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1

On the semi-analytic technique to deal with nonlinear fractional differential equations

Khirsariya Sagar R., Rao Snehal B.

Journal of Applied Mathematics and Computational Mechanics

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2023

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Vol. 22, nr 1

17--30

EN

In this article, we present a novel hybrid approach, by combining the Sawi transform with the homotopy perturbation method, to achieve the approximate and analytic solutions of nonlinear fractional differential equations (ODE as well as PDE) using the time-fractional Caputo derivative. The proposed algorithm is faster and simple compared to other iterative methods. The Sawi transform is used along with the homotopy perturbation method to accelerate the convergence of the series solution. The results discussed using calculations, graphs and tables are compatible for comparison with other known methods like the residual power series method and the exact solution which are discussed in the literature.

2

Solutions of Benjamin-Bona-Mahony, modified Camassa-Holm and Degasperis Procesi equations using an iterative method

Kumar Manoj

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 3

59--72

EN

In the present paper, we solve the non-linear Benjamin-Bona-Mahony, modified Camassa-Holm, and Degasperis-Procesi equations using an iterative method introduced by Daftardar-Gejji and Jafari. Results are compared with those obtained by other iterative methods such as the Adomian decomposition method and homotopy perturbation method. It is observed that the proposed method is computationally inexpensive and yields more accurate solutions than the Adomian decomposition method and the homotopy perturbation method.

3

Homotopy perturbation Shehu transform method for solving fractional models arising in applied sciences

Maitama Shehu, Zhao Weidong

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 1

71--82

EN

Using the recently proposed homotopy perturbation Shehu transform method (HPSTM), we successfully construct reliable solutions of some important fractional models arising in applied physical sciences. The nonlinear terms are decomposed using He’s polynomials, and the fractional derivative is calculated in the Caputo sense. Using the analytical method, we obtained the exact solution of the fractional diffusion equation, fractional wave equation and the nonlinear fractional gas dynamic equation.

4

Heat transfer analysis of non-Newtonian natural convective fluid flow using Homotopy Perturbation and Daftardar-Gejiji & Jafari Methods

Adeleye Olurotimi, Yinusa Ahmed

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 2

5--18

EN

In this paper, the analytical solution of natural convective heat transfer of a non-Newtonian fluid flow between two vertical infinite plates using the Homotopy Perturbation Method (HPM) and Daftardar-Gejiji & Jafari Method (DJM) is presented. The heat transfer problem of natural convection is observed in many engineering fields including geothermal systems, heat exchangers, petroleum reservoirs, nuclear waste reserves, etc. The problem which is modelled as fully coupled nonlinear ordinary differential equations requires special analytical techniques for its solution. The solutions are obtained using an exact analytical method: the Homotopy Perturbation Method (HPM) and a semi-analytical method: the Daftardar-Gejiji & Jafari Method (DJM). These solutions are compared with solutions obtained from the Runge-Kutta numerical method. The results are in good agreement with the numerical solutions. The effects of the Eckert number, Prandtl number and the non-Newtonian fluid viscosity parameter on the non-dimensional temperature and velocity of the fluid are investigated. The results obtained from the analytical method show that the method can be applied to predict excellent results of the problem and can be used for parametric studies of the problem. From the results, it is shown that when the Prandtl number and the Eckert number are increased, there is an increase in both temperature and fluid flow velocity.

5

The stability conditions of the cubic damping Van der Pol-Duffing oscillator using the HPM with the frequency-expansion technology

El-Dib Y. O.

Journal of Applied Mathematics and Computational Mechanics

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2018

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Vol. 17, nr 3

31--44

EN

In this paper, we perform the frequency-expansion formula for the nonlinear cubic damping van der Pol’s equation, and the nonlinear frequency is derived. Stability conditions are performed, for the first time ever, by the nonlinear frequency technology and for the nonlinear oscillator. In terms of the van der Pol’s coefficients the stability conditions have been performed. Further, the stability conditions are performed in the case of the complex damping coefficients. Moreover, the study has been extended to include the influence of a forcing van der Pol’ oscillator. Stability conditions have been derived at each resonance case. Redoing the perturbation theory for the van der Pol oscillator illustrates more of a resonance formulation such as sub-harmonic resonance and super-harmonic resonance. More approximate nonlinear dispersion relations of quartic and quintic forms in the squaring of the extended frequency are derived, respectively.

6

Application of the homotopy perturbation method for calculation of the temperature distribution in the cast-mould heterogeneous domain

Grzymkowski R., Hetmaniok E., Słota D.

Journal of Achievements in Materials and Manufacturing Engineering

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2010

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Vol. 43, nr 1

299--309

EN

Purpose of this paper: In this paper an application of the new method for solving the heat conduction equation in the heterogeneous cast-mould system, with an assumption of the ideal contact at the cast-mould contact point, is introduced. An example illustrating the discussed approach and confirming its usefulness for solving problems of that kind is also presented in the paper. Design/methodology/approach: For solving the discussed problem the homotopy perturbation method is used, which consists in determining the series convergent to the exact solution or enabling to built the approximate solution of the problem. Findings: The paper shows that the homotopy perturbation method, effective in solving many technical problems, is successful also for examining the considered problem. Research limitations/implications: Solution of the problem is provided with the assumption of an ideal contact between the cast and the mould. In further, research of the discussed method shall be employed to solve problems involving the presence of thermal resistance at the cast-mould contact Practical implications: The method allows to determine the solution in form of the continuous function, which is significant for the analysis of the cast cooling in the mould, in order to avoid the defects formation in the cast. Originality/value: Application of the new method for solving the considered problem.