Numerical Solution of Volterra Integral Equations of the Second Kind with Regular and Singular Kernels: A Computational Approach

With an emphasis on regular and unique kernels, this abstract delves into the numerical solution of second class Volterra integral equations. The second kind of Volterra integral equations are a common type of equations in many practical domains, such as engineering and physics. The intricacy of solving these equations stems from the fact that the kernel might be either smooth (regular) or have singularities; the integral involving the unknown function defines them. When analytical solutions are not possible, computational methods become very important in solving these equations, as the abstract emphasizes. For efficient approximation of solutions, methods including numerical integration, iterative schemes, and discretization techniques (e.g., collocation and Galerkin methods) are used. Conventional numerical approaches provide quite accurate results when applied to regular kernels, but singular kernels necessitate additional handling, typically through transformation or adaptive grids, in order to deal with their single behaviour. The computational method addresses the problems presented by solitary kernels and the ways to overcome them, with an emphasis on the importance of accuracy and efficiency, especially for problems of a large scale.