A fast iterative spectral scheme based on novel operational matrices for nonlinear fractional-order singular integral problems

Abstract

Analytic solutions of nonlinear Volterra-Fredholm integral equations with generalized singular kernel arise in atomic scattering, electron emission, microscopy, radio astronomy, radar ranging, plasma diagnostics, and optical fiber evaluation and found a thorough effort among the scientific group. Due to the huge applications of such problems, its analytical solutions become a hot topic among researchers. In this framework, the current article addresses a novel, operational matrix-based iterative spectral scheme to examine the analytic solutions to such problems. First, the novel operational matrix is proposed to approximate the singular integral that appears in the problem, and then a fast scheme is proposed based on developed operational matrices. The well-known Picard iterative schemes are coupled to tackle the nonlinear term present in the problem. To show the reliability, efficiency, and appropriateness of the recommended method the generalized Abel’s, weekly singular integral equations and system are considered. The algorithm is consistent and mathematically validated through error-bound and convergence theorems. Finally, the efficiency of the method endorses that the proposed scheme is well-matched to deal with the solutions of other nonlinear physical mechanisms governed by integral equations. Moreover, it is noted that the convergence control parameters N and r are impacted significantly on the convergence of the proposed method.

Keywords

• Shifted Gegenbauer polynomials
• Abel’s equations
• Operational matrices
• Weekly singular problems