Finite Element Method for Linear Multiterm Fractional Differential Equations

Abstract

We consider the linear multiterm fractional differential equation (fDE). Existence and uniqueness of the solution of such equation are discussed. We apply the finite element method (FEM) to obtain the numerical solution of this equation using Galerkin approach. A comparison, through examples, between our techniques and other previous numerical methods is established.

Lucas polynomials semi-analytic solution for fractional multi-term initial value problems

“…The exact solution of equation (45) is W (z) = z 5 -29z 4 10 + 76z 3 25 -339z 2 250 + 27z 125 . The residual of this equation is z β R(z) = z β E T H (2) which is the exact solution.…”

“…So the fractional calculus investigates the rules, properties of derivatives, and integrals of noninteger orders. For handling these equations, the researchers apply many numerical methods such as finite difference method [1][2][3], finite element method [4][5][6], homotopy analysis method [7,8], variational iteration method [9][10][11], a domain decomposition method [12][13][14][15], and Haar wavelet method [16,17].…”

Stability and Convergence of an Effective Finite Element Method for Multiterm Fractional Partial Differential Equations

“…It is noted that the method in [29] is a special case of the method in our paper for fractional partial differential equation with single fractional order. So, we just need to compare FEM in our paper with other existing methods in [8,28]. We use this example to check the convergence rate (c. rate) and CPU time (CPUT) of numerical solutions with respect to the fractional orders and .…”

“…For the problem (47), our method in this paper is just the DFBDM in Section 3. Therefore, we only need to compare M1 with the FEM in [28] (FEM2). In Table 6, although the convergence rate of FEM2 is higher than that of DFBDM, the error and CPUT of DFBDM are smaller than those of FEM2.…”

“…In [28], Badr investigate the FEM for linear multiterm fractional differential equations with one variable as follows:…”