An iterative spectral strategy for fractional-order weakly singular integro-partial differential equations with time and space delays

Abstract

<abstract> <p>This study aims at extending and implementing an iterative spectral scheme for fractional-order unsteady nonlinear integro-partial differential equations with weakly singular kernel. In this scheme, the unknown function <italic>u</italic>(x, <italic>t</italic>) is estimated by using shifted Gegenbauer polynomials vector Λ(x, <italic>t</italic>), and Picard iterative scheme is used to handle underlying nonlinearities. Some novel operational matrices are developed for the first time in order to approximate the singular integral like, $ \int_0^x {\int_0^y {u(p{a_1} + {b_1}, q{a_2} + {b_2}, t)/{{({x^{{\rho _1}}} - {p^{{\rho _1}}})}^{{\alpha _1}}}{{({y^{{\rho _2}}} - {q^{{\rho _2}}})}^{{\alpha _2}}}{\text{d}}q{\text{d}}p} } $ \end{document} and $ \int_0^t {{u^\gamma }({\bf{x}}, \xi)/{{({t^{{\rho _3}}} - {\xi ^{{\rho _3}}})}^{{\alpha _3}}}{\text{d}}\xi } $, where <italic>ρ</italic>'s > 1, 0 < <italic>α</italic>'s < 1 by means of shifted Gegenbauer polynomials vector. The advantage of this extended method is its ability to convert nonlinear problems into systems of linear algebraic equations. A computer program in Maple for the proposed scheme is developed for a sample problem, and we validate it to compare the results with existing results. Six new problems are also solved to illustrate the effectiveness of this extended computational method. A number of simulations are performed for different ranges of the nonlinearity <italic>n</italic>, <italic>α</italic>, fractional-order, <italic>ρ</italic>, and convergence control <italic>M</italic>, parameters. Our results demonstrate that the extended scheme is stable, accurate, and appropriate to find solutions of complex problems with inherent nonlinearities.</p> </abstract>