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

Malik Zaka Ullah, Stefano Serra-Capizzano and Fayyaz Ahmad

Applied Mathematics and Computation, 2015, vol. 250, issue C, 249-259

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.

Keywords: Nonlinear systems; Nonlinear ordinary differential equations; Higher order Frechet derivative; Multi-step (search for similar items in EconPapers)
Date: 2015
References: View complete reference list from CitEc
Citations: View citations in EconPapers (4)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:250:y:2015:i:c:p:249-259

DOI: 10.1016/j.amc.2014.10.103

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