An efficient and stable Lagrangian matrix approach to Abel integral and integrodifferential equations Abstract   This article studies Abel integral equations (AIEs) and singular integrodifferential equations (SIDEs) and aims to develop two numerical schemes for them. It also emphasises on the comparative analysis of both AIEs u0026 SIDEs which is based on mainly two process namely GaussLegendre roots as collocation node points and random node points over the domain 0,1. For generating interpolating basis functions (IBF), we used Lagrangian interpolating polynomial and for orthonormal Lagrangian basis functions (OLBF), we used GramSchmidt orthogonalization algorithm, respectively. Firstly, we introduced the function approximation by using generated IBF and OLBF, then established the error bounds of these approximations. The constructed approximations by both the schemes convert the AIEs and SIDEs into the system of algebraic equations. We have also established error bounds, stability and convergence analysis of the proposed schemes by considering several mild mathematical conditions. Moreover, the stability of schemes is also established numerically. Finally, the test functions with the support of graphs clearly show the reliability and computational efficiency of the proposed methods.

Crankxe2x80x93NicolsonGalerkin spectral method for solving twodimensional timespace distributedorder weakly singular integropartial differential equation Abstract   The fractional PDEs based upon the distributedorder fractional derivative have several applications in physics. The twodimensional timespace distributedorder weakly singular integropartial differential model is investigated by a combination of finite difference and Galekrin spectral methods. A secondorder finite difference formula is employed to approximate the temporal variable. In this stage, the stability and convergence of the semidiscrete scheme are proved. Then, the Galerkin spectral method based on the modified Jacobi polynomials is applied to discrete the space variable. Also, in this step, the error estimate of the fulldiscrete scheme is studied. Finally, two test problems have been presented to confirm the theoretical results.