Comparison of Approximate Analytical and Numerical Solutions of the Allen Cahn Equation

Hussain, Safdar; Haq, Fazal; Shah, Abdullah; Abduvalieva, Dilsora; Shokri, Ali

doi:https://doi.org/10.1155/2024/8835138

International Journal of Differential Equations

On this page

Abstract Introduction Conclusion Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 8835138 | https://doi.org/10.1155/2024/8835138

Comparison of Approximate Analytical and Numerical Solutions of the Allen Cahn Equation

Safdar Hussain,¹Fazal Haq,¹Abdullah Shah,^2,3Dilsora Abduvalieva,⁴and Ali Shokri⁵

Academic Editor: Sining Zheng

Received20 Oct 2023

Revised07 Mar 2024

Accepted09 Mar 2024

Published23 Mar 2024

Abstract

Allen Cahn (AC) equation is highly nonlinear due to the presence of cubic term and also very stiff; therefore, it is not easy to find its exact analytical solution in the closed form. In the present work, an approximate analytical solution of the AC equation has been investigated. Here, we used the variational iteration method (VIM) to find approximate analytical solution for AC equation. The obtained results are compared with the hyperbolic function solution and traveling wave solution. Results are also compared with the numerical solution obtained by using the finite difference method (FDM). Absolute error analysis tables are used to validate the series solution. A convergent series solution obtained by VIM is found to be in a good agreement with the analytical and numerical solutions.

1. Introduction

The AC equation is a nonlinear parabolic partial differential equation (PDE) which describes natural physical phenomenon in materials and multiphase systems. The model was firstly developed by Allen and Cahn to study the antiphase boundary motion in antiphase domain coarsening [1]. The AC model is mainly used to study various phase separation problems in different fields of sciences such as crystal growth, motion by mean curvature flows, image segmentation, and the mixture of two incompressible fluids [2–6]. It has become a rudimentary mathematical model to discuss the behavior of phase evolutions and interfacial dynamics in material sciences [7]. In order to study the phenomenon of phase transition, a capable and precise method for the solution of AC equation has practical significance. Physical importance of AC equation attracts many researchers to study the solution of this equation. Since the AC equation is highly stiff and nonlinear, therefore, it is not easy to find its exact analytical or numerical solution. Literature review reveals that many researchers use different numerical and analytical schemes and techniques to find the accurate solutions of AC equation. Tascan and Bekir [8] used the first integral method to find traveling wave solution of AC equation. Wazwaz [9] discussed solitons and kink solutions using the tanh-coth method. Tariq and Akram [10] investigated new traveling wave solution to the AC equation. Jeong et al. [11] performed a comparative study of various numerical techniques for solving the AC equation. Gui and Zhao [12] studied the traveling wave solutions of AC equation with a fractional Laplacian. Jeong and Kim [13] developed an explicit hybrid finite difference scheme for the numerical solution of AC equation. Shah et al. [14] proposed an algorithm based on the finite element method and implicit fractional-step θ−scheme to study the numerical solution of AC equation. Asif and Hasan [15] used FDM to obtain numerical solution of AC equation. Shin et al. [16] use hybrid finite element method for solving the AC equation. Bulut [17] implemented the modified exponential function method to find analytical hyperbolic function solutions of AC equation.

In last few years, approximate analytical techniques attract attention of many researchers to find the approximate solutions of nonlinear PDE’s due to their simplicity and easy approach. Series of methods have been established by mathematicians and researchers such as the homotopy perturbation method (HPM) [18, 19], homotopy analysis method (HAM) [20–22], variational iteration method (VIM) [23–25], Adomian decomposition method (ADM) [26–28], differential transform method (DTM) [29–31], tanh method (THM) and extended tanh method [32–34], use of trigamma function [35], Catalan-type numbers [36], and polynomials [37–39]. Traveling wave solutions are also prominent techniques in finding the closed form of solutions. In recent years, many researchers used these techniques to tackle various problems in physics and engineering [40–42]. Among the above stated methods and others, VIM is more dominant and power full due to its simplicity and fast convergence. Literature survey reveals that the AC equation has not been solved using this method. Therefore, in this research work, we have used VIM to obtain series solution of AC equation. The obtained results are compared with numerical solution of [15] and analytical hyperbolic function solution obtained in [17]. We also compare the VIM solution with traveling wave solution obtained in [8].

