Numerical solution of fractional order multi-point boundary value problems using reproducing kernel method with shifted Legendre polynomials

Abstract This article presents an efficient numerical method for obtaining numerical solutions to multi-point boundary problems with multi-fractional orders of Caputo derivatives.The novelty of the suggested method is based on the application of shifted Legendre polynomials in the reproducing kernel theory, as well as the use of extra basis functions for homogeneous or nonhomogeneous boundary conditions. The approximate solution for both linear and nonlinear cases of the considered problem is given in the form of a finite series sum. Convergence and error analysis of the presented method are given. The numerical outcomes obtained with the proposed method are compared with other results found in the literature. The obtained results for linear and nonlinear examples support the method’s efficiency.