A fractional spectral Galerkin method for fuzzy Volterra integral equations with weakly singular kernels: Regularity, convergence, and applications

Fuzzy Volterra integral equations with weakly singular kernels have solutions with singular behavior at the origin. Utilizing spectral methods on such problems with standard (integer-order) basis functions leads to generating approximate solutions with low-order accuracy. This paper deals with improving the accuracy of the spectral Galerkin method which can be applied to such problems by using fractional-order basis functions. New matrix formulation of the proposed method transforms the problem under consideration into a system of algebraic equations with a simple structure. Numerical implementation of the constructed method shows its effectiveness compared to other methods. The convergence analysis of the method is theoretically investigated in a weighted $L^2$-norm.