Abstract
TIn this note, we review homotopy perturbation method (HPM), Discrete HPM, Chebyshev polynomials and its properties. Moreover, the convergences of HPM and error term of Chebyshev polynomials were discussed. Then, linear singular integral equations (SIEs) and hyper-singular integral equations (HSIEs) are solved by combining modified HPM together with Chebyshev polynomials. Convergences of the mixed method for the linear HSIEs are also obtained. Finally, illustrative examples and comparisons with different methods are presented.
Highlights
Integral equations occur naturally in many fields of science and engineering
Main aim of this note is to analysis homotopy perturbation method (HPM) and its applications in different problems of integral equations as well as Chebyshev Polynomials and its applications in SIEs and hyper-singular integral equations (HSIEs)
Chebyshev polynomials are widely used in many areas of numerical analysis for instance approximate solution of a system of singular integral equations (Shahmorad and Ahdiaghdam [40]), numerical solution of nonlinear Volterra integral equations (Maleknejad et al [32]), nonlinear boundary value problems (Shaban et al [41]), Fredhol-Folterra type integro-differential equation (Daschioglu [11] - [12]), singular and hypersingular integral equations (Eshkuvatov et al [14], and Abdulkawi [1]) and so on
Summary
Integral equations occur naturally in many fields of science and engineering. A computational approach to solve integral equation is an essential work in scientific research. HPM has mainly two types of modifications ([17], [18]) and in the recent decades, modified homotopy perturbation method (MHPM) has been used to solve many types of integral equations including SIEs and HSIEs ([42],[13], [45], [31]).
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