Power Series and Finite Element Methods for Solving Cahn-Hilliard Equation

  • Peter Oluwafemi Olatunji Department of Mathematical Sciences, Adekunle Ajasin University, P.M.B. 001, Akungba-Akoko, Ondo State, Nigeria
  • Richard Olu Awonusika Department of Mathematical Sciences, Adekunle Ajasin University, P.M.B. 001, Akungba-Akoko, Ondo State, Nigeria
Keywords: Cahn-Hilliard equation, power series, generalised Cauchy product, nonlinear partial differential equation, finite element method, time-stepping scheme, sparse linear algebra technique

Abstract

The Cahn-Hilliard equation is a nonlinear partial differential equation that describes spinodal decomposition, coarsening phenomena, and the dynamics of phase separation for ternary iron alloys. This article employs a power series technique and the finite element method to obtain analytical and numerical solutions of the Cahn-Hilliard equation, respectively. For the power series method, the nonlinear terms in the proposed problem are dealt with using the generalised Cauchy product of power series, which allows us to obtain an explicit recursion formula for the expansion function coefficient of the series solution. On the other hand, numerical solution to the Cahn-Hilliard equation is obtained using the finite element method that is based on the implicit time-stepping scheme and the sparse linear algebra technique. The obtained analytical and numerical solutions are compared with the exact solution to illustrate the accuracy and reliability of the proposed methods. The absolute errors obtained show that the proposed methods are accurate and reliable. Two and three dimensional graphs of the exact and approximate solutions are presented for comparison purposes.

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Published
2025-04-07
How to Cite
Olatunji, P. O., & Awonusika, R. O. (2025). Power Series and Finite Element Methods for Solving Cahn-Hilliard Equation. Earthline Journal of Mathematical Sciences, 15(4), 473-487. https://doi.org/10.34198/ejms.15425.473487
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