HIGH-ORDER NUMERICAL SCHEME AND THEORETICAL ANALYSIS FOR NONLINEAR TWO-DIMENSIONAL FRACTIONAL VOLTERRA INTEGRAL EQUATIONS WITH INITIAL VALUE SINGULARITY

Various numerical methods have been proposed for solving one-dimensional weakly singular Volterra integral equations (VIEs) with smooth solutions. The main purpose of this paper is to propose and analyze a numerical method for the solution of two-dimensional nonlinear weakly singular VIEs with non-smooth solutions by involving the transformation of variables and modified Block-by-Block method. We rigorously prove that the new scheme can achieve an order of $ O(\tau_s^{3+\alpha}+\lambda_t^{3+\beta}) $ for non-smooth solutions with step size $ \tau_s,\lambda_t $ for $ 0<\alpha,\beta<1 $. Some numerical examples are conducted to support the theoretical results and demonstrate the effectiveness of the proposed method.