A general finite element formulation for fractional variational problems

Abstract

This paper presents a general finite element formulation for a class of Fractional Variational Problems (FVPs). The fractional derivative is defined in the Riemann–Liouville sense. For FVPs the Euler–Lagrange and the transversality conditions are developed. In the Fractional Finite Element Formulation (FFEF) presented here, the domain of the equations is divided into several elements, and the functional is approximated in terms of nodal variables. Minimization of this functional leads to a set of algebraic equations which are solved using a numerical scheme. Three examples are considered to show the performance of the algorithm. Results show that as the number of discretization is increased, the numerical solutions approach the analytical solutions, and as the order of the derivative approaches an integer value, the solution for the integer order system is recovered. For unspecified boundary conditions, the numerical solutions satisfy the transversality conditions. This indicates that for the class of problems considered, the numerical solutions can be obtained directly from the functional, and there is no need to solve the fractional Euler–Lagrange equations. Thus, the formulation extends the traditional finite element approach to FVPs.