## Convergence analysis of the operational Tau method for Abel-type Volterra integral equations

P. Mokhtary and F. Ghoreishi

### Abstract

In this paper, a spectral Tau method based on Jacobi basis functions is proposed and its stability and convergence properties are considered for obtaining an approximate solution of Abel-type integral equations. This work is organized in two parts. First, we present a stable operational Tau method based on Jacobi basis functions that provides an efficient approximate solution for the Abel-type integral equations by using a reduced set of matrix operations. We also provide a rigorous error analysis for the proposed method in the weighted $L^2$- and uniform norms under more general regularity assumptions on the exact solution. It is shown that the proposed method converges, but since the solutions of these equations have a singularity near the origin, a loss in the convergence order of the Tau method is expected. To overcome this drawback we then propose a regularization process, in which the original equation is changed into a new equation which possesses a smooth solution, by applying a suitable variable transformation such that the spectral Tau method can be applied conveniently. We also prove that after this regularization technique, the numerical solution of the new equation based on the operational Tau method has exponential rate of convergence. Some standard examples are provided to confirm the reliability of the proposed method.

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### Key words

Operational Tau method, Abel-type Volterra integral equations

45E10, 41A25