2. Mathematical Model

The AC equation is the mathematical model that describes phase transitions in material science,along with initial conditions (IC’s) and boundary conditions (BC’s)

In equation (1), is the thickness parameter which represents the width of transition region, and is a bounded domain. The term with represents double well potential.

3. Basic Idea of VIM

VIM is established by a Chinese scientist Ji-Huan He in 1999 using the concept of optimization and Lagrange multiplier [23]. In literature, this method is applied to solve large class of PDE’s [23–25]. The key advantage of the suggested technique is that verity of the nonlinear problems can be solved without linearization or small perturbations. It is very straightforward in implementation and iterations can be performed using any symbolic mathematical software like Maple. Later on, many researchers make several modifications and use it to solve several mathematical models [43–51]. In order to elaborate the key notion of VIM, we take the differential equation of the formwhere “L” represents the linear operator, “N” represents the nonlinear operator, and is the forcing function. By using VIM, the constructed correctional functional can be written as [23–25]where “” is the Lagrange multiplier, is the restricted variation, and shows the nth approximate solution. The Lagrange multiplier can be calculated using optimality conditions and integration by parts. There are some alternative ways to calculate the Lagrange multiplier discussed in [52]. The restricted variation gives . The approximate series solution can be obtained after successive iterations for . Finally, we can write the series solution as

4. Numerical Experiments

In this section, we have solved the AC equation considering various IC’s with the help of VIM taking thickness parameter one, i.e., . We compare the converging series solution with the existing numerical and analytical solutions through absolute error tables which validates the applicability of VIM. Furthermore, we also discuss the VIM solution of AC equation with the transition layer parameter. Effect of the transition parameter on concentration is also studied through plots. CPU time for various iterations in case of each example is also presented through tables.

4.1. Example 1

Consider the AC equation of the form [15, 17]with the IC

Taking

The constructed functional for equation (6) using equation (4) iswhere “” represents the Lagrange multiplier which can be calculated via optimality conditions and restricted variation. Here, shows the restricted variation, i.e., . Taking variation on both sides of equation (9), the equation becomesor

Now, using integration by parts, we have

Hence, we have the following stationary conditions:which gives

Equation (9) becomes

For , equation (15) can be written as

Using from equation (8) to equation (16) and after integration, we get

Equation (17) gives

Using equation (18) in equation (19), we get

Similarly, for , we can get

4.1.1. Absolute Error Analysis

Table 1 shows absolute error analysis between the solutions obtained by the modified exponential function method [17], finite difference method [15], and variational iteration method. Absolute error is calculated for fourth and fifth iterations of the variational iteration method. It is clear from Table 1 that error decreases with the increase of the number of iterations. The desired and accurate result can be obtained by performing more iterations.

4.2. Example 2

In this section, we consider the AC equation (8).with the ICwhere is the integrating constant. Taking

Using equation (23) in equation (21) and after integration, we get

Similarly, using equation (24) in equation (19), we get

Similarly, for we can get

4.2.1. Absolute Error Analysis

Table 2 represents the absolute error between traveling wave solution [8] and the series solution obtained by using VIM. Absolute error is estimated for fourth and fifth iterations of VIM solution. The decreasing observation of error indicates the convergence of series solution. Higher accuracy can be attained by performing more iterations.

4.3. Example 3

Here, we consider -version of AC equation of the form [51]with initial profile containing the transition layer thickness parameter.

In order to implement VIM, takingthe correctional functional for equation (26) will take the form

For , equation (29) gives

Further iterations can be performed for up to required accuracy.

4.3.1. Relation between the Transition Parameter and Concentration

Study of relation between the transition parameter and concentration plays an important role in engineering processes, especially formation of alloys and multiphase systems. Figures 1 and 2 depict the effect of the transition parameter on concentration of binary fluids in binary alloys. Figure 1 explores that concentration increases as we increase the values of the transition parameter. Physically larger estimations of transition variable upsurges the rate of diffusion and, therefore, concentration increases. Figure 2 indicates the increasing behavior of concentration with the passage of time for the fixed value of transition parameter. In fact, at fixed diffusion rate, the concentration of molecules in the lower concentrated region increases with the time when diffusion takes place from the higher concentrated region to the lower concentrated region and thus concentration enhances.

