The parareal algorithm for Caputo-Hadamard fractional differential equations

Tingting He; Jian Lu; Min Li

doi:10.3934/cac.2024021

Article Contents

2024, Volume 2, Issue 4: 432-453. Doi: 10.3934/cac.2024021

This issue Previous Article The linearly backward Milstein method with truncated Wiener process for the stochastic SIS epidemic model Next Article

The parareal algorithm for Caputo-Hadamard fractional differential equations

1.
School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China
2.
Center for Mathematical Sciences, China University of Geosciences, Wuhan 430074, China
3.
School of Statistics and Apllied Mathematics, Anhui University of Finance & Economics, Bengbu 233030, China

^*Corresponding author: Min Li
^*Corresponding author: Min Li

Received: October 2024

Revised: November 2024

Early access: November 29, 2024

Published online: November 29, 2024

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Abstract

This paper investigates the parareal algorithm for Caputo-Hadamard fractional differential equations. Due to the nonlocal nature of fractional-order derivatives, the direct application of the parareal algorithm is not feasible. To address this challenge, we decompose the fractional operator into historical and local components and utilize two distinct local time integrators, along with auxiliary variables, to approximate the historical components. The numerical results demonstrate that the parareal algorithm employing a local time integrator based on Gauss-Jacobi kernel compression converges faster than the parareal algorithm utilizing a local time integrator based on kernel compression. Notably, when $ \alpha $ is below a certain critical threshold, the parareal algorithm with a local time integrator based on Gauss-Jacobi kernel compression exhibits faster convergence than the parareal algorithm that employs a local time integrator based on the sum-of-exponentials.

Keywords:

Caputo-Hadamard fractional differential equations,
parareal algorithm,
local time-integrators,
auxiliary variables.

Mathematics Subject Classification: Primary: 65R20, 65Y05; Secondary: 45D05.

Citation:

FullText(HTML)

Figure 1. Compare the effect of parameter $ \alpha $ on the convergence speeds of four parareal algorithms for Example 5.1. Here, $ T = 20 $, $ N = 200 $, $ \mu = 100 $, $ J = 20 $

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Figure 2. Compare the impact of parameters $ \mu $ (left) and $ J( = \frac{\Delta T}{\Delta t}) $ (right) on the convergence speeds of four parareal algorithms for Example 5.1. Here, $ T = 20 $, $ N = 200 $

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Figure 3. Compare the effect of parameter $ \alpha $ on the convergence speeds of four parareal algorithms for Example 5.2. Here, $ T = 20 $, $ N = 200 $, $ m = 3 $, $ J = 20 $

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Figure 4. Compare the effect of parameter $ J $ on the convergence speeds of four parareal algorithms for Example 5.2. Here, $ T = 20 $, $ N = 200 $, $ m = 3 $

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Figure 5. Compare the effect of parameter $ m $ on the convergence speeds of four parareal algorithms for Example 5.2. Here, $ T = 20 $, $ N = 200 $, $ J = 20 $, $ \alpha = 0.1 $

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Figure 6. Compare the effect of parameter $ \gamma $ on the convergence speeds of four parareal algorithms for Example 5.2. Here, $ T = 20 $, $ N = 200 $, $ m = 3 $, $ \alpha = 0.1 $, $ J = 20 $

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Figure 7. The comparison of the convergence speeds of the parareal algorithm based on kernel compression [25] and Parareal Ⅱ-Rectangle in Example 5.1 with parameter $ \mu $ (left) and Example 5.2 with parameter $ \gamma $ (right). Here $ T = 20 $, $ J = 20 $, $ N = 200 $, $ m = 3 $, $ \alpha = 0.1 $

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Figure 8. The comparison of the convergence speeds of the parareal algorithm based on kernel compression [25] and Parareal Ⅱ-Rectangle in Example 5.1 (left) and Example 5.2 (right) with parameter $ \alpha $. Here $ T = 20 $, $ J = 20 $, $ N = 200 $, $ m = 3 $, $ \mu = 100 $

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