Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reaction-diffusion equations described by the fractional Laplacian in bounded rectangular domains of \(\mathbb {R}^n\). The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator.

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Fourier spectral methods for fractional-in-space reaction-diffusion equations

A direct O(Nlog2N) finite difference method for fractional diffusion equations

The Galerkin finite element method for a multi-term time-fractional diffusion equation

We develop an exponentially accurate fractional spectral collocation method for solving steady-state and time-dependent fractional PDEs (FPDEs). We first introduce a new family of interpolants, called fractional Lagrange interpolants, which satisfy the Kronecker delta property at collocation points. We perform such a construction following a spectral theory recently developed in [M. Zayernouri and G. E. Karniadakis, J. Comput. Phys., 47 (2013), pp. 2108--2131] for fractional Sturm--Liouville eigenproblems. Subsequently, we obtain the corresponding fractional differentiation matrices, and we solve a number of linear FODEs in addition to linear and nonlinear FPDEs to investigate the numerical performance of the fractional collocation method. We first examine space-fractional advection-diffusion problem and generalized space-fractional multiterm FODEs. Next, we solve FPDEs, including the time- and space-fractional advection-diffusion equation, time- and space-fractional multiterm FPDEs, and finally the space...

Fractional spectral collocation method

In this paper, we consider spectral approximation of fractional dif- ferential equations (FDEs). A main ingredient of our approach is to define a new class of generalized Jacobi functions (GJFs), which is intrinsically related to fractional calculus and can serve as natural basis functions for properly de- signed spectral methods for FDEs. We establish spectral approximation results for these GJFs in weighted Sobolev spaces involving fractional derivatives. We construct efficient GJF-Petrov-Galerkin methods for a class of prototypical fractional initial value problems (FIVPs) and fractional boundary value prob- lems (FBVPs) of general order, and we show that with an appropriate choice of the parameters in GJFs, the resulting linear systems are sparse and well- conditioned. Moreover, we derive error estimates with convergence rates only depending on the smoothness of data, so true spectral accuracy can be attained if the data are smooth enough. The ideas and results presented in this paper will be useful in dealing with more general FDEs involving Riemann-Liouville or Caputo fractional derivatives.

https://www3.ntu.edu.sg/home/lilian/GJP_FDEs.pdf

Generalized Jacobi functions and their applications to fractional differential equations

. In this paper, we investigate initial boundary value problems of the spacetime fractional diffusion equation and its numerical solutions. Two definitions, i.e., Riemann-Liouville definition and Caputo one, of the fractional derivative are considered in parallel. In both cases, we establish the well-posedness of the weak solution. Moveover, based on the proposed weak formulation, we construct an efficient spectral method for numerical approximations of the weak solution. The main contribution of this work are threefold: First, a theoretical framework for the variational solutions of the space-time fractional diffusion equation is developed. We find suitable functional spaces and norms in which the space-time fractional diffusion problem can be formulated into an elliptic weak problem, and the existence and uniqueness of the weak solution are then proved by using existing theory for elliptic problems. Secondly, we show that in the case of Riemann-Liouville definition, the well-posedness of the space-time fractional diffusion equation does not require any initial conditions. This contrasts with the case of Caputo definition, in which the initial condition has to be integrated into the weak formulation in order to establish the well-posedness. Finally, thanks to the weak formulation, we are able to construct an efficient numerical method for solving the space-time fractional diffusion problem. AMS subject classifications: 35S10, 35A05, 65M70, 65M12

X. J. Li

Papers

Existence and Uniqueness of the Weak Solution of the Space-Time Fractional Diffusion Equation and a Spectral Method Approximation