Mittag–Leffler stability of numerical solutions to time fractional ODEs

Abstract

The asymptotic stable region and long-time decay rate of solutions to linear homogeneous Caputo time fractional ordinary differential equations (F-ODEs) are known to be completely determined by the eigenvalues of the coefficient matrix. Very different from the exponential decay of solutions to classical ODEs, solutions of F-ODEs decay only polynomially, leading to the so-called Mittag-Leffler stability, which was already extended to semi-linear F-ODEs with small perturbations. This work is mainly devoted to the qualitative analysis of the long-time behavior of numerical solutions. By applying the singularity analysis of generating functions developed by Flajolet and Odlyzko (SIAM J. Disc. Math. 3(2), 216–240, 1990), we are able to prove that both $\mathcal {L}$1 scheme and strong A-stable fractional linear multistep methods (F-LMMs) can preserve the numerical Mittag-Leffler stability for linear homogeneous F-ODEs exactly as in the continuous case. Through an improved estimate of the discrete fractional resolvent operator, we show that strong A-stable F-LMMs are also Mittag-Leffler stable for semi-linear F-ODEs under small perturbations. For the numerical schemes based on $\alpha$-difference approximation to Caputo derivative, we establish the Mittag-Leffler stability for semi-linear problems by making use of properties of the Poisson transformation. The new results and analyses provide not only the rigorous justifications and explanations of the Mittag-Leffler stability of numerical solutions with exact decay rate, but also establish some close connection between the continuous and discrete F-ODEs. Numerical experiments are presented for several typical time fractional evolutional equations, including time fractional sub-diffusion equations and semi-linear F-ODEs. All the numerical results exhibit the typical long-time polynomial decay rate, which is fully consistent with our theoretical predictions.

Appendices

Appendix 1. Fractional calculus, Mittag-Leffler function, and Prabhakar function

The Caputo fractional derivative of order $\alpha \in (0, 1)$ is given by [29]

$$\begin{aligned} \mathcal {D}_{t}^{\alpha }y(t):= \mathcal {I}_{t}^{1-\alpha }y'(t)=(k_{1-\alpha }*y')(t)=\frac{1}{\Gamma (1-\alpha )}\int _0^t \frac{y'(s)}{(t-s)^{\alpha }}ds, t>0, \end{aligned}$$

where $\mathcal {I}_{t}^{\alpha }y(t)=(k_{\alpha }*y)(t)=\frac{1}{\Gamma (\alpha )}\int _0^t \frac{y(s)}{(t-s)^{1-\alpha }}ds$ denotes the Riemann-Liouville integral and the stand kernel $k_{\alpha }(t)=\frac{t^{\alpha -1}}{\Gamma (\alpha )}$. We recall the Mittag-Leffler functions $E_{\alpha }(z)$ and $E_{\alpha ,\beta }(z)$: $E_{\alpha }(z)=\sum _{k=0}^{\infty }\frac{z^{k}}{\Gamma (\alpha k+1)},\, E_{\alpha ,\beta }(z)=\sum _{k=0}^{\infty }\frac{z^{k}}{\Gamma (\alpha k+\beta )}$, where $\alpha ,\beta >0$ and $z\in \mathbb {C}$, which can be seen as the fractional generalization of exponential functions and occur naturally in fractional calculus; see more details in [29]. For $\alpha \in (0,1)$, these two functions have the asymptotic expansion

$$\begin{aligned} E_{\alpha ,\beta }(z)=-\sum _{k=1}^{N}\frac{1}{\Gamma (\beta -k\alpha )}\frac{1}{z^{k}}+O\left( \frac{1}{|z|^{N+1}}\right) , ~~N\in \mathbb {N}^{+},\, |z|\rightarrow \infty , \, \theta \le \arg (z)\le \pi \end{aligned}$$

