## Abstract

This paper is concerned with obtaining an approximate solution for a linear multidimensional Volterra integral equation with a regular kernel. We choose the Gauss points associated with the multidimensional Jacobi weight function \(\omega(x) = \prod{_{i=1}^d}(1 - x_i)^\alpha(1 + x_i)^\beta, -1 <\alpha,\beta < \frac{1}{d} - \frac{1}{2}\) (*d* denotes the space dimensions) as the collocation points. We demonstrate that the errors of approximate solution decay exponentially. Numerical results are presented to demonstrate the effectiveness of the Jacobi spectral collocation method.

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## References

Aghazade N, Khajehnasiri A A. Solving nonlinear two-dimensional Volterra integro-differential equations by block-pulse functions. Math Sci, 2013, 7: 1–6

Brunner H. Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge: Cambridge Univ Press, 2004

Canuto C, Hussaini M Y, Quarteroni A, Zang T A. Spectral Methods: Fundamentals in Single Domains. Scientific Computation. Berlin: Springer-Verlag, 2006

Chen Y, Li X, Tang T. A note on Jacobi spectral-collocation methods for weakly singular Volterra integral equations with smooth solutions. J Comput Math, 2013, 31: 47–56

Chen Y, Tang T. Spectral methods for weakly singular Volterra integral equations with smooth solutions. J Comput Appl Math, 2009, 233: 938–950

Chen Y, Tang T. Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Math Comp, 2010, 79: 147–167

Colton D, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory. 2nd ed. Appl Math Sci, Vol 93. Heidelberg: Springer-Verlag, 1998

Darania P, Shali J A, Ivaz K. New computational method for solving some 2-dimensional nonlinear Volterra integro-differential equations. Numer Algorithms, 2011, 57: 125–147

Fedotov A I. Lebesgue constant estimation in multidimensional Sobolev space. J Math, 2004, 14: 25–32

Headley V B. A multidimensional nonlinear Gronwall inequality. J Math Anal Appl, 1974, 47: 250–255

Mckee S, Tang T, Diogo T. An Euler-type method for two-dimensional Volterra integral equations of the first kind. IMA J Numer Anal, 2000, 20: 423–440

Mirzaee F, Hadadiyan E, Bimesl S. Numerical solution for three-dimensional non-linear mixed Volterra-Fredholm integral equations via three-dimensional block-pulse functions. Appl Math Comput, 2014, 237: 168–175

Nemati S, Lima P M, Ordokhani Y. Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials. J Comput Appl Math, 2013, 242: 53–69

Nevai P. Mean convergence of Lagrange interpolation. Trans Amer Math Soc, 1984, 282: 669–698

Ragozin D L. Polynomial approximation on compact manifolds and homogeneous spaces. Trans Amer Math Soc, 1970, 150: 41–53

Ragozin D L. Constructive polynomial approximation on spheres and projective spaces. Trans Amer Math Soc, 1971, 162: 157–170

Shen J, Tang T. Spectral and High-order Methods with Applications. Beijing: Science Press, 2006

Tang T, Xu X, Chen J. On spectral methods for Volterra integral equations and the convergence analysis. J Comput Math, 2008, 26: 825–837

Wei Y X, Chen Y. Legendre spectral collocation methods for pantograph Volterra delay-integro-differential equation. J Sci Comput, 2012, 53: 672–688

Wei Y X, Chen Y. Legendre spectral collocation method for neutral and high-order Volterra integro-differential equation. Appl Numer Math, 2014, 81: 15–29

## Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11671157, 11826212).

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Wei, Y., Chen, Y., Shi, X. *et al.* Spectral method for multidimensional Volterra integral equation with regular kernel.
*Front. Math. China* **14**, 435–448 (2019). https://doi.org/10.1007/s11464-019-0758-8

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DOI: https://doi.org/10.1007/s11464-019-0758-8

### Keywords

- Multidimensional Volterra integral equation
- Jacobi collocation discretization
- multidimensional Gauss quadrature formula