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A high-order multi-resolution wavelet method for nonlinear systems of differential equations

Author

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  • Ahsan, Muhammad
  • Lei, Weidong
  • Bohner, Martin
  • Khan, Amir Ali

Abstract

In this article, the applications of the new Haar wavelet collocation methods called as Haar wavelet collocation method (HWCM) and higher-order Haar wavelet collocation method (H-HWCM) are developed for the solution of linear and nonlinear systems of ordinary differential equations. The proposed H-HWCM is compared with a variety of other methods including the well-known HWCM. The quasi-linearization technique is introduced in the nonlinear cases. The stability and convergence of both techniques is studied in detail, which are the important parts to analyze the proposed methods. The efficiency of the methods is illustrated with certain numerical examples, but the H-HWCM is more accurate with faster convergence than the HWCM and other methods reported in the literature.

Suggested Citation

  • Ahsan, Muhammad & Lei, Weidong & Bohner, Martin & Khan, Amir Ali, 2024. "A high-order multi-resolution wavelet method for nonlinear systems of differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 215(C), pages 543-559.
    • Handle: RePEc:eee:matcom:v:215:y:2024:i:c:p:543-559
    DOI: 10.1016/j.matcom.2023.08.032
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    References listed on IDEAS

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    1. Xuan Liu & Muhammad Ahsan & Masood Ahmad & Muhammad Nisar & Xiaoling Liu & Imtiaz Ahmad & Hijaz Ahmad, 2021. "Applications of Haar Wavelet-Finite Difference Hybrid Method and Its Convergence for Hyperbolic Nonlinear Schr ö dinger Equation with Energy and Mass Conversion," Energies, MDPI, vol. 14(23), pages 1-17, November.
    2. Ahsan, Muhammad & Ahmad, Imtiaz & Ahmad, Masood & Hussian, Iltaf, 2019. "A numerical Haar wavelet-finite difference hybrid method for linear and non-linear Schrödinger equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 165(C), pages 13-25.
    3. Hsiao, Chun-Hui & Wang, Wen-June, 2001. "Haar wavelet approach to nonlinear stiff systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 57(6), pages 347-353.
    4. Hsiao, Chun-Hui, 1997. "State analysis of linear time delayed systems via Haar wavelets," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 44(5), pages 457-470.
    5. Norberg, Ragnar, 1995. "Differential equations for moments of present values in life insurance," Insurance: Mathematics and Economics, Elsevier, vol. 17(2), pages 171-180, October.
    6. Ahsan, Muhammad & Bohner, Martin & Ullah, Aizaz & Khan, Amir Ali & Ahmad, Sheraz, 2023. "A Haar wavelet multi-resolution collocation method for singularly perturbed differential equations with integral boundary conditions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 166-180.
    7. Nazir, Shah & Shahzad, Sara & Wirza, Rahmita & Amin, Rohul & Ahsan, Muhammad & Mukhtar, Neelam & García-Magariño, Iván & Lloret, Jaime, 2019. "Birthmark based identification of software piracy using Haar wavelet," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 166(C), pages 144-154.
    8. Hsiao, C.H., 2004. "Haar wavelet approach to linear stiff systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 64(5), pages 561-567.
    9. Zaheer-ud-Din & Muhammad Ahsan & Masood Ahmad & Wajid Khan & Emad E. Mahmoud & Abdel-Haleem Abdel-Aty, 2020. "Meshless Analysis of Nonlocal Boundary Value Problems in Anisotropic and Inhomogeneous Media," Mathematics, MDPI, vol. 8(11), pages 1-19, November.
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