(A.1)

where $\theta \in (\frac{\alpha \pi }{2}, \pi \alpha )$. According to the expansion (A.1), one can prove that [2, 3]

$$\begin{aligned} \left| E_{\alpha }(\lambda t^\alpha )\right| \le \frac{C_1(\alpha ,\lambda )}{t^{\alpha }}, \qquad \left| E_{\alpha ,\alpha }(\lambda t^\alpha )\right| \le \frac{C_2(\alpha ,\lambda )}{t^{2\alpha }}, ~~\forall t\ge t_0>0, \end{aligned}$$

(A.2)

where $\lambda \in \Lambda _{\alpha }$ and $C_1(\alpha ,\lambda ), C_2(\alpha ,\lambda )$ are real positive constants which are independent of t.

The Prabhakar function is the three-parameter generalization of Mittag-Leffler function defined as

$$\begin{aligned} E_{\alpha ,\beta }^{\gamma }(z)=\sum _{k=0}^{\infty }\frac{(\gamma )_{k}z^{k}}{k! \Gamma (\alpha k+\beta )}, \quad z\in \mathbb {C}, \end{aligned}$$

where $(\gamma )_{k}=\Gamma (\gamma +k)/\Gamma (k)$ is the Pochhammer symbol. It is enough for our purpose to restrict the parameters $\alpha ,\beta , \gamma \in \mathbb {R} ~\text {and}~\alpha >0$. In this case, $E_{\alpha ,\beta }^{\gamma }(z)$ is an entire function of order $\rho = 1/\alpha$. The key historical events and modern development and the main properties of Prabhakar function can be found in the most recent review paper [10]. We have $E_{\alpha ,\beta }^{\gamma }(z)=E_{\alpha ,\beta }(z)$ for $\gamma =1$, which recovers the Mittag-Leffler function. Generally, we have the reduction formula

$$\begin{aligned} E_{\alpha ,\beta }^{\gamma +1}(z)=\frac{ E_{\alpha ,\beta -1}^{\gamma }(z)+(1-\beta +\alpha \gamma ) E_{\alpha ,\beta }^{\gamma }(z) }{\alpha \gamma }. \end{aligned}$$

(A.3)

Appendix 2. Estimate for $D_n$ in (4.14) based on standard resolvent (4.5)

The integral path in (4.14) is firstly deformed to $\Gamma _{(r, \theta )}$ defined in (4.3) and then split into two parts, $\Gamma _{1}$ and $\Gamma _{2}$, where $\Gamma _{1}$ is the circle around the origin and $\Gamma _{2}$ is two line segments. Then, the integral is naturally divided into $D_{n} =D^{1}_{n}+ D^{2}_{n}$, where $D_{n}^{j}$ $(j=1, 2)$ means the integral carried over $z\in \Gamma _{j}$. One of the main reasons for choosing this contour is that the exponential function $e^{z t_{n}}$ will decay as $t_{n}$ increases for $z\in \Gamma _{2}$. In order to estimate $D^{j}_{n}$, we need to estimate three terms in the integral both for $z\in \Gamma _{1}$ and $z\in \Gamma _{2}$, namely, $|e^{z t_{n}}|, \Vert (h^{-\alpha } F_{\mu } (e^{-zh}) I- A)^{-1}\Vert$ and |dz|, where $F_{\mu } (z)=(1-z)^\alpha$ for F-BDF1.

