On the numerical investigations to the Cahn-Allen equation by using finite difference method

Asıf Yokus, Hasan Bulut


In this study, by using the finite difference method (FDM for short) and operators, the discretized Cahn-Allen equation is obtained.  New initial condition for the Cahn-Allen equation is introduced, considering the analytical solution given in Application of the modified exponential function method to the Cahn-Allen equation, AIP Conference Proceedings 1798, 020033 [1]. It is shown that the FDM is stable for the usage of the Fourier-Von Neumann technique. Accuracy of the method is analyzed in terms of the errors in and Furthermore, the FDM is treated in order to obtain the numerical results and to construct a table including numerical and exact solutions as well as absolute measuring error. A comparison between the numerical and the exact solutions is supported with two and three dimensional graphics via Wolfram Mathematica 11.


Cahn-Allen equation; Finite Difference Method; Numerical Solution

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Bulut, H. (2017). Application of the modified exponential function method to the Cahn-Allen equation, AIP Conference Proceedings 1798, 020033.

Villarreal, J. M. (2014). Approximate solutions to the allen-cahn equation using the finite difference method, Thesis, B.S., Texas A & M International University.

Xue, C. X., Pan, E. & Zhang, S. Y. (2011). Solitary waves in a magneto-electro-elastic circular rod, Smart Materials and Structures, 20(105010), 1-7.

Russell, J. S. (1844). Report on waves, 14th Mtg of the British Association for the Advancement of Science.

Yang, Y. J. (2013). New application of the -expansion method to KP equation, Applied Mathematical Sciences, 7(20), 959-967.

Yokus, A. (2011). Solutions of some nonlinear partial differential equations and comparison of their solutions, Ph.D. Thesis, Firat University.

Guo, S. & Zhou, Y. (2011). The extended -expansion method and its applications to the Whitham-Broer-Like equations and coupled Hirota-Satsuma KdV equations, Applied Mathematics and Computation, 215 (9) 3214-3221.

Yokus, A. (2017). Numerical solution for space and time fractional order Burger type equation, Alexandria Engineering Journal, https://doi.org/10.1016/j.aej.2017.05.028.

Yokus, A. & Kaya, D. (2015). Conservation laws and a new expansion method for sixth order Boussinesq equation, AIP Conference Proceedings 1676, 020062.

Jawad, A. J. M., Petkovic, M. D. & Biswas, A. (2010). Modified simple equation method for nonlinear evolution equations, Applied Mathematics and Computation, 217, 869-877.

Su, L., Wang, W. & Yang, Z. (2009). Finite difference approximations for the fractional advection–diffusion equation, Physics Letters A 373, 4405–4408.

Odibat, Z. M. & Shawagfeh, N. T. (2007). Generalized Taylor’s formula. Applied Mathematics and Computation, 186 286-293.2.

Liu, F., Zhuang, P., Anh, V., Turner, I. & Burrage K. (2007). Stability and convergence of the difference methods for the space-time fractional advection–diffusion equation, Applied Mathematics and Computation 191, 2–20.

Su, L., Wang, W. & Yang, Z. (2009). Finite difference approximations for the fractional advection–diffusion equation, Physics Letters A, 373, 4405–4408.

Miura, M. R. (1978). Backlund transformation, Springer, Berlin.

Motsa, S. S., Sibanda, P., Awad, F.G. & Shateyi, S. (2010). A new spectral-homotopy analysis method for the MHD Jeffery-Hamel problem, Computers & Fluids, 39(7), 1219-1225.

Domairry, G., Mohsenzadeh, A. & Famouri, M. (2009). The application of homotopy analysis method to solve nonlinear differential equation governing Jeffery-Hamel flow, Communications in Nonlinear Science and Numerical Simulation, 14(1), 85-95.

Joneidi, A.A., Domairry, G. & Babaelahi, M. (2010). Three analytical methods applied to Jeffery-Hamel flow, Communications in Nonlinear Science and Numerical Simulation, 15(11), 3423–3434.

Alam, M. N., Hafez, M. G., Akbar, M. A. & Roshid, H. O. (2015). Exact Solutions to the (2+1)-Dimensional Boussinesq Equation via exp (Φ(η))-Expansion Method, Journal of Scientific Research, 7(3), 1-10.

Roshid, H. O. & Rahman, Md. A. (2014). The exp (−Φ(η))-expansion method with application in the (1+1)-dimensional classical Boussinesq equations, Results in Physics, 4, 150–155.

Abdelrahman, M. A. E., Zahran, E. H. M. & Khater, M. M. A. (2015). The exp(−ϕ(ξ))-Expansion Method and Its Application for Solving Nonlinear Evolution Equations, International Journal of Modern Nonlinear Theory and Application, 4, 37-47.

Baskonus, H. M., & Bulut, H. (2015). On the complex structures of Kundu-Eckhaus equation via improved Bernoulli sub-equation function method. Waves in Random and Complex Media, 25(4), 720-728.

Bulut, H., Atas, S. S., & Baskonus, H. M. (2016). Some novel exponential function structures to the Cahn–Allen equation. Cogent Physics, 3(1), 1240886.

Wang, M., Li, X., & Zhang, J. (2008). The (G′ G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A, 372(4), 417-423.

Feng, J., Li, W., & Wan, Q. (2011). Using G′ G-expansion method to seek the traveling wave solution of Kolmogorov–Petrovskii–Piskunov equation. Applied Mathematics and Computation, 217(12), 5860-5865.

Yokus, A., Baskonus, H. M., Sulaiman, T. A., & Bulut, H. (2018). Numerical simulation and solutions of the two‐component second order KdV evolutionarysystem. Numerical Methods for Partial Differential Equations, 34(1), 211-227.

Şener, S. Ş., Saraç, Y., & Subaşı, M. (2013). Weak solutions to hyperbolic problems with inhomogeneous Dirichlet and Neumann boundary conditions. Applied Mathematical Modelling, 37(5), 2623-2629.

Subaşı, M., Şener, S. Ş., & Saraç, Y. (2011). A procedure for the Galerkin method for a vibrating system. Computers & Mathematics with Applications, 61(9), 2854-2862.

Rezzolla, L. (2011). Numerical methods for the solution of partial differential equations. Lecture Notes for the COMPSTAR School on Computational Astrophysics, 8-13.

Yokus, A., & Kaya, D. (2017). Numerical and exact solutions for time fractional Burgers’ equation. Journal of Nonlinear Sciences and Applications, 10(7), 3419-3428.

DOI: http://dx.doi.org/10.11121/ijocta.01.2019.00561


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