integral equations and inequalities. Springer, Berlin 9 Ragozin DL (970) Polynomial approximation on compact manifolds and homogeneous spaces. Trans Am Math Soc 50:4 53 Ragozin DL (97) Constructive polynomial approximation on spheres and projective spaces. Trans Am Math Soc 6:57 70 Shen J, Tang T (006) Spectral and high-order methods with applications. Science Press, Beijing Suryanarayana MB (97) On multidimensional integral equations of Volterra type. Pac J Math 4:809 88 Tang T, Xu X, Chen J (008) On spectral methods for Volterra integral equations and the convergence analysis. J Comput Math 6:85 837 van der Houwen PJ, te Riele HJJ (985) Linear multistep methods for Volterra integral and integro-differential equations. Math Comput 45:439 46 Wei YX, Chen Y (0a) Convergence analysis of the spectral methods for weakly singular Volterra integro-differential equations with smooth solutions. Adv Appl Math Mech 4: 0 Wei YX, Chen Y (0b) Legendre spectral collocation methods for pantograph Volterra delay-integro-differential equation. J Sci Comput 53:67 688 Wei YX, Chen Y (03) A spectral method for neutral Volterra integro-differential equation with weakly singular kernel. Numer Math Theory Methods Appl 6:44 446 Wei YX, Chen Y (04) Legendre spectral collocation method for neutral and high-order Volterra integro-differential equation. Appl Numer Math 8:5 9 Yuan W, Tang T (990) The numerical analysis of implicit Runge Kutta methods for a certain nonlinear integro-differential equation. Math Comput 54:55 68