### LOCAL GAUSSIAN-COLLOCATION SCHEME TO APPROXIMATE THE SOLUTION OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS USING VOLTERRA INTEGRAL EQUATIONS

Pouria Assari1, Fatemeh Asadi-Mehregan1, Mehdi Dehghan2

1. 1. Department of Mathematics, Faculty of Sciences, Bu-Ali Sina University, Hamedan 65178, Iran;
2. Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran 15914, Iran
• Received:2019-03-18 Revised:2019-07-17 Published:2021-03-15
• Contact: Pouria Assari,Email:passari@basu.ac.ir
• Supported by:
The authors are very grateful to the reviewers for their valuable comments and suggestions which have improved the paper.

Pouria Assari, Fatemeh Asadi-Mehregan, Mehdi Dehghan. LOCAL GAUSSIAN-COLLOCATION SCHEME TO APPROXIMATE THE SOLUTION OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS USING VOLTERRA INTEGRAL EQUATIONS[J]. Journal of Computational Mathematics, 2021, 39(2): 261-282.

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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