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Approximate Analytical Solutions for Strongly Coupled Systems of Singularly Perturbed Convection–Diffusion Problems

Author

Listed:
  • Essam R. El-Zahar

    (Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
    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-Juaydi

    (Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia)

Abstract

This work presents a reliable algorithm to obtain approximate analytical solutions for a strongly coupled system of singularly perturbed convection–diffusion problems, which exhibit a boundary layer at one end. The proposed method involves constructing a zero-order asymptotic approximate solution for the original system. This approximation results in the formation of two systems: a boundary layer system with a known analytical solution and a reduced terminal value system, which is solved analytically using an improved residual power series approach. This approach combines the residual power series method with Padé approximation and Laplace transformation, resulting in an approximate analytical solution with higher accuracy compared to the conventional residual power series method. In addition, error estimates are extracted, and illustrative examples are provided to demonstrate the accuracy and effectiveness of the method.

Suggested Citation

  • Essam R. El-Zahar & Ghaliah F. Al-Boqami & Haifa S. Al-Juaydi, 2024. "Approximate Analytical Solutions for Strongly Coupled Systems of Singularly Perturbed Convection–Diffusion Problems," Mathematics, MDPI, vol. 12(2), pages 1-24, January.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:2:p:277-:d:1319357
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    References listed on IDEAS

    as
    1. Essam R. El-Zahar, 2016. "Piecewise Approximate Analytical Solutions of High-Order Singular Perturbation Problems with a Discontinuous Source Term," International Journal of Differential Equations, Hindawi, vol. 2016, pages 1-12, November.
    2. Mubashir Qayyum & Qursam Fatima & Muhammad Sohail & Essam R. El-Zahar & K. C. Gokul & Angel Manuel Ramos, 2022. "Extended Residual Power Series Algorithm for Boundary Value Problems," Mathematical Problems in Engineering, Hindawi, vol. 2022, pages 1-14, September.
    3. Lufeng Yang, 2019. "Rational Spectral Collocation Combined with the Singularity Separated Method for a System of Singularly Perturbed Boundary Value Problems," Mathematical Problems in Engineering, Hindawi, vol. 2019, pages 1-12, November.
    4. Fang Chen & Qing-Quan Liu, 2015. "Adomian Decomposition Method Combined with Padé Approximation and Laplace Transform for Solving a Model of HIV Infection of CD4 + T Cells," Discrete Dynamics in Nature and Society, Hindawi, vol. 2015, pages 1-7, September.
    5. Naik, Parvaiz Ahmad & Zu, Jian & Ghoreishi, Mohammad, 2020. "Estimating the approximate analytical solution of HIV viral dynamic model by using homotopy analysis method," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    6. Chein-Shan Liu & Essam R. El-Zahar & Chih-Wen Chang, 2022. "Higher-Order Asymptotic Numerical Solutions for Singularly Perturbed Problems with Variable Coefficients," Mathematics, MDPI, vol. 10(15), pages 1-20, August.
    7. Brahim Benhammouda & Hector Vazquez-Leal & Luis Hernandez-Martinez, 2014. "Modified Differential Transform Method for Solving the Model of Pollution for a System of Lakes," Discrete Dynamics in Nature and Society, Hindawi, vol. 2014, pages 1-12, September.
    8. Brahim Benhammouda & Hector Vazquez-Leal & Arturo Sarmiento-Reyes, 2014. "Modified Reduced Differential Transform Method for Partial Differential-Algebraic Equations," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-9, November.
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