Finite difference methods with non-uniform meshes for nonlinear fractional differential equations

Abstract

In this article, finite difference methods with non-uniform meshes for solving nonlinear fractional differential equations are presented, where the non-equidistant stepsize is non-decreasing. The rectangle formula and trapezoid formula are proposed based on the non-uniform meshes. Combining the above two methods, we then establish the predictor–corrector scheme. The error and stability analysis are carefully investigated. At last, numerical examples are carried out to verify the theoretical analysis. Besides, the comparisons between non-uniform and uniform meshes are given, where the non-uniform meshes show the better performance when dealing with the less smooth problems.

Introduction

In recent years, growing attention has been focused on fractional differential equations because they can provide a better approach to describe the complex phenomena in nature, such as viscoelastic materials, anomalous diffusion, signal processing and control theory, etc., see [1], [16], [17], [28], [29], [32], [38]. Compared to classical integer-order differential equations, the theoretical investigations and establishment of numerical schemes for fractional-order (or fractional for brevity) versions are more complicated due to the special properties of fractional differential operators, such as the non-locality, history dependence, and/or long-range interactions [37].

In this paper, we study the numerical schemes for the following nonlinear fractional initial value problem

$$\{\begin{array}{ccc}\hfill {}_{C}D_{0,t}^{\alpha }y(t)& =f(t,y),\hfill & \hfill T\ge t>0,\\ \hfill y^{(k)}(0)& =y_0^{(k)},\hfill & \hfill k=0,1,\cdots ,n-1,\end{array}$$

where the continuous function $f(t,y)$ is nonlinear with respect to the unknown function y, n is a positive integer such that $n-1<\alpha <n$ and the initial values $y_0^{(k)}$ are assumed to be given. Here ${}_{C}D_{0,t}^{\alpha }y(t)$ is the Caputo derivative, defined by

$${}_{C}D_{0,t}^{\alpha }y(t)=\frac{1}{\mathrm{\Gamma }(n-\alpha )}\underset{0}{\overset{t}{\int }}{(t-s)}^{n-\alpha -1}{y}^{(n)}(s)ds,n-1<\alpha <n\in Z^{+}.$$

In order to ensure the existence of a unique solution of (1.1), we always assume that f satisfies Lipschitz condition with respect to the second variable, that is, $|f(t,y_1)-f(t,y_2)|\le L|y_1-y_2|$ where $L>0$. It is well known that the initial value problem (1.1) is equivalent to the following Volterra integral equation [6], [16]

$$y(t)=\sum _{j=0}^{n-1}\frac{t^j}{j!}{y}_0^{(j)}+\frac{1}{\mathrm{\Gamma }(\alpha )}\underset{0}{\overset{t}{\int }}{(t-s)}^{\alpha -1}f(s,y(s))ds,$$

in the sense that a continuous function is a solution of (1.1) if and only if it solves equation (1.2).

There have existed some studies on the numerical approaches for fractional differential equation (1.1). Early in 1986, Lubich studied the numerical approximation of fractional integrals based on a discrete convolution form and introduced the fractional linear multistep method [26], [27]. Almost at the same time, Brunner and Houwen also presented linear multistep methods for Volterra equations [2]. In 1997, Diethelm proposed an implicit algorithm for the fractional differential equations based on a quadrature formula approach in a finite-part integral sense [5]. For more details about the relationship between fractional derivative and finite-part integral, see [12], [13]. In 2000, Podlubny proposed a unification discrete form based on triangular strip matrices [33], which was further developed in [34]. In 2002, Diethelm et al. discussed an Adams-type predictor–corrector method for equation (1.2) and gave a detailed error analysis that the convergent order was $\min \{2,1+\alpha \}$ if ${}_{C}D_{0,t}^{\alpha }y(t)\in C^2[0,T]$ [7], [8]. Shortly after, Li and Tao further studied the error analysis for the fractional predictor–corrector method [20]. And in [22], Li and Zeng reviewed the finite difference methods for fractional ordinary/partial differential equations. In [24], various kinds of numerical methods for fractional differential equations have been studied thoroughly. There have also been some investigations on the stability analysis for nonlinear fractional differential equations. Ladaci and Moulayin [18] analyzed the $L^p$-stability of fractional nonlinear differential equations. Deng [4] considered the sufficient conditions for the local asymptotical stability of nonlinear fractional differential equations. After that, Li and Zeng [23] studied the numerical stability of the finite difference methods for nonlinear fractional ordinary differential equation (1.1). In very recent, a series of numerical methods for Caputo derivatives is derived and the established algorithms are applied to Caputo-type advection–diffusion equations [3], [19], [21]. In the meantime, a series of numerical methods for Riesz derivatives is established and applied to Riesz-type fractional partial differential equations [9], [10], [11]. Keshtkar et al. [15] investigated the stability of equilibria in the nonlinear fractional-order dynamical systems.

In general, the existence of a weakly singular kernel ${(t-s)}^{\alpha -1}(0<\alpha <1)$ in fractional derivative and integral makes it more difficult to get a higher-order scheme. Particularly when the solution of equation (1.1) is not suitably smooth, those methods on uniform meshes seem to have a poor convergent rate. For these reasons, numerical methods on non-uniform meshes have been placed on the agenda. Especially in recent years, finite difference schemes with non-uniform meshes for fractional differential equations have attracted increasing attention. In [36], [40], Yuste and Quintana-Murillo proposed an L1 method with non-uniform timesteps for fractional diffusion and diffusion–wave equations. In [35], Podlubny et al. studied the matrix approach on non-equidistant grids. Mustapha et al. [30], [31] used the finite difference method to a sub-diffusion equation. Lopez-Fernandez and Sauter [25] presented a generalized convolution quadrature with variable timesteps. In very recent, Zhang et al. [42] investigated the finite difference scheme for the fractional diffusion equation on non-uniform meshes. Finite difference methods with non-uniform meshes often show great advantages when dealing with less smooth problems. However, the theoretical analysis of stability and convergence of the schemes with non-uniform meshes for nonlinear fractional differential equation seems not to be derived thoroughly yet.

In this paper, we mainly focus on the stability and convergence analysis of three kinds of numerical approaches on non-uniform meshes and illustrate their suitability for non-smooth problems through numerical tests. The rest of the paper is organized as follows. In Section 2, we outline the numerical schemes on the non-uniform meshes. Detailed stability and error analysis for the derived schemes are given in Sections 3 and 4, respectively. In Section 5, numerical examples are carried out to verify the theoretical analysis and to check the capability of the derived methods for non-smooth problems. The conclusions are included in the last section.