Singular perturbation problems (SPPs) manifest in various domains of applied mathematics and engineering, including control theory, fluid flow at high Reynolds or Hartmann numbers, heat and mass transfer, chemical reaction in spark ignition turbocharger engine, wave dynamics in coastal and ocean engineering, semiconductor devices, nuclear physics, etc. This study 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 framework. Finally, the study concluded that the proposed algorithm offers a convenient and reliable approach for effectively addressing high-order reaction-diffusion Singular Perturbation Boundary Value Problems (SPBVPs) with a discontinuous source term in a new asymptotic semi-analytical numerical framework. Future research will focus on extending the method’s capabilities to encompass nonlinear problems, a wider variety of higher-order SPBVPs with non-smooth variable coefficients and discontinuous source terms, and more applications with physical interpretation of the parameters and solution behaviour.
Author (s) Details
Essam R. El-Zahar
epartment of Mathematics, College of Science and Humanities in Al-Kharj,
Prince Sattam bin Abdulaziz University, Al-Kharj, 11942, Saudi Arabia and
Department of Basic Engineering Sciences, Faculty of Engineering, Menoufia
University, Shebin El-Kom, 32511, Egypt.
Ghaliah F. Al-Boqami
Department of Mathematics, College of Science and Humanities in Al-Kharj,
Prince Sattam bin Abdulaziz University, Al-Kharj, 11942, Saudi Arabia.
Haifa S. Al-Juayd
Department of Mathematics, College of Science and Humanities in Al-Kharj,
Prince Sattam bin Abdulaziz University, Al-Kharj, 11942, Saudi Arabia.
Please see the book here:- https://doi.org/10.9734/bpi/mcsru/v5/5126
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