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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.
Highlights
The nonlinear partial differential equation (NLPDE) have been considered a fundamental icon in many research ideas
This study is organized as follows: 1. In Sect. 2.1, the modified Khater method is applied to the Cahn–Allen to get the analytical wave solutions of this equation to study the structure of the dynamics for phase separation in Fe–Cr–X (X = Mo, Cu) ternary alloys
Hosseini, Bekir, and Ansari have used the modified Kudryashov method to get the analytical wave solutions of the Cahn–Allen equation, and by focusing on their solutions, we find Eq (30) is similar to u1,2(x, t) when [–α = d, β = 1, e = a]
Summary
The nonlinear partial differential equation (NLPDE) have been considered a fundamental icon in many research ideas. Semi-analytical and numerical schemes have been derived from investigating the physical dynamics of these models such as the Adomian decomposition method, the simplest equation method, modified tanh-function method, B-spline method, iterative method [1–14].
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