A two-stage fourth-order modified explicit Euler/Crank-Nicolson approach for a time-variable fractional transport model

Eric Ngondiep; Areej A. Binsultan; Ibtisam M. Aldawish; Eric Ngondiep; Areej A. Binsultan; Ibtisam M. Aldawish

doi:10.3934/math.2025808

AIMS Mathematics

2025, Volume 10, Issue 8: 18123-18155. doi: 10.3934/math.2025808

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Research article

A two-stage fourth-order modified explicit Euler/Crank-Nicolson approach for a time-variable fractional transport model

Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), 90950 Riyadh, Saudi Arabia

Received: 18 May 2025 Revised: 16 July 2025 Accepted: 22 July 2025 Published: 11 August 2025
MSC : 65M06, 65M12

This paper considered a two-stage fourth-order modified explicit Euler/Crank-Nicolson numerical method for solving a time-variable fractional transport equation subject to suitable initial and boundary conditions. Both stability and error estimates of the new approach were deeply analyzed in the $ L^{\infty}(0, T; L^{2}) $-norm. The analysis suggests that the proposed technique is unconditionally stable and converges with order $ O(\max\{k^{2-\overline{\beta}}, k^{\frac{3}{2}}\}+h^{4}) $, where $ \overline{\beta} $ is a positive constant satisfying $ 0 < \overline{\beta} < 1 $, $ h $ and $ k $ are the space step and time step, respectively. This result indicates that the two-stage fourth-order formulation is fast and efficient. Numerical experiments were performed to verify the unconditional stability and convergence rate of the developed algorithm.
Citation: Eric Ngondiep, Areej A. Binsultan, Ibtisam M. Aldawish. A two-stage fourth-order modified explicit Euler/Crank-Nicolson approach for a time-variable fractional transport model[J]. AIMS Mathematics, 2025, 10(8): 18123-18155. doi: 10.3934/math.2025808

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Abstract

This paper considered a two-stage fourth-order modified explicit Euler/Crank-Nicolson numerical method for solving a time-variable fractional transport equation subject to suitable initial and boundary conditions. Both stability and error estimates of the new approach were deeply analyzed in the $ L^{\infty}(0, T; L^{2}) $-norm. The analysis suggests that the proposed technique is unconditionally stable and converges with order $ O(\max\{k^{2-\overline{\beta}}, k^{\frac{3}{2}}\}+h^{4}) $, where $ \overline{\beta} $ is a positive constant satisfying $ 0 < \overline{\beta} < 1 $, $ h $ and $ k $ are the space step and time step, respectively. This result indicates that the two-stage fourth-order formulation is fast and efficient. Numerical experiments were performed to verify the unconditional stability and convergence rate of the developed algorithm.

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