Cahn-Allen Equation Research Articles

The Efficient Robust Conformable Methods for Solving the Conformable Fractional Cahn-Allen Equation

This study focuses on the novel conformable methods employed to obtain new numerical solutions for the Cahn-Allen equation with conformable fractional derivatives. One of the two distinct methods put forth is the Cq-HATM, a hybrid technique that integrates the q-homotopy analysis transform method with the Laplace transform, utilizing the definition of conformable derivative. The CHPETM is a hybrid technique that combines the homotopy perturbation method with the Elzaki transform (ET). New numerical solutions of the conformal fractional differential Cahn-Allen equation were obtained using CHPETM and Cq-HATM. The computer simulations have been conducted in order to provide validation for the efficacy and reliability of the proposed methods. Upon performing a comparative analysis between the exact solutions and the solutions obtained through the novel methods, it becomes evident that both of these approaches exhibit simplicity, efficacy, and proficiency in addressing nonlinear conformable time-fractional coupled systems.

EXACT SOLUTIONS OF NONLINEAR FRACTIONAL CAHN–ALLEN EQUATION ARIES IN DIFFERENT NONLINEAR PHYSICAL PHENOMENON USING UNIFIED TECHNIQUE

The nonlinear fractional Cahn–Allen (NFCA) equation provides insight into phase modifications and pattern elaboration in natural structures, clarifying how various states of matter have changed throughout time. By using a unified technique, this study aims to present novel, precise solutions to the NFCA problem. This method has led to the discovery of a broad spectrum of soliton solutions, ranging from dark compactons and dark solitons to periodic kink behaviors, rough wave behaviors, and periodic patterns featuring peaked crests and troughs. It has also unveiled anti-kink periodic behaviors, periodic wave behaviors, bright-dark single solitons, periodic patterns displaying anti-peaked crests and anti-troughs, bright solitons, and compound soliton phenomena. Notably, these accomplishments have been made possible through the utilization of Maple software. These findings are important for fractional nonlinear dynamical models (FNLDMs). In order to clarify many dynamic behaviors that solutions of the NFCA equation display in other scientific fields, such as quantum mechanics, mathematical biology, and plasma physics, contour plots and 3D surface representations of these exact solutions are also featured. Through our investigation using the unified technique, we have unveiled a plethora of novel discoveries. Employing this method has revealed a total of 54 fresh findings. In contrast, Javeed et al. [S. Javeed, S. Saif and D. Baleanu, New exact solutions of fractional Cahn–Allen equation and fractional DSW system, Adv. Differential Equations 2018 (2018) 459] applied the first integral technique, resulting in only eight outcomes. Upon comparing their results with ours, our approach has yielded an additional four out of six innovative results. As far as we are aware, the application of our technique to the NFCA equation has not been previously documented. We anticipate that future research will expand this methodology to include other FNLDMs.

Bifurcation analysis, and exact solutions of the two-mode Cahn–Allen equation by a novel variable coefficient auxiliary equation method

This document elaborates on a newly introduced analytical method known as the “Variable Coefficient Generalized Abel Equation Method,” as proposed by Hashemi in Hashemi (2024), designed specifically for addressing the two-mode Cahn–Allen equation. Diverging from conventional techniques that heavily rely on constant coefficient ordinary differential equations and auxiliary ordinary differential equations, our method innovatively incorporates variable coefficient ordinary differential equations within a sub-equation framework. Demonstrating its versatility, we apply this innovative technique to the two-mode Cahn–Allen equation, showcasing its effectiveness and efficiency through the derivation of analytical solutions. Notably, this method emerges as a promising tool for tackling complex nonlinear partial differential equations prevalent in fluid dynamics and wave propagation scenarios. Beyond merely expanding the repertoire of available analytical tools, our approach contributes to advancing solutions for various models within the realm of mathematical physics. Various forms of exact solutions, including exponential-type solutions, Kink solitons, dark solitons, and bright soliton solutions, are obtained for the model under consideration. Moreover, we delve into the analysis of bifurcation, chaotic behavior, and sensitivity within the context of the two-mode Cahn–Allen model, further enhancing the depth and breadth of our study. Three equilibria are analyzed across various classifications, including center point, focus point, saddle point, and node point. Chaotic behavior of the corresponding dynamical system is considered by adding the function ω1sin(ω2ζ). Lastly, sensitivity analysis of the system is conducted by examining different parameters of the model and imposing noise to the initial conditions.

Sensitive visualization, traveling wave structures and nonlinear self-adjointness of Cahn–Allen equation

Sensitive visualization, traveling wave structures and nonlinear self-adjointness of Cahn–Allen equation

Numerical Solution of Cahn-Allen Equations by Wavelet Based Lifting Schemes

The Cahn-Allen equation is a reaction-diffusion equation of mathematical physics that describes the process of phase separation in multi-component alloy systems and order-disorder transitions. This paper presented a numerical solution of Cahn-Allen equations by Lifting scheme using different wavelet filter coefficients. The numerical results obtained using this scheme are compared with the exact solution to demonstrate the accuracy and fast convergence in less computational time than the existing scheme. Some problems are taken to show the validity and applicability of the scheme.

