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Open Access Research

Analysis of the new homotopy perturbation method for linear and nonlinear problems

Ali Demir1*, Sertaç Erman1, Berrak Özgür1 and Esra Korkmaz2

Author Affiliations

1 Department of Mathematics, Kocaeli University Umuttepe, Izmit, Kocaeli, 41380, Turkey

2 Ardahan University, Merkez, Ardahan, 75000, Turkey

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Boundary Value Problems 2013, 2013:61  doi:10.1186/1687-2770-2013-61

Published: 26 March 2013

Abstract

In this article, a new homotopy technique is presented for the mathematical analysis of finding the solution of a first-order inhomogeneous partial differential equation (PDE) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M1">View MathML</a>. The homotopy perturbation method (HPM) and the decomposition of a source function are used together to develop this new technique. The homotopy constructed in this technique is based on the decomposition of a source function. Various decompositions of source functions lead to various homotopies. Using the fact that the decomposition of a source function affects the convergence of a solution leads us to development of a new method for the decomposition of a source function to accelerate the convergence of a solution. The purpose of this study is to show that constructing the proper homotopy by decomposing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> in a correct way determines the solution with less computational work than using the existing approach while supplying quantitatively reliable results. Moreover, this method can be generalized to all inhomogeneous PDE problems.