Local Gaussian-Collocation Scheme to Approximate the Solution of Nonlinear Fractional Differential Equations Using Volterra Integral Equations (Journal of Computational Mathematics, 2020)
This work describes an accurate and effective method for numerically solving a class ofnonlinear fractional differential equations. To start the method, we equivalently convertthese types of differential equations to nonlinear fractional Volterra integral equations ofthe second kind by integrating from both sides of them. Afterward, the solution of thementioned Volterra integral equations can be estimated using the collocation method basedon locally supported Gaussian functions. The local Gaussian-collocation scheme estimatesthe unknown function utilizing a small set of data instead of all points in the solutiondomain, so the proposed method uses much less computer memory and volume computingin comparison with global cases. We apply the composite non-uniform Gauss-Legendrequadrature formula to estimate singular-fractional integrals in the method. Because of thefact that the proposed scheme requires no cell structures on the domain, it is a meshlessmethod. Furthermore, we obtain the error analysis of the proposed method and demonstrate that the convergence rate of the approach is arbitrarily high. Illustrative examplesclearly show the reliability and efficiency of the new technique and confirm the theoreticalerror estimates.