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This work is concerned with the study of explicit solutions for a generalized coupled nonlinear Schrödinger equations (NLS) system with variable coefficients. Indeed, by employing similarity transformations, we show the existence of rogue wave and dark–bright soliton-like solutions for such a generalized NLS system, provided the coefficients satisfy a Riccati system. As a result of the multiparameter solution of the Riccati system, the nonlinear dynamics of the solution can be controlled. Finite-time singular solutions in the L∞ norm for the generalized coupled NLS system are presented explicitly. Finally, an n-dimensional transformation between a variable coefficient NLS coupled system and a constant coupled system coefficient is presented. Soliton and rogue wave solutions for this high-dimensional system are presented as well.

Numerous flow and heat transfer studies have relied on the construction of similarity transformations which map the nonlinear partial differential equations (PDEs) describing the flow and heat transfer, to ordinary differential equations (ODEs). For these reduced equations, one finds multiple analytic and approximate solution procedures as compared to the flow PDEs. Here, we aim at constructing multiple classes of similarity transformations that are different from those already existing in the literature. We adopt the Lie symmetry method to derive these new similarity transformations which reveal new classes of ODEs corresponding to flow equations when applied to them. With these multiple classes of similarity transformations, one finds multiple reductions in the flow PDEs to ODEs. On solving these ODEs analytically or numerically, we obtain different kinds of flow and heat transfer patterns that help in determining optimized solutions in accordance with the physical requirements of a problem. For the said purpose, we derive Lie point symmetries for the magnetohydrodynamic Casson fluid flow and heat transfer in a thin film on an unsteady stretching sheet with viscous dissipation. Linear combinations of these Lie symmetries that are again the Lie symmetries of the flow model are employed here to construct new similarity transformations. We derive multiple Lie similarity transformations through the proposed procedure which lead us to more than one class of reduced ODEs obtained by applying the deduced transformations. We analyze the flow and heat transfer by deriving analytic solutions for the obtained classes of systems of ODEs using the homotopy analysis method. Magnetic parameters and viscous dissipation influences on the flow and heat transports are investigated and presented in graphical and tabulated formats.

A mathematical description of non-Newtonian mixed convective Casson fluid stagnation point flow and transfer of heat exists in terms of partial differential equations. We considered the same to study it further under the effects of unsteadiness and varied film thickness parameters. For inclusion of these parameters in the flow model study we modified the available similarity transformations. The governing equations with three independent variables are converted into ordinary differential equations by employing the modified invertible transformations. For the mass and heat transfer in the mixed convection stagnation point unsteady flow of Casson fluid over a stretching sheet, a detailed comparative analysis is carried out in this paper, of the analytical and numerical approximation techniques. The Homotopy Analysis Method is applied for the analytical solutions while the Runge-Kutta with a Shooting Method (RKF45) and Finite Difference Method are used for obtaining the numerical solutions. With these solution schemes we present an analysis of velocity and temperature profiles under the effects of embedded parameters such as the Casson fluid parameter, unsteadiness parameter, mixed convection parameter, Prandtl number, Eckert number, and stretching ratio. The results are presented in both graphical and tabulated forms and they illustrate the dependence of mass and heat transfer characteristics of Casson fluid upon the embedded parameters. Further, we have shown an agreement between the analytical and the approximate solutions for the considered flow and heat transfer which reflects a validation of the results presented here.

Heat and mass transfer in a viscous film on an unsteady stretching surface in the presence of a variable magnetic field is investigated using Lie symmetry analysis. Combination of translational and scaling Lie point symmetries is used to obtain invariants of the model. The deduced invariants provide a new generalized class of similarity transformations that have not been used before. These transformations restructure the governing problem in a system of eight-parameter nonlinear ordinary differential equations. Analytic series solutions are obtained for the resulting system of equations for different values of these parameters and are illustrated graphically. We found that coefficients of the translational symmetries do not have any effect on the solutions however, coefficients of the scaling symmetries significantly affect the variation of temperature and concentration distribution across the fluid film and help in controlling the heat and mass transfer rates. Also, increasing the value of unsteadiness and magnetic parameter decreases the film thickness and promotes a uniform temperature and concentration across film thickness. Furthermore, we found that at the lower values of Prandtl and Schmidt number, diffusion dominates the heat and mass transfer while at the higher values advection dominates the heat and mass transfer.

