Sylvester Equations and the numerical solution of partial fractional differential equations

Abstract

We develop a new matrix-based approach to the numerical solution of partial differential equations (PDE) and apply it to the numerical solution of partial fractional differential equations (PFDE). The proposed method is to discretize a given PFDE as a Sylvester Equation, and parameterize the integral surface using matrix algebra. The combination of these two notions results in an algorithm which can solve a general class of PFDE efficiently and accurately by means of an $\mathcal{O}(n^3)$ algorithm for solving the Sylvester Matrix Equation (over an $m\times n$ grid with $m\sim n$). The proposed parametrization of the integral surface allows for the solution with the more general Robin boundary conditions, and allows for high-order approximations to derivative boundary conditions. To achieve our ends, we also develop a new matrix-based approximation to fractional order derivatives. The proposed method is demonstrated by the numerical solution of the fractional diffusion equation with fractional derivatives in both the temporal and spatial directions.

Introduction

In the Fractional Calculus the most important problems are perhaps the Partial Fractional Differential Equations (PFDE), since they can model time varying problems in one or more dimensions such as anomalous diffusion. Partial Differential Equations (PDE) are typically difficult to solve [1], and their fractional order relatives, even more so [2], [3], [4]. Regardless of whether the PFDE can be solved analytically or not, of great importance is an efficient and accurate numerical solution. Even when exact solutions are available, it does not imply that they are straightforward or practical to compute. In terms of numerical methods for PFDE, the most common methods appear to be based on the classical methods of finite differences [5]. Some examples of finite difference methods for fractional order equations can be found in [6], [7], [8], [9], [10]. They differ mainly in their difference based approximations to the fractional derivatives. But as with typical finite difference methods, a specific algorithm must be developed for each type of equation, and type of boundary condition. Further, these stepping-type methods also have to deal with the issue of stability. Typically implicit methods for ODE are more stable, but are more difficult to compute. In either case, the minimum step-size for stability can be determined analytically, but typically the step sizes are small. Note also that stability does not imply accuracy, especially when approximating fractional derivatives. Other interesting methods for numerically solving PFDE include the numerical inversion of Laplace Transforms [11], but again such a solution must be developed specifically for one given problem. There are several methods in the literature, which only address a single type of equation, and hence lack generality. An interesting alternative approach which resolves these issues is the matrix-based approach to solving differential equations [12], [13], [14]. The motivation behind their development is the fact that the numerical approximation to fractional order derivatives takes a natural matrix form, specifically, a lower-triangular form. The method in [13] was developed for solving ordinary fractional differential equations (FDE), and solves the problem of stability by formulating the FDE as a least squares minimization subject to linear constraints. The method can therefore be classified as a global-implicit method. There exist matrix methods for ordinary differential equations [15], some for PDE [16], as well as a method for PFDE [14]. Podlubny et al. [14] proposed a matrix based method for PFDE that uses matrices of fractional differences and discretizes the PFDE over an $m\times n$ evenly spaced grid as a linear system of mn equations. The fractional derivative matrices over a single dimension most often have an upper or lower triangular structure, but in the case of certain symmetric operators, such as the Riesz potential, they can also be full matrices. These matrices are thereby non-sparse, in contrast to those one might come across using finite differences for integer order problems. In the integer order case, vectorization of the set of equations yields an $mn\times mn$ sparse coefficient matrix of the set of equations; for the fractional case, this $mn\times mn$ matrix is densely populated. Special sparse methods are used to accelerate the solution of sparse matrix problems (e.g., the LSQR method [17]), however, for the fractional order problems, the dense set of equations must be solved using standard methods. Further, the method depends on the boundary conditions being homogeneous Dirichlet boundary conditions with an initial distribution of $f(x)=0$, and hence any non-homogeneous problem must be analytically transformed beforehand. The method does not treat any other form of boundary condition, such as the Neumann type (derivative). Difficulties arise also when treating a large number of grid points due to the fact that the coefficient matrix of the system of equations is $mn\times mn$, and thereby has $m^2{n}^2$ entries. This leads to long computation times, and also severely limits the resolution of the desired solution. In addition, fractional differences require a relatively small step size to yield accurate results [2].

In this paper, we propose a framework for the numerical solution of PFDE based on the matrix-based parametrization of integral surfaces and the discretization of PFDE as Sylvester Equations. A key component is a new method for the approximation of fractional derivatives which was used successfully in the numerical solution of ordinary fractional differential equations [13]. The method of applying Sylvester Equations has already been applied successfully to the solution of certain inverse problems in PDE over two-dimensional domains [18]. The new method proposed herein makes the following contributions to the numerical solution of Partial Fractional Differential Equations¹:

• The main contribution is the discretization of PFDE in terms of Sylvester Equations. For a grid size of $m\times n$ with $m\sim n$, this leads to an $\mathcal{O}(n^3)$ algorithm for the numerical solution of the PFDE. State-of-the-art methods are $\mathcal{O}(n^6)$, and this therefore represents an orders-of-magnitude improvement in computation time. The Sylvester Equation approach is a global-implicit method and resolves questions of stability which are present in finite-difference type methods.

• Solutions can be computed with the more general Robin boundary conditions, i.e., a linear combination of Dirichlet and Neumann boundary conditions. This is accomplished by a new method for parameterizing integral surfaces, which allows for high-order derivative approximation for boundary conditions. The method can be generalized to other types of boundary conditions, such as constraints on fractional derivatives.

• The method is demonstrated on the diffusion equation, but can be equally well applied to any equation of the evolution-type such as the wave equation. The same approach can also be used for potential-type problems, and it could also be used as a sub-step for solving nonlinear PFDE.

• The method is derived for arbitrary grid spacing, meaning a denser spacing of points can be used in specific regions to yield a more accurate solution without significant increases in computational load.

There are two main sections of the paper: the first describes some of the fundamental numerical concepts used to approximate fractional derivatives, as well as some fundamental properties of Sylvester Equations; the second then describes the discretization of integral surfaces, PFDE, and the solution of the resulting equations. The paper concludes with numerical testing of the new algorithm.