Analytical, semi-analytical, and numerical solutions for the Cahn–Allen equation

Mostafa M.A. Khater, Choonkil Park, Dianchen Lu, Raghda A.M. Attia

Research output: Contribution to journalArticle

Abstract

This paper studies the analytical, semi-analytical, and numerical solutions of the Cahn–Allen equation, which plays a vital role in describing the structure of the dynamics for phase separation in Fe–Cr–X (X= Mo, Cu) ternary alloys. The modified Khater method, the Adomian decomposition method, and the quintic B-spline scheme are implemented on our suggested model to get distinct kinds of solutions. These solutions describe the dynamics of the phase separation in iron alloys and are also used in solidification and nucleation problems. The applications of this model arise in many various fields such as plasma physics, quantum mechanics, mathematical biology, and fluid dynamics. The comparison between the obtained solutions is represented by using figures and tables to explain the value of the error between exact and numerical solutions. All solutions are verified by using Mathematica software.

Original languageEnglish
Article number9
JournalAdvances in Difference Equations
Volume2020
Issue number1
DOIs
StatePublished - 2020 Dec 1

Fingerprint

Phase Separation
Analytical Solution
Numerical Solution
Mathematical Biology
Plasma Physics
Adomian Decomposition Method
Quintic
Mathematica
Solidification
Fluid Dynamics
B-spline
Nucleation
Ternary
Phase separation
Iron
Quantum Mechanics
Tables
Figure
Exact Solution
Distinct

Keywords

  • Adomian decomposition method
  • Analytical, semi-analytical, and numerical solutions
  • Cahn–Allen equation
  • Modified Khater method
  • The quintic B-spline scheme

Cite this

Khater, Mostafa M.A. ; Park, Choonkil ; Lu, Dianchen ; Attia, Raghda A.M. / Analytical, semi-analytical, and numerical solutions for the Cahn–Allen equation. In: Advances in Difference Equations. 2020 ; Vol. 2020, No. 1.
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Analytical, semi-analytical, and numerical solutions for the Cahn–Allen equation. / Khater, Mostafa M.A.; Park, Choonkil; Lu, Dianchen; Attia, Raghda A.M.

In: Advances in Difference Equations, Vol. 2020, No. 1, 9, 01.12.2020.

Research output: Contribution to journalArticle

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AB - This paper studies the analytical, semi-analytical, and numerical solutions of the Cahn–Allen equation, which plays a vital role in describing the structure of the dynamics for phase separation in Fe–Cr–X (X= Mo, Cu) ternary alloys. The modified Khater method, the Adomian decomposition method, and the quintic B-spline scheme are implemented on our suggested model to get distinct kinds of solutions. These solutions describe the dynamics of the phase separation in iron alloys and are also used in solidification and nucleation problems. The applications of this model arise in many various fields such as plasma physics, quantum mechanics, mathematical biology, and fluid dynamics. The comparison between the obtained solutions is represented by using figures and tables to explain the value of the error between exact and numerical solutions. All solutions are verified by using Mathematica software.

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