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This article gives a survey on computational and numerical solutions for the phase separation in Fe‐Cr‐X (X = Mo, Cu) ternary alloys. This separation is mathematically given by the Atangana–Baleanu time‐fractional Cahn–Allen equation. The generalized Kudryashov, sech–tanh, and the Adomian decomposition approaches are employed to get a bundle of analytical and numerical solutions. The obtained analytical solutions are used to evaluate the initial and boundary conditions of the phase ternary alloys separation operation. Moreover, the stability property of solutions is investigated based on the characteristics of the Hamiltonian system. Novel properties of the investigated model are figured out by distinct types of sketches. Finally, the accuracy of both used computational schemes is illustrated. [ABSTRACT FROM AUTHOR]

In this research paper, the modified Khater method, the Adomian decomposition method, and B-spline techniques (cubic, quintic, and septic) are applied to the deoxyribonucleic acid (DNA) model to get the analytical, semi-analytical, and numerical solutions. These solutions comprise much information about the dynamical behavior of the homogenous long elastic rods with a circular section. These rods constitute a pair of the polynucleotide rods of the DNA molecule which are plugged by an elastic diaphragm that demonstrates the hydrogen bond's role in this communication. The stability property is checked for some solutions to show more effective and powerful of obtained solutions. Based on the role of analytical and semi-analytical techniques in the motivation of the numerical techniques to be more accurate, the B-spline numerical techniques are applied by using the obtained exact solutions on the DNA model to show which one of them is more accurate than other, to explain more of the dynamic behavior of the homogenous long elastic rods, and to show the coincidence between the different types of obtained solutions. The obtained solutions verified with Maple 16 & Mathematica 12 by placing them back into the original equations. The performance of these methods shows the power and effectiveness of them for applying to many different forms of the nonlinear evolution equations with an integer and fractional order. [ABSTRACT FROM AUTHOR]

This paper investigates the analytical and semi‐analytical solutions of the time‐fractional Cahn–Allen equation, which describes the structure of dynamic for phase separation in Fe‐Cr‐X (X = Mo, Cu) ternary alloys. We apply a modified auxiliary equation method and the Adomian decomposition method to get distinct solutions to our suggested model. These solutions describe the dynamic of the phase separation in iron alloys and use in solidification and nucleation problems. The applications of this method arise in many various fields such as plasma physics, quantum mechanics, mathematical biology, and fluid dynamics. We apply a conformable fractional derivative to this fractional model to convert it into a nonlinear partial differential equation with integer order. We obtain many analytical wave solutions and also apply a semi‐analytical scheme to calculate the absolute value of error. All solutions are verified by using Mathematica software. [ABSTRACT FROM AUTHOR]

This research paper investigates the computational solutions of the resonant schrödinger's equation. The modified Khater method and Adomian decomposition method are applyied for construct new analytical traveling and semi-analytical wave solutions. This model describes the pulse phenomena and studied in non-linear optics. For further illustration of our obtained solutions, some distinct types of sketches are given. [ABSTRACT FROM AUTHOR]

This research paper extracts novel analytical and semi-analytical wave solutions of the longitudinal wave equation in a magneto-electro-elastic circular rod by using the modified Khater method as one of the most novel and general computational methods and the Adomian decomposition method as a semi-analytical method. The longitudinal waves in metallic thin films are explained for the first time by Nilsson and Lindau, who used the visual evidence. They noted subdued anomalies in the ports of thin (≅ 100 Å) Ag layers deposited on amorphous silica for p-polarized light at frequencies padlock to the dynamic plasma frequency. These properties are studied by our two suggested methods and are explained by sketching some of our obtained solutions. Moreover, the stability property is tested for our obtained solutions by using the features of the Hamiltonian system. The performance of our used methods shows the power and effectiveness of these methods and their ability to apply on many different forms of nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

This paper studies the analytical and semi-analytical wave solutions for the longitudinal wave equation. Moreover, it examines the performance of the modified auxiliary equation method and Adomian decomposition method on this model. This model describes the dispersion in the circular rod that dispersion caused by the transverse Poisson's effect in electro-magneto-elastic. Many explicit wave solutions are found by using the analytical technique. These solutions allow studying the physical properties of this model. The comparison between the analytical and semi-analytical solutions is discussed to show the value of the absolute error between them. [ABSTRACT FROM AUTHOR]