مشخصات پژوهش

An effective technique based on fractional calculus in the sense of Riemann–Liouville has been developed for solving weakly singular Volterra integral equations of the first and second kinds. For this purpose, orthogonal Chebyshev polynomials are applied. Properties and some operational matrices of these polynomials are first presented and then the unknown functions of the integral equations are represented by these polynomials in the matrix form. These matrices are then used to reduce the singular integral equations to some linear algebraic system. For solving the obtained system, Galerkin method is utilized via Chebyshev polynomials as weighting functions. The method is computationally attractive, and the validity and accuracy of the presented method are demonstrated through illustrative examples. As shown in the numerical results, operational matrices, even for first kind integral equations, have relatively low condition numbers, and thus, the corresponding matrices are well posed. In addition, it is noteworthy that when the solution of equation is in power series form, the method evaluates the exact solution.