NUMERICAL APPROXIMATIONS FOR THE FRACTIONAL LAPLACIAN IN SPACE-FRACTIONAL REACTION-DIFFUSION EQUATIONS

Abstract

The systems of non-linear time-dependent space-fractional differential equations have been employed to model important physical phenomena in many fields of engineering and science. The analytical solutions of most of these systems are unknown, and evaluating an analytical solution for some fractional differential equations is complicated and difficult to calculate because it is in the trigonometric series form. Thus, developing numerical solutions for such nonlinear systems is essential. There have been growing interests recently to develop efficient and robust numerical schemes for solving the nonlinear systems of fractional differential equations. In this study, several novel numerical schemes are proposed to solve the systems of multidimensional non-linear space-fractional reaction-diffusion equations efficiently. The non-local nature of the fractional operator adds new features to the mathematical models but also introduces additional difficulties in their implementation where large, dense matrices are required at each time step. To overcome this challenge, the Fourier spectral approach is applied to discretize the fractional Laplacian. This approach gives a diagonal representation of the fractional operator while achieving spectral convergence and the implementation to multi-dimensions is similar to onedimensional problems. Since this approach lacks capability to implement on nonhomogeneous boundary conditions, a second-order matrix transfer technique (MTT) for non-homogeneous boundary conditions is used for the space discretization. A fourth-order MTT based on a compact scheme is also employed for the discretization of the fractional Laplacian. To deal with the nonlinearities, exponential time differencing schemes (ETD) are employed for the reason that while the approaches achieve the expected accuracy, solving nonlinear systems at each time step is no longer needed. The Fourier spectral approach is combined with two second-order ETD schemes to solve space-fractional reaction-diffusion equations with non-smooth initial data and is also combined with a forth-order ETD scheme to provide highly efficient solutions for multidimensional v systems. The second-order MTT is combined with the forth-order ETD scheme to solve problems with non-homogeneous boundary conditions. Moreover, the fourthorder compact scheme MTT is combined with forth-order ETD schemes to show the effectiveness of the L-stable scheme when the initial data is non-smooth and to illustrate that the A-stable scheme is not reliable for some time steps. A novel reliability constraint is introduced to avoid the oscillations present in the solutions when the A-stable scheme is employed. Theoretical and numerical investigation of the convergence and stability of the numerical schemes have been discussed. Extensive numerical experiments are performed on wide well-known systems of time-dependent space-fractional reaction-diffusion equations to demonstrate the reliability, efficiency and accuracy of the developed schemes.​