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Efficient Spectral Galerkin and Collocation Approaches Using Telephone Polynomials for Solving Some Models of Differential Equations with Convergence Analysis

Author

Listed:
  • Ramy Mahmoud Hafez

    (Department of Mathematics, Faculty of Education, Matrouh University, Cairo 51511, Egypt)

  • Hany Mostafa Ahmed

    (Department of Mathematics, Faculty of Technology and Education, Helwan University, Cairo 11281, Egypt)

  • Omar Mazen Alqubori

    (Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia)

  • Amr Kamel Amin

    (Department of Mathematics, Adham University College, Umm Al-Qura University, Makkah 28653, Saudi Arabia)

  • Waleed Mohamed Abd-Elhameed

    (Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt)

Abstract

This study presents Galerkin and collocation algorithms based on Telephone polynomials (TelPs) for effectively solving high-order linear and non-linear ordinary differential equations (ODEs) and ODE systems, including those with homogeneous and nonhomogeneous initial conditions (ICs). The suggested approach also handles partial differential equations (PDEs), emphasizing hyperbolic PDEs. The primary contribution is to use suitable combinations of the TelPs, which significantly streamlines the numerical implementation. A comprehensive study has been conducted on the convergence of the utilized telephone expansions. Compared to the current spectral approaches, the proposed algorithms exhibit greater accuracy and convergence, as demonstrated by several illustrative examples that prove their applicability and efficiency.

Suggested Citation

  • Ramy Mahmoud Hafez & Hany Mostafa Ahmed & Omar Mazen Alqubori & Amr Kamel Amin & Waleed Mohamed Abd-Elhameed, 2025. "Efficient Spectral Galerkin and Collocation Approaches Using Telephone Polynomials for Solving Some Models of Differential Equations with Convergence Analysis," Mathematics, MDPI, vol. 13(6), pages 1-29, March.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:6:p:918-:d:1609264
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    References listed on IDEAS

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    1. Clemente Cesarano & William Ramírez & Stiven Díaz & Adnan Shamaoon & Waseem Ahmad Khan, 2023. "On Apostol-Type Hermite Degenerated Polynomials," Mathematics, MDPI, vol. 11(8), pages 1-13, April.
    2. Francesco Costabile & Anna Napoli, 2012. "Collocation for High‐Order Differential Equations with Lidstone Boundary Conditions," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    3. Singh, Somveer & Patel, Vijay Kumar & Singh, Vineet Kumar, 2018. "Application of wavelet collocation method for hyperbolic partial differential equations via matrices," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 407-424.
    4. De-Gang Wang & Wen-Yan Song & Peng Shi & Hamid Reza Karimi, 2012. "Approximate Analytic and Numerical Solutions to Lane-Emden Equation via Fuzzy Modeling Method," Mathematical Problems in Engineering, Hindawi, vol. 2012, pages 1-15, November.
    5. Francesco Costabile & Anna Napoli, 2012. "Collocation for High-Order Differential Equations with Lidstone Boundary Conditions," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-20, August.
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