A new numerical scheme for solving pantograph type nonlinear fractional integro-differential equations

Abstract

In this work, a general class of pantograph type nonlinear fractional integro-differential equations (PT-FIDEs) with non-singular and non-local kernel is considered. A numerical scheme based on the orthogonal basis functions including the shifted Legendre polynomials (SLPs) is proposed. First, we expand the unknown function and its derivatives in terms of the SLPs with unknown coefficients. Then, we present several theorems based on the SLPs for the help to achieve the approximate solution of the problem under study. Finally, by utilizing these theorems together with the collocation points, the main problem is transformed to a system of linear or nonlinear algebraic equations, which can be simply solved. An investigation for error estimate is discussed. The accuracy and efficiency of the proposed scheme are reported by four illustrative examples.