Stability and Convergence Analysis of a Class of Continuous Piecewise Polynomial Approximations for Time-Fractional Differential Equations

Fractional calculus, as a generalization of classical calculus, has been an intriguing topic for many famous mathematicians since the end of the 17th century. During the last 4 decades, many scholars have been working on the development of theory for fractional derivatives and integrals, found their ways in the world of fractional calculus and their applications. For more detailed information on the historical background, we refer the interested reader to the following books: [6, 21, 22, 34‐36, 38] and [20]. As an application of fractional calculus, differential equations possessing terms with fractional derivatives in the space- or time- or space-time direction have become very important in many areas. Particularly, in recent years a huge amount of interesting and surprising fractional models have been proposed. Here, we mention just a few typical applications: in the theory of Hankel transforms [15], in financial models [40, 42], in elasticity theory [5], in medical applications [23, 39], in geology [8, 27], in physics [7, 10, 33] and many more.

Similar to the work for ordinary differential equations, investigation of numerical methods for time-fractional differential equations (tfDEs) started its development. This paper will consider numerical approaches to tfDEs of the formwith initial condition \(u(0)=u_{0}\), where the operator \({^{C}}D^{\alpha }\) denotes Caputo fractional derivative of order \(\alpha \), whose definition will be given in Definition 2.1 in the next section. As shown in [12], if f(t, u(t)) is continuous and satisfies the Lipschitz condition with respect to u, then equation (1.1) possesses a unique solution in C[0, T]. For this case, (1.1) combined with the initial condition is equivalent to the following Volterra-type integral equation:With respect to numerical approximations for (1.2), two general approaches, called product integration method and fractional linear multistep methods, have been widely discussed. In these cases, a general discrete form of (1.2) is written aswith fixed \(m\in {\mathbb {N}}\). Fractional linear multistep methods were originally proposed in [30] in the mid eighties of the last century. This type of methods construct the convolution quadrature weights \(\{\omega _{j}\}_{j=0}^{\infty }\) satisfyingwhere \((\rho , \sigma )\) denote the classical implicit linear multistep formulae. For the motivation behind this idea we refer to [28]. [29] and [31] discuss the accuracy and stability properties of this type of methods. We can see they highly benefit from those of the corresponding multistep methods. Another more straightforward approach to generate the weights \(\{\omega _{j}\}\) and \(\{w_{n,j}\}\) is based on product integration, that is to replace the integrand \(f(\xi , u(\xi ))\) by some piecewise interpolation polynomials, and construct their fractional integrals of order \(\alpha \) as approximations of the integral in (1.2). On the accuracy and efficiency of these methods applied to some Volterra-type integral equations with irregular kernels, we can refer to [9, 11, 13, 26] and [6, 24]. In addition, [18] applies exponential integrators to fractional order problems. Generalized Adams methods and so-called m-steps methods are utilized by [1, 2].

Under the framework of product integration, recently, some new numerical approximations of the Caputo fractional derivative of order \(\alpha \in (0,1)\), named L1 method [25], L1-2 method [17], L2-\(1_\sigma \) method [4] and method [32], were proposed and applied for solving time-fractional differential equations. These methods are based on piecewise linear or quadratic interpolating polynomials approximations. In this paper, we generalize the approach by improving the degree of the piecewise polynomial to \(3\le k\le 6\) to approximate the function that possesses suitable smoothness. For this situation higher order of accuracy can be obtained. We establish local truncation errors and global errors estimates of the numerical schemes for (1.1) in detail. In addition, we mainly study the numerical stability of the L1 method, L1-2 method, method in [32] and higher-order methods proposed in this paper. We apply the technique in [31] to the investigation of the stability regions of this type of numerical methods. Further, we give rigorous proof that L1 method, L1-2 method and method in [32] possess \(A(\frac{\pi }{2})\)-stability. Numerical experiments confirm our theoretical analyses and show that this class of methods are \(A(\theta )\)-stable uniformly for \(0<\alpha <1\), and for some specific \(\alpha \), A-stability can even be obtained.

The paper is organized as follows. Section 2 introduces numerical approximations of the Caputo fractional derivative of order \(\alpha \in (0,1)\), and applies them to the discretization of problem (1.1). The local truncation errors of the proposed methods are discussed. Sections 3 and 4 respectively treat the stability and convergence of the discrete methods. In Sect. 5, numerical experiments confirm our theoretical considerations with respect to order of convergence and stability restrictions.

