Using He's variational method to seek the traveling wave solution of PHI-four equation

Najafi, Mohammad

doi:10.14419/ijamr.v1i4.411

Cited by 7 publications

(6 citation statements)

References 12 publications

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“…Next, we compare the solutions of the Phi-four equation obtained using the first integral approach with those attained by Najafi using He's variational approach in [35]. Najafi identified four solitary wave solutions; however, using the first integral method in this article, we derived six solitary wave solutions, as presented in table 2.…”

Section: Comparison Of the Resultsmentioning

confidence: 95%

“…Therefore, various researchers have explored soliton and other solutions to the DLW and Phi-four equations using analytical and semi-analytical approaches, such as Khater et al put in use the sech-tanh expansion procedure [31], Qawasmeh and Alquran [34] used the / ¢ ( ) G G -expansion method to determine soliton solutions to the (2+1)-dimensional DLW equation. Najafi [35] applied He's variational approach to find solitary wave solutions to the Phi-4 equation. Additionally, the Lie group approach [36] and the auto and non-auto Bäcklund transformations [37] were among the methods used to address the DLW system and the Phi-4 equation.…”

Section: Introductionmentioning

confidence: 99%

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Analytic soliton solutions to the shallow water dispersive long gravity wave equations: the first integral approach in nonlinear physics

Hussain,

Akbar,

İlhan

2024

Phys. Scr.

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In this article, we investigate the (2+1)-dimensional dispersive long water wave equation and the (1+1)-dimensional PHI-four equation, which describe the behavior of long gravity waves with small amplitudes, long wave propagation in oceans and seas, coastal structures and harbor design, effects of wave motion on sediment transport, quantum field theory, phase transitions of matter, ferromagnetic systems, liquid-gas transitions, and the structure of optical solitons. We use the first integral technique and obtain new and generic solutions for the models under consideration. By setting definite values for the associated parameters, various types of richly structured solitons are generated. The solitons include kink, flat kink, bell-shaped, anti-bell-shaped, and singular kink formations. These solutions allow for a profound understanding of the behavior and properties of the phenomena, offering new insights and potential applications in the associated field. The first integral technique is simpler, directly integrates the models, and the solutions offer clear insights into the underlying phenomena without requiring intermediate steps, making it widely applicable to various other models, including nonlinear equations and those that are challenging to solve using other standard techniques.

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Section: Comparison Of the Resultsmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Analytic soliton solutions to the shallow water dispersive long gravity wave equations: the first integral approach in nonlinear physics

Hussain,

Akbar,

İlhan

2024

Phys. Scr.

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“…An essential NPDE that appears in optics, thermal science, nanofluids, and nonlinear vibration modelling is the PHI-four equation [3]. The PHI-four equation has been studied in earlier research to provide the analytical and numerical solutions by using Adomian decomposition method [4], Sumudu transform method and Homotopy perturbation method [5][6][7], asymptotic iteration method [8], fourth order compact and conservative scheme [9], orthogonal spectral collocation scheme [10] based on Jacobi envelopes, Soliton of solitary wave using Anstaz technique [11], first integral method [12], modified extended direct algebraic method [13], natural decomposition method [14], bifurcation analysis and Anstaz method [15], variational iteration method [16], He's method [17] and others. For instances, recently LADT has been successfully implemented in a variety of differential model like; SDIQR model of Covid-19 [18], Lane emden differential equation [19], model of lassa diseases [20], numerical solution of large deflection beam [21], simulation of unsteady MHD flow in incompressible fluid [22], approximation of time fractional advection dispersion equation [23], approximate solution of fractional order sterile insect technology model [24] etc.…”

Section: Introductionmentioning

confidence: 99%

The kink-antikink single waves in dispersion systems by generalized PHI-four equation in mathematical physics

Sahu,

Jena

2024

Phys. Scr.

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An essential aspect of mathematical physics is the PHI-four equation, which is a specific version of the Klein-Gordon equation that predicts particle physics phenomena. The present paper addresses numerical approaches to generalized PHI-four equation based on Laplace Adomian Decomposition Technique (LADT) which is governed by coupling of Laplace transform and Adomian decomposition method to determine the kink-antikink single waves in dispersion systems arises in mathematical physics. The nonlinear terms in the PHI-four equation are handled using the accelerated polynomial i.e., Adomian polynomial. The approach is extremely interesting computationally and is straightforward to execute. The accuracy and robustness of the current scheme are demonstrated by four test problems. To demonstrate the efficacy of our suggested approach, the current result is contrasted with both the analytical solution and existing solutions in literature. Stability and convergence analysis are well developed to justify the applicability of the current approach.

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“…Sassaman and Biswas employed the soliton perturbation theory to explore solutions for both the Klein-Gordon and PHI-Four equations. Najafi investigated the soliton solution of the aforementioned equation utilizing He's variational method [12].…”

Section: Introductionmentioning

confidence: 99%

Approximate Solution of PHI-Four and Allen–Cahn Equations Using Non-Polynomial Spline Technique

Haq,

Ali

et al. 2024

Mathematics

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The aim of this work is to use an efficient and accurate numerical technique based on non-polynomial spline for the solution of the PHI-Four and Allen–Cahn equations. A recent discovery suggests that the PHI-Four equation focuses on its implications for particle physics and the behavior of scalar fields in the quantum realm. In materials science, ongoing research involves using the Allen–Cahn equation to understand and predict the evolution of microstructures in various materials as well as in biophysics. It depicts pattern formation in biological systems and the dynamics of spatial organization in tissues. To obtain an approximate solution of both equations, this technique uses forward differences for the time and cubic non-polynomial spline function for spatial descretization. The stability of the suggested technique is addressed using the von Neumann technique. Convergence test is carried out theoretically to show the order of convergence of the scheme. Some numerical tests are carried out to confirm accuracy and efficiency in terms of absolute error LR. Convergence rates for different test problems are also computed numerically. Numerical results and simulations obtained are compared with the existing methods.

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Using He's variational method to seek the traveling wave solution of PHI-four equation

Cited by 7 publications

References 12 publications

Analytic soliton solutions to the shallow water dispersive long gravity wave equations: the first integral approach in nonlinear physics

Analytic soliton solutions to the shallow water dispersive long gravity wave equations: the first integral approach in nonlinear physics

The kink-antikink single waves in dispersion systems by generalized PHI-four equation in mathematical physics

Approximate Solution of PHI-Four and Allen–Cahn Equations Using Non-Polynomial Spline Technique

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