Local Gaussian-Collocation Scheme to Approximate the Solution of Nonlinear Fractional Differential Equations Using Volterra Integral Equations
Year: 2021
Author: Pouria Assari, Fatemeh Asadi-Mehregan, Mehdi Dehghan
Journal of Computational Mathematics, Vol. 39 (2021), Iss. 2 : pp. 261–282
Abstract
This work describes an accurate and effective method for numerically solving a class of nonlinear fractional differential equations. To start the method, we equivalently convert these types of differential equations to nonlinear fractional Volterra integral equations of the second kind by integrating from both sides of them. Afterward, the solution of the mentioned Volterra integral equations can be estimated using the collocation method based on locally supported Gaussian functions. The local Gaussian-collocation scheme estimates the unknown function utilizing a small set of data instead of all points in the solution domain, so the proposed method uses much less computer memory and volume computing in comparison with global cases. We apply the composite non-uniform Gauss-Legendre quadrature formula to estimate singular-fractional integrals in the method. Because of the fact that the proposed scheme requires no cell structures on the domain, it is a meshless method. Furthermore, we obtain the error analysis of the proposed method and demonstrate that the convergence rate of the approach is arbitrarily high. Illustrative examples clearly show the reliability and efficiency of the new technique and confirm the theoretical error estimates.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1912-m2019-0072
Journal of Computational Mathematics, Vol. 39 (2021), Iss. 2 : pp. 261–282
Published online: 2021-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 22
Keywords: Nonlinear fractional differential equation Volterra integral equation Gaussian-collocation method Meshless method Error analysis.
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