Tau approximate solution of weakly singular Volterra integral equations

In this paper, we present a numerical solution of weakly singular Volterra integral equations including the Abel’s equations by the Tau method with arbitrary polynomial bases. The Tau method produces approximate polynomial solutions of differential, integral and integro-differential equations. An extension of the Tau method has been done for the numerical solution of weakly singular Volterra integral equations, especially for Abel’s equations. The integral part appearing in the equation is replaced by its operational Tau representation. An efficient error estimation of the Tau method is also introduced. Some numerical experiments are given to demonstrate the superior performance of the Tau method. The Chebyshev and Legendre bases are also introduced to improve the results.