Comparative analysis of the fractional order Cahn-Allen equation

This current work presents a comparative study of the fractional-order Cahn-Allen (CA) equation, where the non-integer derivative is taken in the Caputo sense.The Cahn-Allen equation is an equation that assists in the comprehension of phase transitions and pattern formation in physical systems. This equation describes how different phases of matter, such as solids and liquids, change and interact throughout time. We employ two analytical methods: the Laplace Residual Power Series Method (LRPSM) and the New Iterative Method (NIM), to solve the proposed model. The LRPSM is a combination of the Laplace Transform and the Residual Power Series Method, while the New Iterative Method is a modified form of the Adomian Decomposition Method that does not require any type of polynomial or digitization. For the purpose of accuracy and reliability, we compare our findings with other methods and the exact solution used in the literature. Additionally, 2D and 3D plots are generated for various fractional order values denoted as p. These plots illustrate that as the fractional order p approaches 1, the graph of the approximate solution gradually coincides with the graph of the exact solution.

Exact special solutions of space–time fractional Cahn–Allen equation by beta and M-truncated derivatives

In this paper, we consider the nonlinear space–time fractional form of Cahn–Allen equation (FCAE) with beta and M-truncated derivatives. Cahn–Allen equation (CAE) is commonly used in many problems of physics and engineering, such as, solidification problems, phase separation in iron alloys and others. We apply the improved [Formula: see text]-expansion method (ITEM). We obtain four types of traveling wave solutions, including, trigonometric, hyperbolic, rational and exponential function solutions. We demonstrate some of the extracted solutions using definitions of the beta (BD) and M-truncated derivatives (MTD) to understand their dynamical behavior. We observe the fractional effects of the aforementioned derivatives on the related physical phenomena up to possible extent.

Global solutions of nonlinear fractional diffusion equations with time-singular sources and perturbed orders

In a Hilbert space, we consider a class of nonlinear fractional equations having the Caputo fractional derivative of the time variable t and the space fractional function of the self-adjoint positive unbounded operator. We consider various cases of global Lipschitz and local Lipschitz source with time-singular coefficient. These sources are generalized of the well–known fractional equations such as the fractional Cahn–Allen equation, the fractional Burger equation, the fractional Cahn–Hilliard equation, the fractional Kuramoto–Sivashinsky equation, etc. Under suitable assumptions, we investigate the existence, uniqueness of maximal solution, and stability of solution of the problems with respect to perturbed fractional orders. We also establish some global existence and prove that the global solution can be approximated by known asymptotic functions as \(t\rightarrow \infty \).

Fractional View Analysis of Cahn–Allen Equations by New Iterative Transform Method

In this article, the new iterative transform method is applied to evaluate the time-fractional Cahn–Allen model solution. In this technique, Elzaki transformation is a mixture of the new iteration technique. Two problems are studied to demonstrate and confirm the accuracy of the proposed technique. The current technique’s mathematical analysis showed that the method is simple to understand and reliable. These solutions indicate that the proposed technique is advantageous and simple to apply in science and engineering problems.

Lie symmetry analysis and conservation laws of two-mode Cahn-Allen equation

The Lie group method analyzes this study's two-mode Cahn-Allen (TMCAE) equation. The wave propagation form of TMCAE is defined by the mobility of two symmetric waves at the same instant. The Lie scheme is successfully employed on the considered model to find the infinitesimal generators using the invariance condition. Then, we convert the assumed nonlinear TMCAE into an ordinary differential equation utilizing obtained generators and similarity reduction concepts. Using the two analytical methods, the tanh method and the new auxiliary method, to find some new wave solutions. To show the importance of obtained results, we draw the graphical structure of some results by selecting some appropriate values of involving parameters. We find the multipliers depending on independent variables, dependent variables, and derivatives of dependent variables up to fixed order. The conserved quantities fluxes are found corresponding to each conservation law. The local conserved quantities are constructed for all understudying systems employing the multiplier approach.

Regarding on the Results for the Fractional Clannish Random Walker’s Parabolic Equation and the Nonlinear Fractional Cahn-Allen Equation

In this research, the Ψ , Φ -expansion scheme has been implemented for the exact solutions of the fractional Clannish Random Walker’s parabolic (FCRWP) equation and the nonlinear fractional Cahn-Allen (NFCA) equation. Some new solutions of the FCRWP equation and the NFCA equation have been obtained by using this method. The diverse variety of exact outcomes such as intersection between rough wave and kinky soliton wave profiles, intersection between lump wave and kinky soliton wave profiles, soliton wave profiles, kink wave profiles, intersection between lump wave and periodic wave profiles, intersection between rough wave and periodic wave profiles, periodic wave profiles, and kink wave profiles are taken. Comparing our developed answers and that got in previously written research papers presents the novelty of our investigation. The above techniques could also be employed to get exact solutions for other fractional nonlinear models in physics, applied mathematics, and engineering.

