Volume 39, Issue 2
Local Gaussian-Collocation Scheme to Approximate the Solution of Nonlinear Fractional Differential Equations Using Volterra Integral Equations

J. Comp. Math., 39 (2021), pp. 261-282.

Published online: 2020-11

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• 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.

• Keywords

Nonlinear fractional differential equation, Volterra integral equation, Gaussian-collocation method, Meshless method, Error analysis.

34A08, 45D05, 65G99, 65L60

passari@basu.ac.ir (Pouria Assari)

mdehghan@aut.ac.ir (Mehdi Dehghan)

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@Article{JCM-39-261, author = {Assari , Pouria and Asadi-Mehregan , Fatemeh and Dehghan , Mehdi}, title = {Local Gaussian-Collocation Scheme to Approximate the Solution of Nonlinear Fractional Differential Equations Using Volterra Integral Equations}, journal = {Journal of Computational Mathematics}, year = {2020}, volume = {39}, number = {2}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1912-m2019-0072}, url = {http://global-sci.org/intro/article_detail/jcm/18374.html} }
TY - JOUR T1 - Local Gaussian-Collocation Scheme to Approximate the Solution of Nonlinear Fractional Differential Equations Using Volterra Integral Equations AU - Assari , Pouria AU - Asadi-Mehregan , Fatemeh AU - Dehghan , Mehdi JO - Journal of Computational Mathematics VL - 2 SP - 261 EP - 282 PY - 2020 DA - 2020/11 SN - 39 DO - http://doi.org/10.4208/jcm.1912-m2019-0072 UR - https://global-sci.org/intro/article_detail/jcm/18374.html KW - Nonlinear fractional differential equation, Volterra integral equation, Gaussian-collocation method, Meshless method, Error analysis. AB -

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.

Pouria Assari, Fatemeh Asadi-Mehregan & Mehdi Dehghan. (2020). Local Gaussian-Collocation Scheme to Approximate the Solution of Nonlinear Fractional Differential Equations Using Volterra Integral Equations. Journal of Computational Mathematics. 39 (2). 261-282. doi:10.4208/jcm.1912-m2019-0072
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