We first introduce fractional derivatives in the Caputo sense:

Next, we will derive a class of numerical approximations of the Caputo fractional derivative of order \(\alpha \in (0,1)\) by constructing a series of continuous piecewise polynomials. The main idea is as follows.

Let \({\mathscr {I}}=[0, T]\) be an interval and the \(M+1\) nodes \(\{t_{i}\}_{i=0}^{M}\) define a partitionAssume that \(p_{j, q}^{k}(t)\) are a class of polynomials of degree \(k\ge 1\) with compact supports \({\mathscr {I}}_{j}=[t_{j-1}, t_{j}]\). Their coefficients are uniquely determined by the following \(k+1\) conditionsHere the index q records the number of shifts of the \(k+1\) interpolating nodes \(\{t_{n}\}_{n=j-1-k}^{j-1}\), and the sign of q indicates the direction of the shift. Then we haveIf the partition (2.3) is equidistant, i.e., \(t_{n}=n\Delta t,~0\le n\le M\) and \(\Delta t=\frac{T}{M}\), then (2.5) can be written asFor convenience of notation, let \(t=t_{j-1}+s\Delta t\), we getwhere \(\left( {\begin{array}{c}s-q+r-1\\ r\end{array}}\right) \) denotes a binomial coefficient, and the r-th order backward difference operators \(\nabla ^{r}\) satisfyLetbe the space of continuous piecewise polynomials of degree at most k. On the uniform grid, we construct a class of polynomials of the formwhere \(1\le i\le k\le 6\) and \(t\in (t_{n-1}, t_{n}]\) for \(1\le n\le M\). \(\sum _{j=1}^{k-i}p_{j,k-j}^{k-1}(t)=0\) and \(\sum _{j=n-i+2}^{n}p_{j,n+1-j}^{k}(t)=0\) if \(k-i<1\) and \(n-i+2>n\), respectively. Then \(P_{i}^{k}(t)\) are considered as approximations of the function u(t) in (2.2) in the space \(C_{p}^{k}({\mathscr {I}})\).

Correspondingly, we propose the operatorfor \(t\in {\mathscr {I}}\) as an approximation to (2.2). If \(t=t_{n}\), we rewritten (2.9) aswhere \(u_{n}:=u(t_{n})\).

In the following part, we present the explicit representations of the weight coefficients \(\{w_{n,j}^{(k,i)}\}\) and \(\{\omega _{j}^{(k,i)}\}\) for \(1\le i\le k\le 3\) as examples. Note that in the case \(1\le i\le k\le 2\), the weight coefficients have been derived by [17, 25, 32] in a similar way. Here we rewrite them into the form of integrals for convenience of further theoretical analyses. First, we define a class of integrals of the formwhere \(q, r\in {\mathbb {N}}^{+}\) and \(n\in {\mathbb {Z}}\). If we denotethen the weight coefficients can be expressed asand It can be observed that when \(\alpha \rightarrow 1\), the operator \(D_{k,i}^{\alpha }u_{n}\) in (2.10) recovers the k-step BDF method.

Using (2.10), we construct the discrete schemesas approximations of Eq. (1.1). If starting values are given, then we define the local truncation errors of the n-th step bywhere u(t) is the exact solution of (1.1).

To analyse the stability of discrete schemes (2.17) with initial value \(u(0)=u_{0}\), we apply (2.9) to the test equationand obtainWe rewrite (3.2) as the formal power series formReplaced by (2.10), formula (3.3) becomeswhere \(z:=\lambda (\Delta t)^{\alpha }\). The above notations are defined as

Analogous to the \(A(\theta )\)-stability for classical ODE mentioned in [19], we define \(A(\theta )\)-stability of methods for fractional ODE.

