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An Efficient Alternative Kernel of Gaussian Radial Basis Function for Solving Nonlinear Integro-Differential Equations

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Abstract

This paper focuses on improving radial basis function (RBF) method for solving nonlinear Volterra–Fredholm integro-differential equations. The RBF method is one of the most important numerical techniques for approximating the solution of problems and interpolating scattered data in any dimensions. Infinitely smooth RBFs such as Gaussians (GA) have the spectral convergence rate. These kernels depend on an auxiliary parameter, called shape parameter that plays a significant role to specify the accuracy of RBF method. When this parameter is close to zero, we achieve the highest accuracy, here the standard RBF interpolant matrix becomes severely ill-conditioned. To overcome this problem, we use a stable alternative base made on the eigenfunction expansion with the help of the properties of the orthogonal Hermite polynomials for GA-RBFs. This idea produces a stable cardinal interpolation function that lacks eigenvalues as the factors of the ill-conditioning of the interpolation matrix. We generalize the upper bound of Volterra integro-differential from x to a function in terms of x with specific conditions. By applying this approach and the Legendre–Gauss–Lobatto quadrature formula, the solution of problem under study is reduced to the solution of a nonlinear system of algebraic equations. We also examine the effect of the number of Legendre–Gauss–Lobatto nodes on the accuracy of the calculations. Furthermore, we find an upper bound for the error function. Several numerical experiments indicate that the present method is more accurate and stable in comparison with the standard GA-RBFs and some other methods.

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Author notes

  1. Haman Deilami Azodi

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

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 & Mohammad Reza Yaghouti

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  1. Farnaz Farshadmoghadam
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  2. Haman Deilami Azodi
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Correspondence to Mohammad Reza Yaghouti.

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Farshadmoghadam, F., Azodi, H.D. & Yaghouti, M.R. An Efficient Alternative Kernel of Gaussian Radial Basis Function for Solving Nonlinear Integro-Differential Equations. Iran J Sci Technol Trans Sci 46, 869–881 (2022). https://doi.org/10.1007/s40995-022-01286-6

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