Theoretical analysis of physical characteristics of unsteady, squeezing nanofluid flow is studied. The flow of nanofluid between two plates that placed parallel in a rotating system by keeping the variable physical properties: viscosity and thermal conductivity. It is analyzed by using Navier Stokes Equation, Energy Equation and Concentration equation. The prominent equations are transformed by virtue of suitable similarity transformation. Nanofluid model includes the important effects of Thermophoresis and Brownian motion. For analysis graphical results are drawn for verity parameters of our interest i.e., Injection parameter, Squeezing number, Prandtle number and Schmidt number are investigated for the Velocity field, Temperature variation and Concentration profile numerically. The findings underline that the parameter of skin friction increases when the Squeezing Reynolds number, Injection parameter and Prandtle number increases. However, it shows inverse relationship with Schmidt number and Rotation parameter. Furthermore, direct relationship of Nusselt number with injection parameter and Reynolds number is observed while its relation with Schmidt number, Rotation parameter, Brownian parameter and Thermophoretic parameter shows an opposite trend. The results are thus obtained through Parametric Continuation Method (PCM) which is further validated through BVP4c. Moreover, the results are tabulated and set forth for comparison of findings through PCM and BVP4c which shows that the obtained results correspond to each other.

This paper focuses on the theoretical analysis of blood flow in the presence of thermal radiation and chemical reaction under the influence of time dependent magnetic field intensity. Unsteady non linear partial differential equations of blood flow consider time dependent stretching velocity, the energy equation also accounts time dependent temperature of vessel wall and the concentration equation includes the time dependent blood concentration. The governing non linear partial differential equations of motion, energy and concentration are converted into ordinary differential equations using similarity transformations solved numerically by applying ode45. The effect of physical parameters, viz., the permeability parameter, unsteadiness parameter, Prandtl number, Hartmann number, thermal radiation parameter, chemical reaction parameter and Schmidt number on flow variables, viz., velocity of blood flow in vessel, temperature and concentration of blood, has been analyzed and discussed graphically. From the simulation study the following important results are obtained: velocity of blood flow increases with the increment of both permeability and unsteadiness parameter. The temperature of blood increases at the vessel wall as the Prandtl number and Hartmann number increase. Concentration of blood decreases as time dependent chemical reaction parameter and Schmidt number increases.

We apply the theory of Lie point symmetries for the study of a family of partial differential equations which are integrable by the hyperbolic reductions method and are reduced to members of the Painlevé transcendents. The main results of this study are that from the application of the similarity transformations provided by the Lie point symmetries, all the members of the family of the partial differential equations are reduced to second-order differential equations, which are maximal symmetric and can be linearized.

In this study, an exact solution of the Navier-Stokes and energy equations is obtained for the problem of unsteady three-dimensional stagnation point flow and heat transfer of viscous, incompressible fluid on a flat plate. An external flow with strain rate impinges obliquely on the flat plate when the plate is assumed to be with transpiration. This flow consists of an irrotational stagnation-point flow (Hiemenz) and a tangential component. The relative importance of these two flows is measured by a parameter . Appropriate similarity transformations are introduced, for the first time, to reduce the governing Navier-Stokes and energy equations to a coupled system of ordinary differential equations. The fourth-order Runge-Kutta method along with a shooting technique is applied to numerically solve the ordinary differential equations. The results obtained from numerical procedure are presented and discussed for a wide range of parameters characterizing the problem. The results achieved reveal that the transpiration rate has a considerable effect on the distributions of velocity components, temperature and pressure. Moreover, it is shown that the main consequence of the free stream obliqueness is to move the stagnation point away from the origin of the coordinate system.

A numerical study based on finite difference approximation is attempted to analyze the bulk flow, micro spin flow and heat transfer phenomenon for micropolar fluids dynamics through Darcy porous medium. The fluid flow mechanism is considered over a moving permeable sheet. The heat transfer is associated with two different sets of boundary conditions, the isothermal wall and isoflux boundary. On the basis of porosity of medium, similarity functions are utilized to avail a set of ordinary differential equations. The non-linear coupled ODE’s have been solved with a very stable and reliable numerical scheme that involves Simpson’s Rule and Successive over Relaxation method. The accuracy of the results is improved by making iterations on three different grid sizes and higher order accuracy in the results is achieved by Richardson extrapolation. This study provides realistic and differentiated results with due considerations of micropolar fluid theory. The micropolar material parameters demonstrated reduction in the bulk fluid speed, thermal distribution and skin friction coefficient but increase in local heat transfer rate and couple stress. The spin behavior of microstructures is also exhibited through microrotation vector N ( η ) .

In this paper, a new analytical approach based on homotopy perturbation Sumudu transform method (HPSTM) to a two-dimensional viscous flow with a shrinking sheet is presented. The series solution is obtained by HPSTM coupled with Padé approximants to handle the condition at infinity. The HPSTM is a combined form of the Sumudu transform method, homotopy perturbation method and He’s polynomials. This scheme finds the solution without any discretization or restrictive assumptions and avoids round-off errors. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and computationally very attractive. doi:10.14456/WJST.2014.39