Piecewise approximate analytical solutions of high-order reaction-diffusion singular perturbation problems with boundary and interior layers

Essam R. El-Zahar; Ghaliah F. Al-Boqami; Haifa S. Al-Juaydi; Essam R. El-Zahar; Ghaliah F. Al-Boqami; Haifa S. Al-Juaydi

doi:10.3934/math.2024756

AIMS Mathematics

2024, Volume 9, Issue 6: 15671-15698. doi: 10.3934/math.2024756

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Research article

Piecewise approximate analytical solutions of high-order reaction-diffusion singular perturbation problems with boundary and interior layers

1.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia; Galiahfr@gmail.com, hs.aljuaydi@psau.edu.sa
2.
Department of Basic Engineering Sciences, Faculty of Engineering, Menoufia University, Shebin El-Kom 32511, Egypt

Received: 03 March 2024 Revised: 02 April 2024 Accepted: 09 April 2024 Published: 30 April 2024
MSC : 34B08, 34D15, 35A22, 44A10

This work aims to present a reliable algorithm that can effectively generate accurate piecewise approximate analytical solutions for third- and fourth-order reaction-diffusion singular perturbation problems. These problems involve a discontinuous source term and exhibit both interior and boundary layers. The original problem was transformed into a system of coupled differential equations that are weakly interconnected. A zero-order asymptotic approximate solution was then provided, with known asymptotic analytical solutions for the boundary and interior layers, while the outer region solution was obtained analytically using an enhanced residual power series approach. This approach combined the standard residual power series method with the Padé approximation to yield a piecewise approximate analytical solution. It satisfies the continuity and smoothness conditions and offers higher accuracy than the standard residual power series method and other numerical methods like finite difference, finite element, hybrid difference scheme, and Schwarz method. The algorithm also provides error estimates, and numerical examples are included to demonstrate the high accuracy, low computational cost, and effectiveness of the method within a new asymptotic semi-analytical numerical framewor.
- singular perturbation theory,
- boundary value problems,
- discontinuous source term asymptotic approximation,
- residual power series method,
- Padé approximant
Citation: Essam R. El-Zahar, Ghaliah F. Al-Boqami, Haifa S. Al-Juaydi. Piecewise approximate analytical solutions of high-order reaction-diffusion singular perturbation problems with boundary and interior layers[J]. AIMS Mathematics, 2024, 9(6): 15671-15698. doi: 10.3934/math.2024756

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Abstract

This work aims to present a reliable algorithm that can effectively generate accurate piecewise approximate analytical solutions for third- and fourth-order reaction-diffusion singular perturbation problems. These problems involve a discontinuous source term and exhibit both interior and boundary layers. The original problem was transformed into a system of coupled differential equations that are weakly interconnected. A zero-order asymptotic approximate solution was then provided, with known asymptotic analytical solutions for the boundary and interior layers, while the outer region solution was obtained analytically using an enhanced residual power series approach. This approach combined the standard residual power series method with the Padé approximation to yield a piecewise approximate analytical solution. It satisfies the continuity and smoothness conditions and offers higher accuracy than the standard residual power series method and other numerical methods like finite difference, finite element, hybrid difference scheme, and Schwarz method. The algorithm also provides error estimates, and numerical examples are included to demonstrate the high accuracy, low computational cost, and effectiveness of the method within a new asymptotic semi-analytical numerical framewor.

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