Shifted Fractional-Order Jacobi Collocation Method for Solving Variable-Order Fractional Integro-Differential Equation with Weakly Singular Kernel

We propose a fractional-order shifted Jacobi–Gauss collocation method for variable-order fractional integro-differential equations with weakly singular kernel (VO-FIDE-WSK) subject to initial conditions. Using the Riemann–Liouville fractional integral and derivative and fractional-order shifted Jacobi polynomials, the approximate solutions of VO-FIDE-WSK are derived by solving systems of algebraic equations. The superior accuracy of the method is illustrated through several numerical examples.

A numerical study on the non-smooth solutions of the nonlinear weakly singular fractional integro-differential equations

The solutions of weakly singular fractional integro-differential equations involving the Caputo derivative have singularity at the lower bound of the domain of integration. In this paper, we design an algorithm to prevail on this non-smooth behaviour of solutions of the nonlinear fractional integro-differential equations with a weakly singular kernel. The convergence of the proposed method is investigated. The proposed scheme is employed to solve four numerical examples in order to test its efficiency and accuracy.

Solution of Fractional Autonomous Ordinary Differential Equations

Autonomous differential equations of fractional order and non-singular kernel are solved. While solutions can be obtained through numerical, graphical, or analytical solutions, we seek an implicit analytical solution.