We first consider $D^{1}_{n}$ for $z\in \Gamma _{1}$. Let $z=re^{i\theta }=\frac{1}{t_{n}}e^{i\theta }$, where $-\phi \le \theta \le \phi$ and $\phi \in (\pi /2, \pi )$. By direct calculation, we can get that $|e^{z t_{n}}|=|e^{\frac{1}{t_{n}}(\cos (\theta )+i\sin (\theta )) t_{n}}|= |e^{\cos (\theta )} |\le e$ and $\int _{\Gamma _{1}}|dz|=\ell (\Gamma _{1}) \le 2\pi r\le \frac{C}{t_{n}}$, where $\ell (\Gamma ^{1})$ is the length of $\Gamma _{1}$. According to (4.5), we have $\Vert (h^{-\alpha } F_{\mu } (e^{-zh}) I- A)^{-1}\Vert \le C |\frac{h^{\alpha }}{(1-e^{-zh})^{\alpha }}|$. On the other hand, there exist constants $c_1, c_2>0$ such that $c_{1}|zh|\le |1-e^{-zh}|\le c_{2}|zh|$ for $z\in \Gamma _{1}$ [13], which yields that $|\frac{h^{\alpha }}{(1-e^{-zh})^{\alpha }}| \le \frac{C}{|z|^{\alpha }}=\frac{C}{t_{n}^{\alpha }}.$ Combining the above bounds lead to

$$\begin{aligned} \Vert D^{1}_{n} \Vert \le \frac{e}{2\pi } \cdot \frac{C}{t_{n}^{\alpha }}\cdot \frac{C}{t_{n}} =O(t_{n}^{-\alpha -1}). \end{aligned}$$

(B.1)

We now estimate $D^{2}_{n}$. Let $z=re^{i\phi }$, where ${1}/{t_{n}}\le r\le \pi /(h\sin (\phi ))$ and $\phi \in (\pi /2, \pi )$. It follows by direct calculation that $|e^{z t_{n}}|=|e^{r(\cos (\phi )+i\sin (\phi )) t_{n}}| =|e^{r\cos (\phi ) t_{n}}|= e^{r\cos (\phi ) t_{n}}$. Similarly to the above estimation of $D^{1}_{n}$, there exist constants $c_1, c_2>0$ such that $c_{1}|zh|\le |1-e^{-zh}|\le c_{2}|zh|$ for $z\in \Gamma _{2}$, which yields that $|\frac{h^{\alpha }}{(1-e^{-zh})^{\alpha }}| \le \frac{C}{|z|^{\alpha }}=\frac{C}{r^{\alpha }}.$ Combining the above bounds and the simple transformation $s=rt_{n}$, we can derive

$$\begin{aligned} \Vert D^{2}_{n} \Vert\le & {} \frac{1}{2\pi } \int _{\frac{1}{t_{n}}}^{\frac{\pi }{h\sin (\phi )}} e^{r\cos (\phi ) t_{n}} \cdot \frac{C}{r^{\alpha }} dr = \frac{C}{2\pi } \int _{\frac{t_{n}}{t_{n}}}^{\frac{\pi }{h\sin (\phi )} t_{n}} e^{s\cos (\phi )} \cdot \frac{t_{n}^\alpha }{s^{\alpha }} \cdot \frac{1}{t_n}ds\nonumber \\= & {} \frac{C}{2\pi } \cdot \frac{1}{t_{n}^{1-\alpha }} \int _{1}^{\frac{\pi }{h\sin (\phi )} t_{n}} e^{s\cos (\phi )} s^{-\alpha } ds=O(t_{n}^{\alpha -1}). \end{aligned}$$

(B.2)

Therefor, we conclude that $\Vert D_{n}\Vert \le \Vert D^{1}_{n} \Vert +\Vert D^{2}_{n} \Vert =O(t_{n}^{\alpha -1})$ as $t_{n}\rightarrow \infty .$

From the above analysis, we see that $\Vert D^{1}_{n} \Vert =O(t_{n}^{-\alpha -1})$. However, for $D^{2}_{n}$ with $z\in \Gamma _{2}$, this method only gives that $\Vert D^{2}_{n} \Vert =O(t_{n}^{\alpha -1})$, hence resulting in the estimate $\Vert D_{n}\Vert =O(t_{n}^{\alpha -1})$ or a reduction of the decay rate. In this case, the sum of the series $\sum _{n=1}^{\infty }\Vert D_{n}\Vert$ diverges. This is not satisfactory and cannot be used to establish the long-time decay of numerical solutions.