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

• Published in: International journal of differential equations Vol. 2024; pp. 1 - 9
• Main Authors: Hussain, Safdar, Haq, Fazal, Shah, Abdullah, Abduvalieva, Dilsora, Shokri, Ali
• Format: Journal Article
• Language: English
• Published: New York Hindawi 23-03-2024; John Wiley & Sons, Inc; Hindawi Limited
• Subjects: Error analysis; Exact solutions; Finite difference method; Finite element analysis; Hyperbolic functions; Lagrange multiplier; Mathematical models; Partial differential equations; Phase transitions; Researchers; Traveling waves
• ISSN: 1687-9643; 1687-9651
• DOI: 10.1155/2024/8835138

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.