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

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 sch...

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Published in:Applied mathematics and computation Vol. 250; pp. 249 - 259
Main Authors: Ullah, Malik Zaka, Serra-Capizzano, Stefano, Ahmad, Fayyaz
Format: Journal Article Publication
Language:English
Published: Elsevier Inc 01.01.2015
Subjects:
Computation
Derivatives
Differential equations
Dynamical systems
Equacions diferencials i integrals
Equacions diferencials ordinàries
Frechet derivative
Higher order
Higher order Frechet derivative
Iterative methods
Jacobians
Matemàtiques i estadística
Mathematical models
Multi-step
Nonlinear dynamics
Nonlinear ordinary differential equations
Nonlinear systems
Ordinary differential equations
Polynomials
Sistemes no lineals
Solving systems
Àrees temàtiques de la UPC
Multi-step
Nonlinear systems
Nonlinear ordinary differential equations
Higher order Frechet derivative
ISSN:0096-3003, 1873-5649, 1873-5649
Online Access:Get full text
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Summary: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.
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ISSN:0096-3003
1873-5649
1873-5649
DOI:10.1016/j.amc.2014.10.103