5. Conclusion

In the present work, the VIM is effectively applied to obtain an approximate analytical solution of the AC equation. The AC model is solved for different kinds of initial conditions. A series solution obtained using VIM is compared with the numerical results obtained by using FDM and analytical solutions obtained by the traveling wave solution method and the tanh function method. Computed results are illustrated via absolute error tables, and a good agreement of the results is observed. Decreasing behavior of error in Tables 1 and 2 validates the application of VIM for solving highly nonlinear PDE’s. Results are compared after fourth and fifth iterations of VIM. The error tables show the congregating behavior of series solution to the analytical solution and numerical solution. CUP time for different iterations in case of every example is tabulated through Tables 3–5. A mathematical software Maple is used to perform iterations and absolute error calculations.

Data Availability

Data sharing is not applicable to this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Safdar Hussain and Fazal Haq contributed to formal Analysis, methodology, and writing the original draft. Abdullah Shah supervised over all activities. Dilsora Abduvalieva contributed to formal analysis and code validation. Ali Shokri analyzed and interpreted the data and contributed to funding acquisition.

References

M. Allen and J. W. Cahn, “A microscopic theory for antiphase boundary motion and its application to its applicationto antiphase domain coarsening,” Acta Metallurgica et Materialia, vol. 27, 1979.
View at: Publisher Site | Google Scholar
A. Shah, M. Sabir, and P. Bastian, “An efficient time-stepping scheme for numerical simulation of dendritic crystal growth,” European Journal of Computational Mechanics, vol. 25, no. 6, pp. 475–488, 2016.
View at: Publisher Site | Google Scholar
M. Benes, V. Chalupecký, and K. Mikula, “Geometrical image segmentation by the Allen–Cahn equation,” Applied Numerical Mathematics, vol. 51, no. 2-3, pp. 187–205, 2004.
View at: Publisher Site | Google Scholar
O. Chodosh and C. Mantoulidis, “Minimal surfaces and the Allen-Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates,” Annals of Mathematics, vol. 191, no. 1, 2020.
View at: Publisher Site | Google Scholar
C. Liu and J. Shen, “A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-Spectral method,” Physica D: Nonlinear Phenomena, vol. 179, no. 3-4, pp. 211–228, 2003.
View at: Publisher Site | Google Scholar
D. Jeong and J. Kim, “Conservative Allen–Cahn–Navier–Stokes system for incompressible two-phase fluid flows,” Computers & Fluids, vol. 156, pp. 239–246, 2017.
View at: Publisher Site | Google Scholar
L. Q. Chen, “Phase-field models for microstructure evolution,” Annual Review of Materials Research, vol. 32, no. 1, pp. 113–140, 2002.
View at: Publisher Site | Google Scholar
F. Tascan and A. Bekir, “Travelling wave solutions of the Cahn-Allen equation by using first integral method,” Applied Mathematics and Computation, vol. 207, no. 1, pp. 279–282, 2009.
View at: Publisher Site | Google Scholar
A.-M. Wazwaz, “The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1467–1475, 2007.
View at: Publisher Site | Google Scholar
H. Tariq and G. Akram, “New traveling wave exact and approximate solutions for the nonlinear Cahn–Allen equation: evolution of a nonconserved quantity,” Nonlinear Dynamics, vol. 88, pp. 581–594, 2017.
View at: Publisher Site | Google Scholar
D. Jeong, S. Lee, D. Lee, J. Shin, and J. Kim, “Comparison study of numerical methods for solving the Allen–Cahn equation,” Computational Materials Science, vol. 111, pp. 131–136, 2016.
View at: Publisher Site | Google Scholar
C. Gui and M. Zhao, “Traveling wave solutions of Allen–Cahn equation with a fractional Laplacian,” Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol. 32, 2014.
View at: Publisher Site | Google Scholar
D. Jeong and J. Kim, “An explicit hybrid finite difference scheme for the Allen–Cahn equation,” Journal of Computational and Applied Mathematics, vol. 340, pp. 247–255, 2018.
View at: Publisher Site | Google Scholar
