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

Yanping Chen,Yuanyuan Zhang,Xiulian Shi,Yunxia Wei

doi:10.1186/s40064-016-3358-z

Yanping Chen, Yuanyuan Zhang + Show 2 more

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https://doi.org/10.1186/s40064-016-3358-z

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Journal: SpringerPlus	Publication Date: Oct 4, 2016
Citations: 2	License type: CC BY 4.0

Affiliation: South China Normal University, Yantai University

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 omega ({mathbf{x}})=Pi _{i=1}^d(1-x_i)^alpha (1+x_i)^beta ,; -1<alpha , beta <frac{1}{d}-frac{1}{2} (d denotes the space dimensions) as the collocation points. The error analysis in L^infty-norm and L_omega ^2-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.

Highlights

We observe that there are many numerical approaches for solving one-dimensional Volterra integral equation, such as Runge–Kutta method (Brunner 1984; Yuan and Tang 1990), polynomial collocation method (Brunner 1986; Brunner et al 2001; Brunner and Tang 1989), multistep method (Mckee 1979; Houwen and Riele 1985), hp-discontinuous Galerkin method (Brunner and Schötzau 2006) and Taylor series method (Goldfine 1977)
We study the multidimensional nonlinear Volterra integral equation of the form t1 t2 td y(t1, t2, . . . , td) +
D 2 and denote (IN u)(x) its interpolation polynomial associated with the multidimensional Jacobi Gauss points {xj, j ≤ N }

Summary

Introduction

We observe that there are many numerical approaches for solving one-dimensional Volterra integral equation, such as Runge–Kutta method (Brunner 1984; Yuan and Tang 1990), polynomial collocation method (Brunner 1986; Brunner et al 2001; Brunner and Tang 1989), multistep method (Mckee 1979; Houwen and Riele 1985), hp-discontinuous Galerkin method (Brunner and Schötzau 2006) and Taylor series method (Goldfine 1977). The error analysis in L∞-norm and L2ω-norm theoretically justifies the exponential convergence of spectral collocation method in multidimensional space. In the literature (Tang et al 2008), the authors proposed a Legendre spectral collocation method for Volterra integral equation with a regular kernel in one-dimensional space. Chen and Tang (2009, 2010), Chen et al (2013), developed the spectral collocation method for one-dimensional weakly singular Volterra integral equation.

Results

Conclusion