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We use a general transformation, to find exact solutions for the Manakov system with variable coefficients (depending on the time ε\varepsilon ) using an improved tanh–coth method. The solutions obtained in this work are more general compared to those in other works because they involve variable coefficients. The implemented computational method is applied in a direct way on the reduced system, avoid in this way the reduction to only one equation, as occurs in the works respect to exact solutions, made by other authors. Clearly, from the solutions obtained here, new solutions are derived for the standard model (constant coefficients), complementing in this way the results obtained by other authors mentioned here. Finally, we give some discussion on the results and give the respective conclusions.

The variant Boussinesq equation has significant application in propagating long waves on the surface of the liquid layer under gravity action. In this article, the improved Bernoulli subequation function (IBSEF) method and the new auxiliary equation (NAE) technique are introduced to establish general solutions, some fundamental soliton solutions accessible in the literature, and some archetypal solitary wave solutions that are extracted from the broad-ranging solution to the variant Boussinesq wave equation. The established soliton solutions are knowledgeable and obtained as a combination of hyperbolic, exponential, rational, and trigonometric functions, and the physical significance of the attained solutions is speculated for the definite values of the included parameters by depicting the 3D profiles and interpreting the physical incidents. The wave profile represents different types of waves associated with the free parameters that are related to the wave number and velocity of the solutions. The obtained solutions and graphical representations visualize the dynamics of the phenomena and build up the mathematical foundation of the wave process in dissipative and dispersive media. It turns out that the IBSEF method and the NAE are powerful and might be used in further works to find novel solutions for other types of nonlinear evolution equations ascending in physical sciences and engineering.

In this article, we first consider approach of optical spherical magnetic antiferromagnetic model for spherical magnetic flows of ϒ\Upsilon -magnetic particle with spherical de-Sitter frame in the de-Sitter space S12{{\mathbb{S}}}_{1}^{2}. Hence, we establish new relationship between magnetic total phases and spherical timelike flows in de-Sitter space S12{{\mathbb{S}}}_{1}^{2}. In other words, the applied geometric characterization for the optical magnetic spherical antiferromagnetic spin is performed. Moreover, this approach is very useful to analyze some geometrical and physical classifications belonging to ϒ\Upsilon -particle. Besides, solutions of fractional optical systems are recognized for submitted geometrical designs. Geometrical presentations for fractional solutions are obtained to interpret the model. These obtained results represent that operation is a compatible and significant application to restore optical solutions of some fractional systems. Components of models are described by physical assertions with solutions. Additionally, we get solutions of optical fractional flow equations with designs of our results in de-Sitter space S12{{\mathbb{S}}}_{1}^{2}.

In the past few decades, many models have been proposed to address the shortcomings found in the classical theories of thermoelasticity and to allow limited speeds of heat waves. In this context, in the current paper a new generalized model of thermoelasticity based on the Moore–Gibson–Thompson (MGT) equation has been introduced. This new model can be derived by introducing the relaxation time factor into the third type of Green–Naghdi model (GN-III). In contrast to the previous works, it was taken into account that the physical properties of the material are dependent on temperature and on the viscous type. The viscoelastic medium has been assumed to obey the Kelvin–Voigt model. On the basis of the present model, thermo-viscoelastic interactions have been investigated in an unbounded orthotropic body with a cylindrical cavity. The surface of the cavity is restricted and exposed to a pulse-formed heat flow that dissolves exponentially. The characteristic thermal modulus of the material is assumed to be a linear function of temperature. The Laplace transform can be used to eliminate time dependency from control equations. Using a suitable approximate method, the transformed equations have been finally inverted by numerical inversion of the Laplace transform. Certain comparisons have been introduced to estimate the effects of the viscosity, pulsed heat, and thermal temperature-independent properties on all studied fields. A comparison with previous models of thermoelasticity is also performed in tables to verify the accuracy of the proposed model. We found from the results that the physical fields strongly depend on the viscoelastic parameter, the change of the thermal conductivity, and pulsed heat, so it is not possible to neglect their effect on the manufacturing process of machines and devices.

In this article, contact problem with fractional derivatives is studied. We use fractional derivative in the sense of Caputo. We deploy penalty function method to degenerate the obstacle problem into a system of fractional boundary value problems (FBVPs). The series solution of this system of FBVPs is acquired by using the variational iteration method (VIM). The performance as well as precision of the applied method is gauged by means of significant numerical tests. We further study the convergence and residual errors of the solutions by giving variation to the fractional parameter, and graphically present the solutions and residual errors accordingly. The outcomes thus obtained witness the high effectiveness of VIM for solving FBVPs.

In this analysis, the baffling method is used to increase the efficiency of channel heat exchangers (CHEs). The present CFD (computational fluid dynamics)-based work aims to analyze the constant property, steady, turbulent, Newtonian, and incompressible fluid flow (air), in the presence of transverse-section, arc-shaped vortex generators (VGs) with two various geometrical models, i.e., arc towards the inlet section (called arc-upstream) and arc towards the outlet section (called arc-downstream), attached to the hot lower wall, in an in-line situation, through a horizontal duct. For the investigated range of Reynolds number (from 12,000 to 32,000), the order of the thermal exchange and pressure loss went from 1.599–3.309 to 3.667–21.103 times, respectively, over the values obtained with the unbaffled exchanger. The arc-downstream configuration proved its superiority in terms of thermal exchange rate by about 14% than the other shape of baffle. Due to ability to produce strong flows, the arc-downstream baffle has given the highest outlet bulk temperature.

The main aim of this article is to use a new and simple algorithm namely the modified variational iteration algorithm-II (MVIA-II) to obtain numerical solutions of different types of fifth-order Korteweg-de Vries (KdV) equations. In order to assess the precision, stability and accuracy of the solutions, five test problems are offered for different types of fifth-order KdV equations. Numerical results are compared with the Adomian decomposition method, Laplace decomposition method, modified Adomian decomposition method and the homotopy perturbation transform method, which reveals that the MVIA-II exceptionally productive, computationally attractive and has more accuracy than the others.