A New Piecewise-Spectral Homotopy Analysis Method for Solving Chaotic Systems of Initial Value Problems

A New Piecewise-Spectral Homotopy Analysis Method for Solving Chaotic Systems of Initial Value Problems

An accurate algorithm for solving initial value problems (IVPs) which are highly oscillatory is proposed. The proposed method is based on a novel technique of extending the standard spectral homotopy analysis method (SHAM) and adapting it to a sequence of multiple intervals. In this new application...

Full description

Bibliographic Details
Main Authors: H. Saberi Nik, Sohrab Effati, Sandile S. Motsa, Stanford Shateyi
Format: Article
Language:English
Published: Hindawi Limited 2013-01-01
Series:Mathematical Problems in Engineering
Online Access:http://dx.doi.org/10.1155/2013/583193
id doaj-e9f8b239665f4e34a65d80dc3d843d79
record_format Article
spelling doaj-e9f8b239665f4e34a65d80dc3d843d792020-11-24T23:52:32ZengHindawi LimitedMathematical Problems in Engineering1024-123X1563-51472013-01-01201310.1155/2013/583193583193A New Piecewise-Spectral Homotopy Analysis Method for Solving Chaotic Systems of Initial Value ProblemsH. Saberi Nik0Sohrab Effati1Sandile S. Motsa2Stanford Shateyi3Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, IranDepartment of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, IranSchool of Mathematics, Computer Science and Statistics, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South AfricaDepartment of Mathematics, University of Venda, P Bag X5050, Thohoyandou 0950, South AfricaAn accurate algorithm for solving initial value problems (IVPs) which are highly oscillatory is proposed. The proposed method is based on a novel technique of extending the standard spectral homotopy analysis method (SHAM) and adapting it to a sequence of multiple intervals. In this new application the method is referred to as the piecewise spectral homotopy analysis method (PSHAM). The applicability of the proposed method is examined on the differential equation system modeling HIV infection of CD4+ T cells and the Genesio-Tesi system which is known to be chaotic and highly oscillatory. Also, for the first time, we present here a convergence proof for SHAM. We treat in detail Legendre collocation and Chebyshev collocation. The method is compared to MATLAB’s ode45 inbuilt solver as a measure of accuracy and efficiency.http://dx.doi.org/10.1155/2013/583193
collection DOAJ
language English
format Article
sources DOAJ
author H. Saberi Nik
Sohrab Effati
Sandile S. Motsa
Stanford Shateyi
spellingShingle H. Saberi Nik
Sohrab Effati
Sandile S. Motsa
Stanford Shateyi
A New Piecewise-Spectral Homotopy Analysis Method for Solving Chaotic Systems of Initial Value Problems
Mathematical Problems in Engineering
author_facet H. Saberi Nik
Sohrab Effati
Sandile S. Motsa
Stanford Shateyi
author_sort H. Saberi Nik
title A New Piecewise-Spectral Homotopy Analysis Method for Solving Chaotic Systems of Initial Value Problems
title_short A New Piecewise-Spectral Homotopy Analysis Method for Solving Chaotic Systems of Initial Value Problems
title_full A New Piecewise-Spectral Homotopy Analysis Method for Solving Chaotic Systems of Initial Value Problems
title_fullStr A New Piecewise-Spectral Homotopy Analysis Method for Solving Chaotic Systems of Initial Value Problems
title_full_unstemmed A New Piecewise-Spectral Homotopy Analysis Method for Solving Chaotic Systems of Initial Value Problems
title_sort new piecewise-spectral homotopy analysis method for solving chaotic systems of initial value problems
publisher Hindawi Limited
series Mathematical Problems in Engineering
issn 1024-123X
1563-5147
publishDate 2013-01-01
description An accurate algorithm for solving initial value problems (IVPs) which are highly oscillatory is proposed. The proposed method is based on a novel technique of extending the standard spectral homotopy analysis method (SHAM) and adapting it to a sequence of multiple intervals. In this new application the method is referred to as the piecewise spectral homotopy analysis method (PSHAM). The applicability of the proposed method is examined on the differential equation system modeling HIV infection of CD4+ T cells and the Genesio-Tesi system which is known to be chaotic and highly oscillatory. Also, for the first time, we present here a convergence proof for SHAM. We treat in detail Legendre collocation and Chebyshev collocation. The method is compared to MATLAB’s ode45 inbuilt solver as a measure of accuracy and efficiency.
url http://dx.doi.org/10.1155/2013/583193
work_keys_str_mv AT hsaberinik anewpiecewisespectralhomotopyanalysismethodforsolvingchaoticsystemsofinitialvalueproblems
AT sohrabeffati anewpiecewisespectralhomotopyanalysismethodforsolvingchaoticsystemsofinitialvalueproblems
AT sandilesmotsa anewpiecewisespectralhomotopyanalysismethodforsolvingchaoticsystemsofinitialvalueproblems
AT stanfordshateyi anewpiecewisespectralhomotopyanalysismethodforsolvingchaoticsystemsofinitialvalueproblems
AT hsaberinik newpiecewisespectralhomotopyanalysismethodforsolvingchaoticsystemsofinitialvalueproblems
AT sohrabeffati newpiecewisespectralhomotopyanalysismethodforsolvingchaoticsystemsofinitialvalueproblems
AT sandilesmotsa newpiecewisespectralhomotopyanalysismethodforsolvingchaoticsystemsofinitialvalueproblems
AT stanfordshateyi newpiecewisespectralhomotopyanalysismethodforsolvingchaoticsystemsofinitialvalueproblems
_version_ 1725473272256004096

Similar Items