A novel Petrov-Galerkin method for a class of linear systems of fractional differential equations

This paper presents a novel Petrov-Galerkin method for a class of linear systems of fractional differential equations. New fractional-order generalized Jacobi functions are introduced, and their approximation properties are investigated. We show that these functions satisfy the given supplementary conditions and have the same asymptotic behavior as the exact solution, which are essential properties to design a high-order spectral Petrov-Galerkin method. For implementing our scheme, we represent the approximate solution by a linear combination of fractional-order generalized Jacobi functions and minimize the residual using shifted fractional Jacobi functions. The numerical solvability of the resultant algebraic system is justified as well as convergence and stability properties of the proposed scheme are explored. Finally, the reliability of the method is evaluated using various analytical and realistic problems. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.