A. Shah, M. Sabir, M. Qasim, and P. Bastian, “Efficient numerical scheme for solving the Allen-Cahn equation,” Numerical Methods for Partial Differential Equations, vol. 34, no. 5, pp. 1820–1833, 2018.
View at: Publisher Site | Google Scholar
A. Yokus and H. Bulut, “On the numerical investigations to the Cahn-Allen equation by using finite difference method,” An International Journal of Optimization and Control: Theories & Applications, vol. 9, no. 1, pp. 18–23, 2018.
View at: Publisher Site | Google Scholar
J. Shin, S.-K. Park, and J. Kim, “A hybrid FEM for solving the Allen-Cahn equation,” Applied Mathematics and Computation, vol. 244, pp. 606–612, 2014.
View at: Publisher Site | Google Scholar
H. Bulut, “Application of the modified exponential function method to the Cahn-Allen equation,” AIP Conference Proceedings, vol. 1798, Article ID 020033, 2017.
View at: Publisher Site | Google Scholar
J. H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-linear Mechanics, vol. 35, no. 1, p. 37, 2000.
View at: Publisher Site | Google Scholar
J.-H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters A, vol. 350, no. 1-2, pp. 87-88, 2006.
View at: Publisher Site | Google Scholar
S. J. Liao, The Proposed Homotopy Snalysis Technique for the Solution of Nonlinear Problems, Shanghai Jiao Tong University, Shanghai, China, 1992, Ph.D. thesis.
S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC Press, Boca Raton, FL, USA, 2003.
S. J. Liao, “Comparison between the homotopy analysis method and homotopy perturbation method,” Applied Mathematics and Computation, vol. 169, no. 2, pp. 1186–1194, 2005.
View at: Publisher Site | Google Scholar
J. H. He, “Variational iteration method—some recent results and new interpretations,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 3–17, 2007.
View at: Publisher Site | Google Scholar
A. M. Wazwaz, “The variational iteration method for rational solutions for KdV, K(2,2), Burgers, and cubic Boussinesq equations,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 18–23, 2007.
View at: Publisher Site | Google Scholar
A. M. Wazwaz, “The variational iteration method: a reliable analytic tool for solving linear and nonlinear wave equations,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 926–932, 2007.
View at: Publisher Site | Google Scholar
G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, London, UK, 1994.
G. Adomian, “A review of the decomposition method in applied mathematics,” Journal of Mathematical Analysis and Applications, vol. 135, no. 2, pp. 501–544, 1988.
View at: Publisher Site | Google Scholar
W. Li and Y. Pang, “Application of Adomian decomposition method to nonlinear systems,” Advances in Differential Equations, vol. 2020, no. 1, p. 67, 2020.
View at: Publisher Site | Google Scholar
J. K. Zhou, Differential Transformation and its Applications for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986.
M. Zaid, “Odibat, Differential transform method for solving Volterra integral equation with separable kernels,” Mathematical and Computer Modelling, vol. 7, no. 8, pp. 1144–1149, 2008.
View at: Publisher Site | Google Scholar
B. Benhammouda and H. Vazquez-Leal, A New Multi-step Technique with Differential Transform Method for Analytical Solution of Some Nonlinear Variable Delay Differential Equations, Springer Plus, Berlin, Germany, 2016.
T. Antczak and M. Arana-Jimenez, “Optimality and duality results for new classes of nonconvex quasidifferentiable vector optimization problems,” Applied and Computational Mathematics, vol. 21, no. 1, pp. 21–34, 2022.
View at: Google Scholar
W. Malfliet, “The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations,” Journal of Computational and Applied Mathematics, vol. 164-165, pp. 529–541, 2004.
View at: Publisher Site | Google Scholar
K. Ali, Khalid, M. A. Mehanna, and P. M. Ali, “Approach to a (2+1)-dimensional time-dependent date-jimbo-kashiwara-miwa equation in real physical phenomena,” Applied and Computational Mathematics, vol. 21, pp. 193–206, 2022.
View at: Publisher Site | Google Scholar
F. Qi, “Necessary and sufficient conditions for a difference defined by four derivatives of a function containing trigamma function to be completely monotonic,” Applied and Computational Mathematics, vol. 21, no. 1, pp. 61–70, 2022.
