Existence and uniqueness of strong solution for a generalized fractional coupled transport system

Jinsheng Du; Lijie Li; Van Thien Nguyen

doi:10.3934/dcdss.2024124

Article Contents

2026, Volume 22: 29-59. Doi: 10.3934/dcdss.2024124

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Existence and uniqueness of strong solution for a generalized fractional coupled transport system

1.
College of Mathematics and Information Science, Guangxi University, Nanning, 530004, China
2.
Center for Applied Mathematics of Guangxi and Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, Guangxi, China
3.
Departement of Mathematics, FPT University, Education zone, Hoa Lac high tech park, Km29 Thang Long highway, Thach That ward, Hanoi, Vietnam

^*Corresponding author: Lijie Li
^*Corresponding author: Lijie Li

Received: May 2024

Revised: July 2024

Early access: July 23, 2024

Published online: July 23, 2024

This paper is supported by the Natural Science Foundation of Guangxi Grant Nos. 2022GXNSFAA035617 and 2021GXNSFFA196004, the NNSF of China Grant Nos. 12371312 and 12371179, and Natural Science Foundation of Chongqing Grant No. CSTB2024NSCQ-JQX0033. It is also supported by the project cooperation between Guangxi Normal University and Yulin Normal University.

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Abstract

This paper investigates a generalized fractional coupled transport system in a Hilbert space, in which such system contains Caputo-Katugampola differential terms and Caputo-Katugampola integral terms, as well as fully nonlinear coupling functions. First, we transform this system into an iterative system by using a semi-discrete method in time based on the backward Eulerian difference format (i.e., Rothe method). Second, employing Gronwall's inequality and the properties of $ m $-accretive operators, we establish the unique solvability and several priori estimates of solutions for the iterative system. Finally, we present two illustrative examples to demonstrate the obtained theoretical results.

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Mathematics Subject Classification: Primary: 35K57, 35R11; Secondary: 35A01, 47B44.

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Figure 1. Numerical solution of $ u $

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Figure 2. Numerical solution of $ v $

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Figure 3. Numerical solution of $ u $

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Figure 4. Numerical solution of $ v $

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Table 1. The maximum errors of $ u $ and $ v $ with $ \alpha = 0.5, \rho = 0.5, \beta = 0.4, \gamma = 1.2, \alpha_1 = 0.8, \rho_1 = 2, \alpha_2 = 1.2, \rho_2 = 1, \beta_1 = 0.4, \gamma_1 = 1.2, J = $ 1024


$ N $	$ { }^u E_{\infty}(\tau, h) $	$ { }^v E_{\infty}(\tau, h) $
64	2.3358e-01	4.6569e-01
128	1.1464e-01	2.2773e-01
256	5.6872e-02	1.1300e-01
512	2.8362e-02	5.6461e-02
1024	1.4183e-02	2.8308e-02

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Table 2. The maximum errors of $ u $ and $ v $ with $ \alpha = 0.4, \rho = 2, \beta = 0.8, \gamma = 0.6, \alpha_1 = 0.5, \rho_1 = 3, \alpha_2 = 1.25, \rho_2 = 0.75, \beta_1 = 0.8, \gamma_1 = 0.8, J = $ 1024


$ N $	$ { }^u E_{\infty}(\tau, h) $	$ { }^v E_{\infty}(\tau, h) $
64	1.2912e-01	3.5486e-01
128	6.3160e-02	1.7175e-01
256	3.1145e-02	8.4132e-02
512	1.5416e-02	4.1457e-02
1024	7.6448e-03	2.0491e-02

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