In this section, we will establish the error estimate for (2.17). Assume that u(t) is the exact solution of (1.1), then it satisfieswhere the difference operator \(D_{k,i}^{\alpha }\) and the local truncation error \(\tau _{n}^{(k,i)}\) are defined by (2.10) and (2.18), respectively. Suppose that \(u_{n}^{(k,i)}\) is the solution of (2.17) for each k, i, we denote global errors bySubtracting (2.17) by (4.1) yieldswhere \(\delta f_{n}^{(k,i)}=f(t_{n}, u(t_{n}))-f(t_{n}, u_{n}^{(k,i)})\). From (2.10) and (4.3), we haveMultiplying \(\xi ^{n-k}\) on both sides of (4.4) and summing up for all \(n\ge k\), we obtainIt follows thatwhere

In this section, we utilize (2.10) to approximate the equations in Examples 5.1 and 5.2, and prescribe starting values exactly.

In Fig. 1a–d, we plot the truncated boundary locus curves \(\sum _{n=0}^{6000}\omega _{n}^{(k,i)}e^{\sqrt{-1}\theta n}~(0\le \theta \le 2\pi )\) for \(1\le i\le k\le 3\) and some \(\alpha \in (0,1)\). It is already known from Theorem 3.1 that the stability regions of methods (3.2) lie outside their boundary locus curves. Here, we introduce the points \(z_{n}=\lambda (\Delta t_{n})^{\alpha }, 1\le n\le 5\), where \(\Delta t_{n}=1/2^{n+6}\) denote different time steps. Tables 1 and 2 show the accuracy and convergence rates of the error \(|u(t_{M})-u_{M}^{(k,i)}|\) for Example 5.1, where \(t_{M}=1\) is fixed and \(M=2^{j}\) for \(7\le j\le 11\), \(u(t_{M})\) and \(u_{M}^{(k,i)}\) are the exact solution and computed solution for (5.1), respectively.

From Fig. 1a–d and Tables 1, 2, we can see the influence of the stability of a numerical method on global error. In Fig. 1a, the points \(z_{n}\) with \(1\le n\le 5\) all lie in the stability regions for \(\alpha =0.5\) and \(\lambda =-50\), we get \((k+1-\alpha )\)-th order of accuracy shown in Tables 1, 2. In Fig. 1b, c, \(\{z_{n}\}_{n=1}^{5}\) fall on the half line with angle \(\frac{\pi \alpha }{2}\). It is observed that when all \(\{z_{n}\}_{n=1}^{5}\) fall out of the instability region (cf. Fig. 1b), correspondingly, as shown in Tables 1, 2, the global error agrees with \((k+1-\alpha )\)-th order of accuracy. On the other hand, due to the points \(z_{4}\) and \(z_{5}\) outside the stability regions for \(k=3\) (cf. Fig. 1c), perturbation errors are magnified and accumulated significantly, which are shown in Tables 1, 2 as well. In Fig. 1d, \(\{z_{n}\}\) are chosen on the imaginary axis with pure imaginary number \(\lambda \), Theorem 3.2 tells us that all \(\{z_{n}\}\) are in the stability region for \(k=1, 2\). The error and convergence order are obtained (cf. Table 1).

As a counter example, in Fig. 1d, the point \(z_{3}\) doesn’t belong to the stability region for \(\alpha =0.98\), for this case, the errors shown in Table 2 blow up. In fact, it can be observed that for \(k=3\), methods (3.2) don’t possess \(A(\frac{\pi }{2})\)-stability when \(\alpha \) tends to 1, as it is known that BDF3 method for ODEs is not \(A(\frac{\pi }{2})\)-stable.

We use (2.17) in combination with Newton’s method for solving the nonlinear equation (5.2). Tables 3, 4, 5 and 6 show the global error \(|e_{M}^{(k,i)}|=|u(t_{M})-u_{M}^{(k,i)}|\) and orders of accuracy for Example 5.2 with different \(\mu \) and \(\alpha \), where \(t_{M}=1\) is fixed and \(\Delta t=1/M\) with \(M=2^{j},~5\le j\le 9\). Further, it is observed that \(|e_{M}^{(k,i)}|=O(\Delta t^{k+1-\alpha })\) for \(1\le i\le k\le 3\).

We have proposed a class of new high-order approximations for solving time-fractional initial value models of order \(0<\alpha <1\). Furthermore, the local truncation error estimate in terms of a smooth solution is presented. Additionally, stability and convergence analysis of these numerical methods are discussed in detail. This will promote further investigation of the proposed methods for solving time-fractional partial differential equations.