Homotopy Perturbation Method for Time-Fractional Shock Wave Equation

Singh, Mithilesh; Gupta, Praveen Kumar

doi:10.4208/aamm.10-m1137

Cited by 8 publications

(4 citation statements)

References 13 publications

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“…Now, the solution to this simple differential equation with proper initial conditions is then promised. This process is directed in such a way that an approximate solution to equation (4) is introduced as 53–55 …”

Section: Preliminariesmentioning

confidence: 99%

Hypersingular integral equations of the first kind: A modified homotopy perturbation method and its application to vibration and active control

Novin

Araghi

2019

Journal of Low Frequency Noise, Vibration and Active Control

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This paper attempts to propose and investigate a modification of the homotopy perturbation method to study hypersingular integral equations of the first kind. Along with considering this matter, of course, the novel method has been compared with the standard homotopy perturbation method. This method can be conveniently fast to get the exact solutions. The validity and reliability of the proposed scheme are discussed. Different examples are included to prove so. According to the results, we further state that new simple homotopy perturbation method is so efficient and promises the exact solution. The modification of the homotopy perturbation method has been discovered to be the significant ideal tool in dealing with the complicated function-theoretic analytical structures within an analytical method.

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Section: Preliminariesmentioning

confidence: 99%

Hypersingular integral equations of the first kind: A modified homotopy perturbation method and its application to vibration and active control

Novin

Araghi

2019

Journal of Low Frequency Noise, Vibration and Active Control

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“…Previously Shock wave and wave equations have been studied by using Adomian decomposition method [8] and homotopy perturbation method [9]. Further, time-fractional Shock wave and wave equations have been investigated by using homotopy perturbation method [10]. Fractional differential equations have gained importance and popularity, mainly due to its demonstrated applications in science and engineering.…”

Section: Introductionmentioning

confidence: 99%

A New Fractional Model of Nonlinear Shock Wave Equation Arising in Flow of Gases

Singh

Kumar²,

Kumar

2014

Nonlinear Engineering

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In this paper, we present a numerical algorithm based on new homotopy perturbation transform method (HPTM) to solve a time-fractional nonlinear shock wave equation which describes the flow of gases. The fractional derivative is considered in the Caputo sense. The HPTM is combined form of Laplace transform, homotopy perturbation method and He's polynomials. The nonlinear terms can be easily handled by the use of He's polynomials. The technique finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The obtained results are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, efficient, easy to implement and computationally very attractive.

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“…The pursuit of solutions to this intricate equation has spurred the efforts of numerous scientists and researchers, leading to the development of both analytical and numerical schemes for approximating solutions. Singh and Gupta [ 21 ] introduced the homotopy perturbation scheme to study shock wave problems with time-fractional order, presenting results in a series form. Allan and Khaled [ 22 ] employed the Adomian decomposition strategy, providing successive iterations for the shock wave problem and expressing results in a series context.…”

Section: Introductionmentioning

confidence: 99%

An efficient scheme for nonlinear shock wave model in a fractal domain under Caputo fractional operator

Nadeem,

Alsayaad

2024

PLoS ONE

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This paper introduces a refined approach for obtaining the analytical solution of the nonlinear shock wave model incorporating fractal derivatives. The Fractal Yang Variational Iteration Strategy (FYVIS) is utilized to obtain the approximate solution of a fractal model in the form of a series under Caputo fractional operator. The suggested method is the composition of the fractal Yang transform and the variational iteration approach. By using the two-scale fractal theory, we transform the fractal model into its traditional problem and then apply the yang transform to generate a recurrence relation. The variational iteration approach is now suitable to handle this recurrence relation without imposing any hypotheses or restrictions on variables. The derived results by the proposed scheme are shown in terms of series solution. Numerical calculations verify the accuracy and consistency of the suggested approach, demonstrating its excellent performance. The dynamic behavior of fractal components is explored by evaluating absolute errors and presenting two-dimensional diagrams across the fractal domain. This investigation underscores that the suggested technique offers an efficient and user-friendly solution for solving the nonlinear shock wave model involving fractal derivatives.

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Homotopy Perturbation Method for Time-Fractional Shock Wave Equation

Cited by 8 publications

References 13 publications

Hypersingular integral equations of the first kind: A modified homotopy perturbation method and its application to vibration and active control

Hypersingular integral equations of the first kind: A modified homotopy perturbation method and its application to vibration and active control

A New Fractional Model of Nonlinear Shock Wave Equation Arising in Flow of Gases

An efficient scheme for nonlinear shock wave model in a fractal domain under Caputo fractional operator

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