Bivariate barycentric rational interpolation method for two dimensional fractional Volterra integral equations

Abstract The advantages of the barycentric rational interpolation (BRI) introduced by Floater and Hormann include the stability of interpolation, no poles, and high accuracy for any sufficiently smooth function. In this paper we design a transformed BRI scheme to solve two dimensional fractional Volterra integral equation (2D-FVIE), whose solution may be non-smooth since its derivatives may be unbounded near the integral domain boundary. The transformed BRI method is constructed based on bivariate BRI and some smoothing transformations, hence inherits the advantages of the BRI even for a singular function. First, the smoothing transformations are employed to change the original 2D-FVIE into a new form, so that the solution of the new transformed 2D-FVIE has better regularity. Then the transformed equation can be solved efficiently by using the bivariate BRI together with composite Gauss-Jacobi quadrature formula. Last, some inverse transformations are used to obtain the solution of the original equation. The whole algorithm is easy to be implemented and does not require any integral computation. Besides, we analyze the convergence behaviour via the transformed equation. Several numerical experiments are provided to illustrate the features of the proposed method.

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