A general fractional finite element formulation

Abstract Differential equations governing the behaviors of systems can be obtained through minimizations of energy functionals. In many applications, finite element formulations for a system can be developed directly using energy or some other types of functionals. One of the advantages of these formulations is that they do not require the natural boundary conditions to be imposed explicitly. In recent years, there has been a growing interest in the area of fractional variational calculus. Recently developed fractional variational formulations allow one to obtain the necessary Euler-Lagrange equations and the natural boundary conditions. However, closed form solutions for these problems may not be available, and a numerical technique may be necessary. In this paper, we present a general fractional finite element formulation for a class of fractional variational problems defined in terms of the Riemann-Liouville fractional derivatives. Specifically, we consider a functional quadratic in function y and its fractional derivative, where α is the order of the derivative. In this formulation, the domain of the integral is divided into several elements, and y and over the elements are written in terms of nodal variables and some shape functions. Various choices of shape functions are being considered. Here we approximate y and over each element using a linear function and a constant function, respectively. The constant function considered here represents at the center of the element. These approximations are used to reduce the original functional into a discretized quadratic form. Two approaches are presented to approximate at the center of an element. In the first scheme, 1) the number of node points is doubled; 2) the function values at the new node points are computed using the linear function; and 3) is expressed using Grunwald-Letnikov formula. In the second scheme, the Grunwald-Letnikov formula is modified to write at the center of an element in terms of nodal functional values. This modified Grunwald-Letnikov formula, when written for α = 1, gives the central difference formula for the derivative. These approximations are substituted into the above quadratic equation, the resulting equation is minimized, and the terminal kinematic conditions are imposed to obtain a set of algebraic equations. These algebraic equations can be solved using a direct or an iterative scheme. Here we use a direct scheme. Solution of these equations provides the response of the system. To demonstrate the applications of the formulations developed here, two fractional variational problems are solved using the scheme developed here, and the results are compared with analytical solutions and the solutions obtained using other schemes. Results show that 1) solutions converge as the number of node points is increased. However, in some cases, this convergence may be slow. 2) As α approaches 1, the classical solutions are recovered as expected. For integer order systems, finite element is a well established technique, and it is widely being used in academia and industries. It is hoped that this research will initiate a similar course for fractional order system.