Preconditioned Pseudo-Spectral Gradient Flow for Computing the Steady-State of Space Fractional Cahn-Allen Equations With Variable Coefficients

The aim of this paper is to propose some efficient and accurate numerical methods to compute the steady-state of variable coefficients space fractional Cahn-Allen equations. The approach combines an adaptive time stepping semi-implicit gradient flow method to minimize the fractional energy functional and pseudo-spectral approximation schemes. Based on the use of a preconditioned GMRES, the space fractional Cahn-Allen equation is then solved efficiently. The full methodology is supported by the numerical solution of a one-dimensional problem.

Exact soliton solutions to the Cahn–Allen equation and Predator–Prey model with truncated M-fractional derivative

The current research is about new exact soliton solutions to the Cahn–Allen equation and Predator–Prey model with the most general fractional derivative operator. The novel truncated M-fractional derivative is applied to study the aforementioned models to secure soliton solutions via the modified integration scheme known as the extended Sinh–Gordon equation expansion method (ShGEEM). This study provides the dark, singular, singular-dark, kink solitons and other solutions with certain conditions. The gained results are also verified with the use of symbolic software. The obtained solutions may be applied in the demonstration of the Cahn–Allen equation and Predator–Prey model in some better way using a fractional derivative. These solutions suggest that this method is more effective, straight forward and reliable as compared to other methods.

Convex-periodic, kink-periodic, peakon-soliton and kink bidirectional wave-solutions to new established two-mode generalization of Cahn–Allen equation

In this paper, the evolutionary Cahn–Allen equation is upgraded into the generalized two-mode version via Korsunsky’s sense. The new established model is characterized in terms of the wave propagation form represented by the movement of two simultaneous waves of symmetric type. The Kudryashov-expansion method is used to extract some solitary wave solutions to the suggested model. Moreover, from mathematical point of view, some physical properties are explored to the two-mode Cahn–Allen equation via examining, graphically, the effect of the phase velocity and the dispersion parameters acting on the propagations of the obtained bidirectional wave-solutions. Finally, very interesting types of solutions are obtained to the new model by implementing a modified versions of the rational sine–cosine and sinh–cosh methods.

Extraction of new exact soliton solutions and Painlevé-test for fractional Cahn–Allen equation

Extraction of new exact soliton solutions and Painlevé-test for fractional Cahn–Allen equation

Stable soliton solutions to the nonlinear low-pass electrical transmission lines and the Cahn-Allen equation

The low-pass nonlinear electrical transmission lines and the Cahn-Allen equation are important nonlinear model equations to figure out different tangible systems, namely, electrical engineering, fluid dynamics etc. The contrivance of this study is to introduce advanced Bernoulli sub-equation function method to search for stable and effective solitary solutions of the described wave equations. Stable solitary solutions are reported as an integration of exponential functions, hyperbolic functions, etc., and the graphical implications for specific values of the corresponding parameters are explained in the solutions obtained in order to uncover the inmost structure of the tangible phenomena. It is establish that the IBSEF method is reliable, contented and might be used in further works to found ample novel soliton solutions for other types of NLEEs arising in physical science and engineering.

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.

On the phase separation in the ternary alloys: Numerical and computational simulations of the Atangana–Baleanu time‐fractional Cahn–Allen equation

AbstractThis article gives a survey on computational and numerical solutions for the phase separation in Fe‐Cr‐X (X = Mo, Cu) ternary alloys. This separation is mathematically given by the Atangana–Baleanu time‐fractional Cahn–Allen equation. The generalized Kudryashov, sech–tanh, and the Adomian decomposition approaches are employed to get a bundle of analytical and numerical solutions. The obtained analytical solutions are used to evaluate the initial and boundary conditions of the phase ternary alloys separation operation. Moreover, the stability property of solutions is investigated based on the characteristics of the Hamiltonian system. Novel properties of the investigated model are figured out by distinct types of sketches. Finally, the accuracy of both used computational schemes is illustrated.

Analytical and semi‐analytical solutions for time‐fractional Cahn–Allen equation

This paper investigates the analytical and semi‐analytical solutions of the time‐fractional Cahn–Allen equation, which describes the structure of dynamic for phase separation in Fe‐Cr‐X (X = Mo, Cu) ternary alloys. We apply a modified auxiliary equation method and the Adomian decomposition method to get distinct solutions to our suggested model. These solutions describe the dynamic of the phase separation in iron alloys and use in solidification and nucleation problems. The applications of this method arise in many various fields such as plasma physics, quantum mechanics, mathematical biology, and fluid dynamics. We apply a conformable fractional derivative to this fractional model to convert it into a nonlinear partial differential equation with integer order. We obtain many analytical wave solutions and also apply a semi‐analytical scheme to calculate the absolute value of error. All solutions are verified by using Mathematica software.

On finite series solutions of conformable time-fractional Cahn-Allen equation

AbstractThe aim of this article is to derive new exact solutions of conformable time-fractional Cahn-Allen equation. We have achieved this aim by hyperbolic function and expa function methods with the aid of symbolic computation using Mathematica. This idea seems to be very easy to employ with reliable results. The time fractional Cahn-Allen equation is reduced to respective nonlinear ordinary differential equation of fractional order. Also, we have depicted graphically the constructed solutions.