Solvability of two-dimensional nonlinear singular Volterra integral equations with fractional order in Banach space and approximation of the solution of it

In this article, existence and uniqueness of the solution for two-dimensional non-linear singular Volterra integral equations with fractional orders in a Banach space is discussed by utilizing the concept of the measure of non-compactness and fixed-point theorem. In fact, this kind of equations is a generalization of two-dimensional Riemann-Liouville fractional non-linear integral equations. To approximate the solution of the above problem, we use modified homotopy perturbation with the help of Adomian polynomials. To validity of the derived results, we introduce an example in the field of singular non-linear integral equations. Hence, a semianalytic solution for given example is obtained ensuring satisfactory accuracy. Also, to ensure the effectiveness of the proposed method the results are compared with some other works.