Numerical Solution of a Nonlinear Fractional Integro-Differential Equation by a Geometric Approach

Abstract

Numerical solution of a Riemann–Liouville fractional integro-differential boundary value problem with a fractional nonlocal integral boundary condition is studied based on a numerical approach which preserve the geometric structure on the Lorentz Lie group. A fictitious time $$\tau $$ is used to transform the dependent variable y(t) into a new one $$u(t,\tau ):=(1+\tau )^{\gamma }y(t)$$ in an augmented space, where $$0<\gamma \le 1$$ is a parameter, such that under a semi-discretization method and use of a Newton-Cotes quadrature rule the original equation is converted to a system of ODEs in the space $$(t,\tau )$$ and the obtained system is solved by the Group Preserving Scheme (GPS). Some illustrative examples are given to demonstrate the accuracy and implementation of the method.