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

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.

New optical solitons of conformable resonant nonlinear Schrödinger’s equation

“…In the literature, there are several methods for acquiring traveling solutions to the NLEEs, such as first integral [1,2], exp(−φ(χ))-expansion [3,4], Jacobi elliptic functions [5,6], modified Khater [7,8], generalized Kudryashov [9,10], modified auxiliary equation [11,12], new extended direct algebraic method [13,14]], functional variable [15,16], sub-equation [17,18], (G′/G)-expansion [19,20] and others .…”

Numerical investigation for the fractional nonlinear space‐time telegraph equation via the trigonometric Quintic B‐spline scheme

“…Thus, several divisional derivative operators, such as Riemann–Liouville, Caputo, and the conformal fractional, Atangana–Baleanu fractional operator, 20‐22 have been dependent to transform the nonlinear divisive differential equations into ordinary differential equations 23 . These operators have been used for many nonlinear evolutional equations and have shown their usefulness and their strength 24,25 …”

“…23 These operators have been used for many nonlinear evolutional equations and have shown their usefulness and their strength. 24,25 In the area of electromagnetic waves, this manuscript treats obedient fractional non-linear model time-space, namely, the telegraph equation. This model illustrates the modern or voltage standard of an electrical transport system for the gap between electrical propagation and the application of an electromagnetic wave.…”

An efficient finite element method and error analysis for eigenvalue problem of Schrödinger equation with an inverse square potential on spherical domain

“…Therefore new tools and techniques, different from linear elliptic operators with bounded coefficients, are urgently needed for such an eigenvalue problem with the IS potential, both in analysis and in numerics. Ghanbari et al [12][13][14][15] and Khater et al [16][17][18][19][20] discussed some effective numerical methods to remove the singularity of nonlocal operators. In addition, some other works [21][22][23][24][25][26] mainly focus on studying exact solitary wave solutions.…”