Subordination based approximation of Caputo fractional propagator and related numerical methods

TL;DR

This work develops an exponentially convergent numerical method for the Caputo fractional propagator $S_\alpha(t)$ by combining a subordination identity with contour-based sinc-quadrature, enabling the representation of $S_\alpha(t)$ in terms of $S_\beta(t)$ for $\beta\in[\alpha,2(1-\varphi_s/\pi))$. The approach decouples time dependence from the operator $A$, allowing resolvent reuse across multiple $\alpha$ and improving convergence, especially for small $\alpha$, while maintaining stability, parallelism, and robustness to limited spatial smoothness. A provable error bound $\|S_\alpha(t)x - \widetilde{S}_{\alpha}^{N}(\beta,t)x\| \le C_1 \frac{(1+e^{a_m t})}{\kappa \beta} e^{-c\sqrt{\kappa\beta N}} \|A^{\kappa}x\|$ demonstrates exponential convergence in $\sqrt{N}$, with constants depending on angular size $\omega(\Theta)$ and spectral data of $A$. Applications to direct and inverse problems for fractional Cauchy problems show rapid convergence and substantial reductions in resolvent evaluations, including large gains in fractional-order identification tasks, highlighting the method's practical impact for efficient and accurate fractional dynamics simulations.

Abstract

In this work, we propose an exponentially convergent numerical method for the Caputo fractional propagator $S_α(t)$ and the associated mild solution of the Cauchy problem with time-independent sectorial operator coefficient $A$ and Caputo fractional derivative of order $α\in (0,2)$ in time. The proposed methods are constructed by generalizing the earlier developed approximation of $S_α(t)$ with help of the subordination principle. Such technique permits us to eliminate the dependence of the main part of error estimate on $α$, while preserving other computationally relevant properties of the original approximation: native support for multilevel parallelism, the ability to handle initial data with minimal spatial smoothness, and stable exponential convergence for all $t \in [0, T]$. Ultimately, the use of subordination leads to a significant improvement of the method's convergence behavior, particularly for small $α< 0.5$, and opens up further opportunities for efficient data reuse. To validate theoretical results, we consider applications of the developed methods to the direct problem of solution approximation, as well as to the inverse problem of fractional order identification.

Figures (4)

• Figure 1: Schematic plot of the domain $D \equiv D_d$ in which the parametrized integrand $\mathcal{F}_{\alpha,1}(\beta,t,\xi)$ remains analytic and exponentially decaying for any $t \in [0, T]$ (a); and the image of $D_d$ under the mapping $v \to z(v)$ defined by $\Gamma_I$, along with the "forbidden" regions of complex plane indicated by "beige" color (b). The parameters of $\Gamma_I$: $\alpha =0.7$, $\beta=1$, $\varphi_s = {\pi}/{6}$, $\phi_c=0.35\pi$.
• Figure 1: Sup-norm error of the approximate solution to fractional Cauchy problem \ref{['eq:FCP_DEBC']}, \ref{['eq:FCP_ex1_hom_A']}, \ref{['eq:FCP_ex1_hom_IV']} with $\delta = 0$, $f(t) = 0$, $T=1$ and the angular size: (a) $\omega(\Theta) = \omega_c$; (b) $\omega(\Theta) = \omega_\star$.
• Figure 2: Sup-norm error of the approximate solution to fractional Cauchy problem \ref{['eq:FCP_DEBC']}, \ref{['eq:FCP_ex1_hom_A']}, with $T=1$, $\omega(\Theta) = \omega_\star$ and: (a) $u_0 = u_1 = 0$, $f(t) = \sin{\pi x} + t\sin{4 \pi x}$; (b) $u_0$, $u_1$ and $f(t)$ calculated from the exact solution $u(t) = x^2 (x - 1) \left(x - t ^ 2 + \tfrac{1}{2} \right )$.
• Figure 3: Result of the fractional order identification experiments. (a) Absolute fitting error $\mathrm{err}_{\alpha}$ for 1000 values of $\alpha$, uniformly randomly distributed on $[0.1, 1.6]$; (b) Measured data along with the fitted approximate solution for $x = \pi/10$, $\mathcal{T} = \{0, 1/40, \ldots, 1\}$.

Theorems & Definitions (19)

• Definition 1.1
• Definition 1.2: Pruess2013
• Lemma 2.1
• Proof 1
• Proposition 2.2: McLean2010Sytnyk2023
• Corollary 2.3
• Proof 2
• Lemma 3.1
• Proof 3
• Lemma 3.2