Solution of Newell-Whitehead-Segel equation by variational iteration method with He's polynomials

Nadeem, Muhammad; Yao, Shao-Wen; Parveen, Nusrat

doi:10.22436/jmcs.020.01.03

Cited by 10 publications

(6 citation statements)

References 32 publications

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“…and where the values of λ c and λ m can be identified by Nadeem and Yao (2019), Nadeem et al (2020).The inverse Laplace transforms of equations (5) and (6) becomes as: …”

Section: Analysis Of the Methodsmentioning

confidence: 99%

Analytical approach for the temperature distribution in the casting-mould heterogeneous system

Nadeem

Habib

et al. 2021

HFF

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Purpose The main purpose of this paper is to calculate the analytical solution or a closed-form solution for the temperature distribution in the heterogeneous casting-mould system. Design/methodology/approach First, the authors formulate and analyze the mathematical formulation of heat conduction equation in the heterogeneous casting-mould system, with an arbitrary assumption of the ideal contact at the cast-mould contact point. Then, He-Laplace method, based on variational iteration method (VIM), Laplace transform and homotopy perturbation method (HPM), is used to elaborate the analytical solution of this system. The main focus of He-Laplace method is to find the Lagrange multiplier with an easy approach which enables the implementation of HPM very smoothly and provides the series solution very close to the exact solution. Findings An example is considered to show that He-Laplace method provides the efficient results for calculating the temperature distribution in the casting-mould heterogeneous system. Graphical representation and error distribution represents that He-Laplace method is very simple to implement and effective for casting-mould heterogeneous system. Originality/value The work in this paper is original and advanced. Specially, calculation of Lagrange multiplier for casting-mould system has not been reported in the literature for this work.

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“…and where the values of λ c and λ m can be identified by Nadeem and Yao (2019), Nadeem et al (2020).The inverse Laplace transforms of equations (5) and (6) becomes as: …”

Section: Analysis Of the Methodsmentioning

confidence: 99%

Analytical approach for the temperature distribution in the casting-mould heterogeneous system

Nadeem

Habib

et al. 2021

HFF

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“…Prakash and Verma (2019) obtained the numerical results of time-fractional Newell–Whitehead–Segel (TFNWS) model by using the idea of Adomian decomposition method and compared with exact solution for different values of fractional order. Nadeem et al (2022) used the strategy of natural transform with homotopy perturbation method for the analytical solution of fractional problems. Various researchers (Areshi et al , 2022; Ali et al , 2019) have studied numerous schemes and obtained the approximate results of TFNWS model but their approaches have some difficult steps such as they use evolution theorem and integration to derive the series solution.…”

Section: Introductionmentioning

confidence: 99%

Laplace residual power series method for the numerical solution of time-fractional Newell–Whitehead–Segel model

Luo

Nadeem²

2023

HFF

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Purpose This study aims to investigate the approximate solution of the time fractional time-fractional Newell–Whitehead–Segel (TFNWS) model that reflects the appearance of the stripe patterns in two-dimensional systems. The significant results of plot distribution show that the proposed approach is highly authentic and reliable for the fractional-order models. Design/methodology/approach The Laplace transform residual power series method (ℒT-RPSM) is the combination of Laplace transform (ℒT) and residual power series method (RPSM). The ℒT is examined to minimize the order of fractional order, whereas the RPSM handles the series solution in the form of convergence. The graphical results of the fractional models are represented through the fractional order α. Findings The derived results are obtained in a successive series and yield the results toward the exact solution. These successive series confirm the consistency and accuracy of ℒT-RPSM. This study also compares the exact solutions with the graphical solutions to show the performance and authenticity of the visual solutions. The proposed scheme does not require the restriction of variables and produces the numerical results in terms of a series. This strategy is capable to handle the nonlinear terms very easily for the TFNWS model. Originality/value This paper presents the original work. This study reveals that ℒT can perform the solution of fractional-order models without any restriction of variables.

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“…In [15], the iterative variational method is used to find the results for a system of linear and nonlinear differential equations with the help of He's Polynomials. Numerical solutions of the biological population model have been founded with the help of homotopy perturbation, and He's Polynomial by [16].In [17], the numerical of the Newell-Whitehead-Segal equation with the help of the variational iteration method and He's Polynomials. With the help of He's Polynomial and homotopy perturbation, the iterative variation method has been used to solve the KS equation [18].…”

Section: Introductionmentioning

confidence: 99%

α-Sumudu Transformation Homotopy Perturbation Technique on Fractional Gas Dynamical Equation

Ali

Shokat

Kuffi

2023

IHJPAS

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Transformation and many other substitution methods have been used to solve non-linear differential fractional equations. In this present work, the homotopy perturbation method to solve the non-linear differential fractional equation with the help of He’s Polynomials is provided as the transformation plays an essential role in solving differential linear and non-linear equations. Here is the α-Sumudu technique to find the relevant results of the gas dynamics equation in fractional order. To calculate the non-linear fractional gas dynamical problem, a consumer method created on the new homotopy perturbation a-Sumudu transformation method (HP TM) is suggested. In the Caputo type, the derivative is evaluated. a-Sumudu homotopy perturbation technique and He’s polynomials are all incorporated in the HPSaTM. The availability of He’s polynomials could be used to conveniently manage the non-linearity. The suggested approach shows that the strategy is simple to implement and provides results that can be compared to the results gained from any other transformation technique.

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Solution of Newell-Whitehead-Segel equation by variational iteration method with He's polynomials

Cited by 10 publications

References 32 publications

Analytical approach for the temperature distribution in the casting-mould heterogeneous system

Analytical approach for the temperature distribution in the casting-mould heterogeneous system

Laplace residual power series method for the numerical solution of time-fractional Newell–Whitehead–Segel model

α-Sumudu Transformation Homotopy Perturbation Technique on Fractional Gas Dynamical Equation

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