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An improved radial basis functions method for the high-order Volterra–Fredholm integro-differential equations

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Abstract

This paper is aimed at rectifying the numerical solution of linear Volterra–Fredholm integro-differential equations with the method of radial basis functions (RBFs). In this method, the spectral convergence rate can be acquired by infinitely smooth radial kernels such as Gaussian RBF (GA-RBF). These kernels are made by a free shape parameter, and the highest accuracy can often be achieved when this parameter is small, but herein the coefficient matrix of interpolation is ill-conditioned. Alternative bases can be used to improve the stability of method. One of them is based on the eigenfunction expansion for GA-RBFs which is utilized in this study. The Legendre–Gauss–Lobatto integration rule is applied to estimate the integral parts. Moreover, the error analysis is discussed. The results of numerical experiments are presented to demonstrate stable solutions with high accuracy compared to the standard GA-RBFs, the analytical solutions, and the other methods.

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Authors and Affiliations

  1. Department of Applied Mathematics and Computer Sciences, Faculty of Mathematical Sciences, University of Guilan, P.O. Box 1914, Rasht, Iran

    Farnaz Farshadmoghadam, Haman Deilami Azodi & Mohammad Reza Yaghouti

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  1. Farnaz Farshadmoghadam
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  2. Haman Deilami Azodi
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Farshadmoghadam, F., Deilami Azodi, H. & Yaghouti, M.R. An improved radial basis functions method for the high-order Volterra–Fredholm integro-differential equations. Math Sci 16, 445–458 (2022). https://doi.org/10.1007/s40096-021-00432-2

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