A Galerkin Finite Element Method to Solve Fractional Diffusion and Fractional Diffusion-Wave Equations

Abstract

In the present study, numerical solutions of the fractional diffusion and fractional diffusion-wave equations where fractional derivatives are considered in the Caputo sense have been obtained by a Galerkin finite element method using quadratic B-spline base functions. For the fractional diffusion equation, the L1 discretizaton formula is applied, whereas the L2 discretizaton formula is applied for the fractional diffusion-wave equation. The error norms L 2 and L ∞ are computed to test the accuracy of the proposed method. It is shown that the present scheme is unconditionally stable by applying a stability analysis to the approximation obtained by the proposed scheme.

Numerical solution of time fractional Burgers equation

“…Applications of differentiation and integration with non-integer orders can be traced back to premature in history, so it can be said that it is not new [16]. Many different techniques and methods of dealing with fractional differential equations resulting analytical and numerical solutions can be found in a wide variety of studies in the literature [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31].…”

“…In this paper, to achieve a finite element layout of the time fractional Burgers equation, Caputo fractional derivative formulation can be discretizated through L1 formulae [17]:…”

Solving time fractional Burgers’ and Fisher’s equations using cubic B-spline approximation method

“…In open literature, a lot of work is available on solving linear/nonlinear partial/fractional partial differential equations numerically by means of different techniques. The authors in [4] used the Galerkin finite element method to solve fractional diffusion and fractional diffusion wave equations. A B-spline collocation method was used by [5] for numerical treatment of fractional diffusion and fractional diffusion wave equations.…”

Galerkin Finite Element Approximations for Stochastic Space-Time Fractional Wave Equations

“…Even in the kingdom of diffusion, the central pillars of anomalous diffusion, including sub-diffusion and super-diffusion, long-range dependence (LRD), 1/f noise, and Lévy statistics, have intrinsic power-law structures [12]; and the corresponding fractional diffusion equations have been much better investigated than the power law wave equation [5,10,16,17,32,37]. For the diffusion-wave equations, there are also some discussions, including analysis, algorithm, and applications; the fundamental solutions and their properties are considered in [1,24,25,30]; and different kinds of numerical methods and approximation schemes have also been developed, e.g., the finite difference method [33,38], spectral method [4], and finite element method [13,18], etc.…”