The asymptotic approximations to linear weakly singular Volterra integral equations via Laplace transform

In this paper, the asymptotic expansions for the solution about zero and infinity are formulated via Laplace transform for linear Volterra integral equation with weakly singular convolution kernel. The expansions about zero and infinity, as well as their Pade approximations, are used to approximate the solution when the argument is small and large, respectively, and the Pade approximations are more accurate. The methods are also valid to solve some other Volterra type integral equations including linear Volterra integro-differential equations, fractional integro-differential equations, and system of singular Volterra integral equations of the second kind with convolution kernels. Some examples confirm the correctness of the methods and the effectiveness of the asymptotic expansions. They show that numerical methods are only necessary in a small interval in practical computation when uniform high precision evaluations are needed for solving these kinds of Volterra integral equations.