Abstract 

Fractional-order unsteady nonlinear integro-partial differential equations with weakly singular kernel ensue in numerous scientific process. Accurate solutions of such kind of problems become the challenging task among the scientific community due to the involvment of weakly singular kernel, nonlinear terms, time and space delay terms, and fractional-order derivative. In this context, this paper is dedicated to the extension and implementation of an iterative spectral scheme for fractional-order unsteady nonlinear integro-partial differential equations with weakly singular kernel. In this scheme, the unknown function u(x, t) is estimated by using shifted Gegenbauer polynomial vector Λ(x, t) and Picard iterative scheme used to tackle the nonlinearity. Some novel operational matrices are developed first the time in literature to approximate the singular integral. The advantage of the extended method is it convert the considered nonlinear problem into system of linear algebraic equations. Mathematical code of the proposed scheme is developed for the suggested problem and we validate it to compare the results with existing results. The attained results demostrate that the extended scheme is stable, accurate and appropriate to seek the solutions of problems under study.