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An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs

Author

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  • Ullah, Malik Zaka
  • Serra-Capizzano, Stefano
  • Ahmad, Fayyaz

Abstract

We developed multi-step iterative method for computing the numerical solution of nonlinear systems, associated with ordinary differential equations (ODEs) of the form L(x(t))+f(x(t))=g(t): here L(·) is a linear differential operator and f(·) is a nonlinear smooth function. The proposed iterative scheme only requires one inversion of Jacobian which is computationally very efficient if either LU-decomposition or GMRES-type methods are employed. The higher-order Frechet derivatives of the nonlinear system stemming from the considered ODEs are diagonal matrices. We used the higher-order Frechet derivatives to enhance the convergence-order of the iterative schemes proposed in this note and indeed the use of a multi-step method dramatically increases the convergence-order. The second-order Frechet derivative is used in the first step of an iterative technique which produced third-order convergence. In a second step we constructed matrix polynomial to enhance the convergence-order by three. Finally, we freeze the product of a matrix polynomial by the Jacobian inverse to generate the multi-step method. Each additional step will increase the convergence-order by three, with minimal computational effort. The convergence-order (CO) obeys the formula CO=3m, where m is the number of steps per full-cycle of the considered iterative scheme. Few numerical experiments and conclusive remarks end the paper.

Suggested Citation

  • Ullah, Malik Zaka & Serra-Capizzano, Stefano & Ahmad, Fayyaz, 2015. "An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs," Applied Mathematics and Computation, Elsevier, vol. 250(C), pages 249-259.
  • Handle: RePEc:eee:apmaco:v:250:y:2015:i:c:p:249-259
    DOI: 10.1016/j.amc.2014.10.103
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    Citations

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    Cited by:

    1. Fayyaz Ahmad & Shafiq Ur Rehman & Malik Zaka Ullah & Hani Moaiteq Aljahdali & Shahid Ahmad & Ali Saleh Alshomrani & Juan A. Carrasco & Shamshad Ahmad & Sivanandam Sivasankaran, 2017. "Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPs," Complexity, Hindawi, vol. 2017, pages 1-30, May.
    2. R. H. Al-Obaidi & M. T. Darvishi, 2022. "Constructing a Class of Frozen Jacobian Multi-Step Iterative Solvers for Systems of Nonlinear Equations," Mathematics, MDPI, vol. 10(16), pages 1-13, August.
    3. Howk, Cory L., 2016. "A class of efficient quadrature-based predictor–corrector methods for solving nonlinear systems," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 394-406.
    4. Chein-Shan Liu & Essam R. El-Zahar & Chih-Wen Chang, 2024. "Optimal Combination of the Splitting–Linearizing Method to SSOR and SAOR for Solving the System of Nonlinear Equations," Mathematics, MDPI, vol. 12(12), pages 1-24, June.

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