Solving system of partial differential equations using variational iteration method with He's polynomials

Nadeem, Muhammad; Yao, Shao-Wen

doi:10.22436/jmcs.019.03.07

Cited by 14 publications

(7 citation statements)

References 22 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

Self Cite

View full text Add to dashboard Cite

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“…In the latest paper, [14] Appreciated consideration of homotopy perturbation Sumudu transform method (HPSTM) has been given to analyze results of partial differential nonlinear and linear equations. 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.…”

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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“…In [4][5][6], the authors gave a dynamic analysis of a fractional-order Lorenz chaotic system. Although some numerical and analytical methods of the FDEs have been announced, such as spectral method [7][8][9][10][11], reproducing kernel method [12][13][14][15][16][17][18][19], homotopy perturbation method [20][21][22][23], high-precision numerical approach [24][25][26][27], and so on numerical and analytical methods [28][29][30][31][32][33][34][35][36]. These researchers all say their own approach can accurately simulate chaotic systems.…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation of the Fractional-Order Lorenz Chaotic Systems with燙aputo Fractional Derivative

Dai¹,

Li²,

Li³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

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Although some numerical methods of the fractional-order chaotic systems have been announced, high-precision numerical methods have always been the direction that researchers strive to pursue. Based on this problem, this paper introduces a high-precision numerical approach. Some complex dynamic behavior of fractional-order Lorenz chaotic systems are shown by using the present method. We observe some novel dynamic behavior in numerical experiments which are unlike any that have been previously discovered in numerical experiments or theoretical studies. We investigate the influence of α 1 , α 2 , α 3 on the numerical solution of fractional-order Lorenz chaotic systems. The simulation results of integer order are in good agreement with those of other methods. The simulation results of numerical experiments demonstrate the effectiveness of the present method.

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Solving system of partial differential equations using variational iteration method with He's polynomials

Cited by 14 publications

References 22 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

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

Numerical Simulation of the Fractional-Order Lorenz Chaotic Systems with燙aputo Fractional Derivative

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