View at: Google Scholar
I. Kucukoglu and Y. Simsek, “Formulas and combinatorial identities for Catalan-type numbers and polynomials: their analysis with computational algorithms,” Applied and Computational Mathematics, vol. 21, no. 2, pp. 158–177, 2022.
View at: Google Scholar
M. A. Rufai, A. Shokri, and E. O. Omole, “A one-point third derivative hybrid multistep technique for solving second-order oscillatory and periodic problems,” Journal of Mathematics, vol. 2023, Article ID 2343215, 12 pages, 2023.
View at: Publisher Site | Google Scholar
D. A. Juraev, A. Shokri, and D. Marian, “Regularized solution of the Cauchy problem in an unbounded domain,” Symmetry, vol. 14, no. 8, p. 1682, 2022.
View at: Publisher Site | Google Scholar
A. Shokri and A. Shokri, “The new class of implicit L-stable hybrid Obrechkoff method for the numerical solution of first order initial value problems,” Computer Physics Communications, vol. 184, no. 3, pp. 529–531, 2013.
View at: Publisher Site | Google Scholar
M. A Almatrafi, “New soliton wave solutions to a nonlinear equation arising in plasma physics,” Computer Modeling in Engineering and Sciences, vol. 137, pp. 1–15, 2023.
View at: Publisher Site | Google Scholar
T. Antczak, “On optimality conditions and duality results in a class of nonconvex quasidifferentiable optimization problems,” Computational and Applied Mathematics, vol. 36, no. 3, pp. 1299–1314, 2017.
View at: Publisher Site | Google Scholar
M. B. Almatrafi, “Abundant traveling wave and numerical solutions for Novikov-Veselov system with their stability and accuracy,” Applicable Analysis, vol. 102, no. 8, pp. 2389–2402, 2023.
View at: Publisher Site | Google Scholar
T. A. Abassy, “Modified variational iteration method (non-homogeneous initial value problem),” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1222–1232, 2012.
View at: Publisher Site | Google Scholar
T. A. Abassy, M. A. El-Tawil, and H. El Zoheiry, “Toward a modified variational iteration method,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 137–147, 2007.
View at: Publisher Site | Google Scholar
W. He, H. Kong, and Y. M. Qin, “Modified variational iteration method for analytical solutions of nonlinear oscillators,” Journal of Low Frequency Noise, Vibration and Active Control, vol. 38, no. 3-4, pp. 1178–1183, 2019.
View at: Publisher Site | Google Scholar
V. K. Sinha and P. Maroju, “New development of variational iteration method using quasilinearization method for solving nonlinear problems,” Mathematics, vol. 11, no. 4, p. 935, 2023.
View at: Publisher Site | Google Scholar
S. Iskandarov and E. Komartsova, “On the influence of integral perturbations on the boundedness of solutions of a fourth-order linear differential equation,” TWMS Journal of Pure and Applied Mathematics, vol. 13, no. 1, pp. 3–9, 2022.
View at: Google Scholar
S. I. Hamidov, “Optimal trajectories in reproduction models of economic dynamics,” TWMS Journal of Pure and Applied Mathematics, vol. 13, no. 1, pp. 16–24, 2022.
View at: Google Scholar
A. Akbay, N. Turgay, and M. Ergüt, “On space-like generalized constant ratio hypersufaces in minkowski spaces,” TWMS Journal of Pure and Applied Mathematics, vol. 13, no. 1, pp. 25–37, 2022.
View at: Google Scholar
M. B. Almatrafi, “Solitary wave solutions to a fractional model using the improved modified extended tanh-function method,” Fractal and Fractional, vol. 7, no. 3, p. 252, 2023.
View at: Publisher Site | Google Scholar
J. W. Choi, H. G. Lee, D. Jeong, and J. Kim, “An unconditionally gradient stable numerical method for solving the Allen–Cahn equation,” Physica A: Statistical Mechanics and Its Applications, vol. 388, no. 9, pp. 1791–1803, 2009.
View at: Publisher Site | Google Scholar
S. S. Samaee, O. Yazdanpanah, D. D. Ganji, and Davood, “New approaches to identification of the Lagrange multiplier in the variational iteration method,” Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 37, no. 3, pp. 937–944, 2014.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2024 Safdar Hussain et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

145

Downloads

100

Citations