Phi-4 Equation Research Articles

Analytic Approach Solution to Time-Fractional Phi-4 Equation with Two-Parameter Fractional Derivative

This paper is devoted to building a general framework for constructing a solution to fractional Phi-4 differential equations using a Caputo definition with two parameters. We briefly introduce some definitions and properties of fractional calculus in two parameters and the Phi-4 equation. By investigating the homotopy analysis method, we built the solution algorithm. The two parameters of the fractional derivative gain vary the behavior of the solution, which allows the researchers to fit their data with the proper parameter. To evaluate the effectiveness and accuracy of the proposed algorithm, we compare the results with those obtained using various numerical methods in a comprehensive comparative study.

Analytical and numerical solutions to the Klein–Gordon model with cubic nonlinearity

In this paper, the nonlinear Klein–Gordon equation’s exact solutions are obtained through the application of an appropriate transformation based on He’s semi-inverse approach. This equation considered a generalization of other famous models in applied science, such as Phi-4 equation, Duffing equation, Fisher–Kolmogorov model through population dynamics and Hodgkin–Huxley equation that characterizes the propagation of electrical signals via nervous system. The suggested approach is simple, robust, and efficient, and its application in other partial differential equations in applied science seems promising. Numerical solution of the nonlinear Klein–Gordon equation is presented using finite difference method. The method’s accuracy is demonstrated by contrasting it with the exact solution that we obtained. The Von Neumann stability technique is applied to obtain the stability condition and a convergent test is presented Some 2D and 3D graphs matching to chosen solutions are simulated by taking into account appropriate values for the parameters.

Exploring soliton solutions of stochastic Phi-4 equation through extended Sinh-Gordon expansion method

Exploring soliton solutions of stochastic Phi-4 equation through extended Sinh-Gordon expansion method

Novel kink and multi wave soliton solutions to the stochastic Phi-4 equation driven by the Weiner process

Novel kink and multi wave soliton solutions to the stochastic Phi-4 equation driven by the Weiner process

Precise Travelling Wave Solutions to Space-Time Fractional Phi-4 Equation Using a New Method

Precise Travelling Wave Solutions to Space-Time Fractional Phi-4 Equation Using a New Method

Numerical study of fractional phi-4 equation

<abstract><p>In this paper, we established an analytical solution for the fractional phi-4 model within the Caputo derivative using the homotopy analysis method. This equation known for its nonlinear characteristics often describes various physical phenomena like solitons, wave propagation, and field theories. The fractional version introduces fractional derivatives, making it even more challenging. The homotopy analysis method can effectively handle these nonlinearities. Our objective was to illustrate the reliability and accuracy of our proposed algorithm, which we achieved through a comparative analysis against results obtained using the Yang transform decomposition method. Using the residual error to determine the optimal value of the convergence control parameter $ \hbar $, the results presented underscored the remarkable efficiency and accuracy of this approach.</p></abstract>

Investigation of nonlinear problems governed by stochastic phi-4 type equations in nuclear and particle physics

In this article, the stochastic Phi-4 model is under consideration analytically. The stochastic Phi-4 equation has been crucial to nuclear and particle physics. The randomness of the nuclear particle at the micro level is represented by the stochastic phenomena which are shown by the white noise. The stochastic version of the Phi-4 model is investigated for the first time analytically to gain the soliton solutions by using the two techniques such as Sardar sub-equation and modified exponential rational function. The stochastic soliton solutions are successfully constructed in the form of exponential, trigonometric, and hyperbolic forms. Additionally, the stability analysis of this model is observed. Lastly, the stochastic behavior of some solutions is shown in the form of 2D, and 3D by choosing the different values of parameters.

Comparison of fractional effects for Phi-4 equation using beta and M-truncated derivatives

Comparison of fractional effects for Phi-4 equation using beta and M-truncated derivatives

The deterministic and stochastic solutions for the nonlinear Phi-4 equation

Abstract In the present work, the exp(−φ(ξ))-expansion method is applied for solving the deterministic and stochastic Phi-4 equation. Namely, we introduce hyperbolic, trigonometric, and rational function solutions. The computational study shows that the offered method is pretentious, robust, and influential in applications of interesting analysis, observations of particle physics, plasma physics, quantum field theory, and fluid dynamics. The control on the randomness input (the coefficients are random variables) is studied in order to obtain stability stochastic process solution with beta distribution. In this work, we will deal with stability moment method and then we apply the mean square calculus for the stability concept.

Observations of fractional effects of [formula omitted]-derivative and M-truncated derivative for space time fractional Phi-4 equation via two analytical techniques

The presented work deals with the investigation of space time fractional Phi-4 model which has great importance in particle and nuclear physics. Two eminent techniques, namely, the extended (G′G2)-expansion method and the modified auxiliary equation (MAE) method are utilized to construct the traveling wave solutions of fractional Phi-4 model. The fractional effects on the model are studied using the β-derivative and M-truncated derivative. The obtained results include trigonometric and hyperbolic functions with dark and periodic traveling wave behaviors. Comparison of fractional effects for the β-derivative and M-truncated derivative is illustrated through graphs by assigning particular values to fractional parameter. The obtained results will be helpful to understand the dynamical framework of time fractional Phi-4 model in the related physical phenomena.

