Application of reproducing kernel method for solving a class of two-dimensional linear integral equations with weakly singular kernel

Abstract

In this paper‎, ‎we will present a new method for solving a class of two-dimensional linear Volterra integral equation of the second kind with weakly singular kernel from Abel type in the reproducing kernel space‎. The reproducing kernel function is discussed in detail. Weak singularity of problem is removed by applying integration by parts. Further, improper integral belongs to L_2 (Ω). ‎In our method the exact solution ϕ(x,t) is represented in the form of series in the reproducing kernel space W(ω), and the approximate solution ϕ_n (x,t) is constructed via truncating the series to n terms. ‎Convergence analysis of the method is proved in detail‎. ‎Some numerical examples are also studied to demonstrate the efficiency and accuracy of the presented method‎. ‎The obtained results show that the error of the approximate solution is monotone decreasing in the sense of the norm of W(ω), when increasing the number of the nodes. Also, that indicate the method is simple and effective. It turns out that this method is valid.