Local radial basis function scheme for solving a class of fractional integro‐differential equations based on the use of mixed integral equations

Abstract

Integro‐differential equations with non‐integer order derivatives are an all‐purpose subdivision of fractional calculus. In the current paper, we present a numerical method for solving fractional Volterra‐Fredholm integro‐differential equations of the second kind. To establish the scheme, we first convert these types of integro‐differential equations to mixed integral equations by fractional integrating from both sides of them. Then, the discrete collocation method by combining the locally supported radial basis functions are used to approximate the mentioned integral equations. Since the local method proposed in this paper estimates an unknown function via a small set of data instead of all points in the solution domain, it uses much less computer memory in comparison with the global cases. We apply the nonuniform Gauss‐Legendre integration rule to compute the singular‐fractional integrals appearing in the scheme. As the scheme does not need any background meshes, it can be recognized as a meshless method. Numerical results are included to show the validity and efficiency of the new technique and confirm that the algorithm of the presented approach is attractive and easy to implement on computers.

A discrete collocation technique using the thin plate splines for solving a certain class of integro-differential equations arising in biology models

“…The computational algorithms have been obtained via the reproducing kernel method for several types of integro-differential equations [1] with periodic boundary conditions [2,3]. Locally supported RBFs have been developed for solving a type of integro-differential equations of fractional differential order [7]. This paper contains a numerical scheme to solve the equation (1).…”