Two Reliable Techniques for Solving Conformable Space-Time Fractional PHI-4 Model Arising in Nuclear Physics via β-derivative

Nowadays, nonlinear fractional partial differential equations have been highly using for modelling of physical phenomena. Therefore, it is very important to achieve exact solutions of fractional differential equations for understanding complex phenomena in mathematical physics. In this study, new exact traveling wave solutions are reached of space-time fractional Phi-4 equation indicated by Atangana’s conformable derivative using two powerful different techniques. These are the functional variable method and the first integral method. Obtaining new solutions of this equation show that method is effective to understanding other nonlinear complex problems in particle and nuclear physics.

The fractional analysis of fusion and fission process in plasma physics

The space-time fractional non-linear Phi-4 equation is a significant equation to describes the fission and fusion process that ensued in chemical kinematics, solid-state physics, astrophysical fusion plasma, plasma physics, and electromagnetic interactions, etc. The Phi-4 non-linear partial differential equation is reshaped utilizing the three different fractional-order derivatives and constructed transformations corresponding to every fractional-order derivative to convert the partial differential equation into an ordinary differential equation. A new extended direct algebraic equation method was successfully applied to extract the solitons solutions. The solitons solutions are developed with the exponential, trigonometric, rational, and hyperbolic functions including different unknown constant parameters. The graphical interpretations of obtained solutions are also depicted by allocating the feasible values to unknown constant parameters. The proposed scheme is an effective and functional scientific technique to investigate different fractional systems and models in engineering and physics referenced to real physical problems.

Search for adequate closed form wave solutions to space–time fractional nonlinear equations

The nonlinear space–time fractional Phi-4 equation and density dependent fractional reaction–diffusion equation (FRDE) are important models to interpret the fusion and fission phenomena ensued in solid state physics, plasma physics, chemical kinematics, astrophysical fusion plasma, electromagnetic interactions etc. In this study, we search advanced and wide-ranging wave solutions to the formerly reported nonlinear fractional evolution equations in diverse family through the new generalized (G′∕G)-expansion technique. The solutions are developed with trigonometric, hyperbolic, exponential and rational functions including parameters. The technique is a compatible, functional and effective scientific scheme to examine diverse space–time fractional models in physics and engineering concerned with the real life problems.

Numerical simulation for fractional phi-4 equation using homotopy Sumudu approach

In this study, we utilise the homotopy analysis method and the Sumudu transform to find the solution of fractional order partial differential equations. We focus primarily on the employment of the method for solving the fractional phi-4 equation in one-dimensional spatial domain. The method can be extended easily for more general nonlinear equations. Convergence and error analysis are given. Numerical experiments show that the method is both effective and accurate, and suits such type of applications.

Periodic solution of fractal Phi-4 equation

This paper focuses on a fractal Phi-4 equation with time-space fractal derivatives, though its solitary solutions have been deeply studied, its periodic solution was rarely revealed due to its strong non-linearity. Now the condition is completely changed, He?s frequency formulation provides with a universal tool to having a deep insight into the periodic property of the fractal Phi-4 equation. The two-scale transform is used to convert approximately the fractal Phi-4 equation a differential model, and a criterion is suggested for the existence of a periodic solution of the equation, the effect of fractal orders on the periodic property is also elucidated.

Numerical simulation for fractional phi-4 equation using homotopy Sumudu approach

Numerical simulation for fractional phi-4 equation using homotopy Sumudu approach

A modified exp-function method for fractional partial differential equations

This paper proposes a novel exponential rational function method, a modification of the well-known exp-function method, to find exact solutions of the time fractional Cahn-Allen equation and the time fractional Phi-4 equation. The solution procedure is reduced to solve a system of algebraic equations, which is then solved by Wu?s method. The results show that the present method is effective, and can be applied to other fractional differential equations.

Bifurcation and new exact traveling wave solutions for time-space fractional Phi-4 equation

In this paper, the dynamical behavior of a time-space fractional Phi-4 equation is investigated by utilizing the bifurcation method of a planar dynamical system. Under the given parameter conditions, phase portraits and bifurcations are obtained with the help of the mathematical software Maple. Moreover, some new exact traveling wave solutions are obtained, such as Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions, kink solitary wave solutions, and periodic wave solutions.

Closed-form solutions to the conformable space-time fractional simplified MCH equation and time fractional Phi-4 equation

The time-fractional problem is a class of important models to represent the real phenomena. We construct new solitary waves for the space-time fractional simplified modified Camassa-Holm (MCH) and the time fractional Phi-4 equations using the unified solver. The fractional derivatives are defined in the sense of the new conformable fractional derivative. The acquired solutions may be useful for various vital observations in nuclear and particle physics and fluid mechanics. The proposed unified solver is a sturdy mathematical tool for solving various classes of fractional partial differential equations in applied science. Moreover, the simulation of some selected solutions has been demonstrated with the aid of matlab software.

New Analytical Solutions for Space and Time Fractional Phi-4 Equation

In this paper, by utilizing an analytical method based on the Jacobi elliptic functions, the largest set in the literature of time, space and space-time conformable fractional Phi-4 equations is found. These solutions are obtained in general form including rational, trigonometric, hyperbolic and complex functions. Four problems are presented to illustrate the application of the method and some of the solutions are also demonstrated by the graphics.