Modified homotopy perturbation method for solving linear second-order Fredholm integro-differential equations

Elbeleze, Asma Ali; Kılıçman, Adem; Taib, Bachok M.

doi:10.2298/fil1607823e

Cited by 8 publications

(4 citation statements)

References 15 publications

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“…There are many modifications of the method, 23–28 according to web of science there are 338 hits by search for the topic of “modified homotopy perturbation method” and there are 113 articles on sciencedirect.com using a modified homotopy perturbation method until 26 December 2017.…”

Section: Introductionmentioning

confidence: 99%

Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators

García

2018

Journal of Low Frequency Noise, Vibration and Active Control

122

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A brief introduction to the development of the homotopy perturbation method is given, and the main milestones are elucidated with more than 90 references. This paper further improves the method by constructing a homotopy equation with one or more auxiliary parameters embedding in the linear term with a clear advantage in accelerating and controlling the approximation convergence speed. Moreover, a revision of a recent amplitude-period approximation formula is presented providing an answer to an open problem related to the optimal approximation along with a new universal formula. Duffing equation is used as an example to illustrate the solution process for the homotopy perturbation method, and only one or few iterations are needed in practical applications, making the method much attractive. From the side of amplitude-period formulation, the nonlinear pendulum, the Duffing equation and an oscillator with discontinuity are analyzed providing an asymptotic exact equivalence for bigger parameter values in the case of Duffing's system. This mini review gives a tutorial guideline for practical applications of the homotopy perturbation method, the references are not exhaustive.

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Section: Introductionmentioning

confidence: 99%

Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators

García

2018

Journal of Low Frequency Noise, Vibration and Active Control

122

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“…Since the fractional calculus has attracted much more interest among mathematicians and other scientists, the solutions of the fractional integrodifferential equations have been studied frequently in recent years (Alkan & Hatipoglu 2017;Hamoud & Ghadle 2018a, 2018bIbrahim et al 2015;Kumar et al 2017;Ma & Huang 2014;Nemati et al 2016;Ordokhani & Dehestani 2016;Parand & Nikarya 2014;Pedas et al 2016;Shahooth et al 2016;Turmetov & Abdullaev 2017;Wang & Zhu 2016;Yi et al 2016). The methods that are used to find the solutions of the linear fractional Fredholm integro-differential equations are given as fractional pseudospectral integration matrices (Tang & Xu 2016), least squares with shifted Chebyshev polynomials Mohammed 2014), least squares method using Bernstein polynomials (Oyedepo et al 2016), fractional residual power series method (Syam 2017), Taylor matrix method (Gülsu et al 2013), reproducing kernel Hilbert space method (Bushnaq et al 2016), second kind Chebyshev wavelet method (Setia et al 2014), open Newton method (Al-Jamal & Rawashdeh 2009), modified Homotopy perturbation method (Elbeleze et al 2016), Sinc collocation method (Emiroglu 2015).…”

Section: Introductionmentioning

confidence: 99%

Solving Fractional Fredholm Integro-Differential Equations by Laguerre Polynomials

Daşçıoğlu¹,

Bayram²

2019

JSM

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The main purpose of this study was to present an approximation method based on the Laguerre polynomials to obtain the solutions of the fractional linear Fredholm integro-differential equations. This method transforms the integro-differential equation to a system of linear algebraic equations by using the collocation points. In addition, the matrix relation for Caputo fractional derivative of Laguerre polynomials is also obtained. Besides, some examples are presented to illustrate the accuracy of the method and the results are discussed. ABSTRAKTujuan utama kajian ini adalah untuk mengemukakan kaedah penghampiran berdasarkan polinomial Laguerre untuk mendapatkan penyelesaian pecahan linear persamaan pembezaan-kamiran Fredholm. Kaedah ini menjelmakan persamaan pembezaan-kamiran ke sistem persamaan aljabar linear dengan menggunakan titik-titik kolokasi. Di samping itu, hubungan matriks untuk terbitan pecahan Caputo polinomial Laguerre juga diperoleh. Selain itu, beberapa contoh dibentangkan untuk menggambarkan ketepatan kaedah dan hasilnya dibincangkan.Kata kunci: Persamaan pembezaan-kamiran Fredholm; persamaan pembezaan-kamiran pecahan; polinomial Laguerre

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“…Different approaches in literature have been used for the solutions of these equations. Some of them are Variational Iterative Method (VIM) [1][2], Modified Homotopy Perturbation Method (MHPM) [3][4], Homotopy Analysis Transform Method (HAM) [5][6][7][8], Conjugate Gradient Method (CGM) [9], Adomian Decomposition Method (ADM) [10][11] and Block Pulse Functions (BPF) [12] etc. Optimal Homotopy Asymptotic method (OHAM) is one of the most power full techniques for finding the approximate solutions of differential and integro differential equations.…”

Section: Introductionmentioning

confidence: 99%

Optimum Solutions of Fredholm and Volterra Integro-differential Equations

Akbar¹,

Nawaz²,

Ahsan³

2019

IJTAM

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Integro-differential equations arise in modeling various physical and engineering problems. Several numerical and analytical methods have been developed for solving integro-differential equations. In this paper, a powerful semi analytical technique known as Optimal Homotopy Asymptotic Method (OHAM) has been used for finding the approximate solutions of Fredholm type integro-differential equations and Volterra type integro-differential equations. The proposed method does not required discretization like other numerical and approximate method, and it is also free from any small/large parameters. The presented technique provides better accuracy at lower order of approximation, the accuracy of the method can further be increases with higher order of approximation. Moreover, we can easily adjust and control the convergence region. The ability of the method is checked with different problems in literature. The results obtained through OHAM are compared with solutions of Adomian Decomposition Method. It is observed that solutions obtained through the proposed method is more accurate than existing techniques, which proves the validity and stability of the proposed method for solving integrodifferential equations. The presented technique is more consistent, effective, suitable and rapidly convergent. The use of Optimal Homotopy Asymptotic Method is simple and straight forward. For the computation of problems, we have used Mathematica 9.0.

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Modified homotopy perturbation method for solving linear second-order Fredholm integro-differential equations

Cited by 8 publications

References 15 publications

Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators

Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators

Solving Fractional Fredholm Integro-Differential Equations by Laguerre Polynomials

Optimum Solutions of Fredholm and Volterra Integro-differential Equations

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