Reduction Of Partial Differential Equations Research Articles

A numerical study of heat and mass transfer characteristic of three-dimensional thermally radiated bi-directional slip flow over a permeable stretching surface

Within fluid mechanics, the flow of hybrid nanofluids over a stretching surface has been extensively researched due to their influence on the flow and heat transfer properties. Expanding on this concept by introducing porous media, the current study explore the flow and heat and mass transport characteristics of hybrid nanofluid. This investigation includes the effect of magnetohydrodynamic (MHD) with chemical reaction, thermal radiation, and slip effects. The nanoparticles, copper, and alumina are combined with water for the formation of a hybrid nanofluid. Using the self-similar method for the reduction of Partial differential equations (PDEs) to the system of Ordinary differential equations (ODEs). These nonlinear equation systems are solved numerically using the bvp4c (boundary value solver) technique. The effect of the different physical non-dimensional flow parameters on different flow profiles such as velocity, temperature, concentration, skin friction, Nusselt and mass transfer rate are depicted through graphs and tables. The velocity profiles diminish with the effect of magnetic and slip parameters. The temperature and concentration slip parameters reduce the temperature and concentration profile respectively. The higher values of magnetic factor lessened the skin friction coefficient for both slip and no-slip conditions. An elevation in the thermal slip parameter reduced the boundary layer thickness and the heat transfer from the surface to the fluid. The Nusselt number amplified with the climbing values of the radiation parameter. The mass transfer rate depressed with the solutal slip parameter. Comparison is made with the published work in the literature and there is excellent agreement between them.

Convolution Representation of Traveling Pulses in Reaction-Diffusion Systems

Convolution representation manifests itself as an important tool in the reduction of partial differential equations. In this study, we consider the convolution representation of traveling pulses in reaction-diffusion systems. Under the adiabatic approximation of inhibitor, a two-component reaction-diffusion system is reduced to a one-component reaction-diffusion equation with a convolution term. To find the traveling speed in a reaction-diffusion system with a global coupling term, the stability of the standing pulse and the relation between traveling speed and bifurcation parameter are examined. Additionally, we consider the traveling pulses in the kernel-based Turing model. The stability of the spatially homogeneous state and most unstable wave number are examined. The practical utilities of the convolution representation of reaction-diffusion systems are discussed.

Symmetry Analysis, Exact Solutions and Conservation Laws of a Benjamin–Bona–Mahony–Burgers Equation in 2+1-Dimensions

The Benjamin–Bona–Mahony equation describes the unidirectional propagation of small-amplitude long waves on the surface of water in a channel. In this paper, we consider a family of generalized Benjamin–Bona–Mahony–Burgers equations depending on three arbitrary constants and an arbitrary function G(u). We study this family from the standpoint of the theory of symmetry reductions of partial differential equations. Firstly, we obtain the Lie point symmetries admitted by the considered family. Moreover, taking into account the admitted point symmetries, we perform symmetry reductions. In particular, for G′(u)≠0, we construct an optimal system of one-dimensional subalgebras for each maximal Lie algebra and deduce the corresponding (1+1)-dimensional nonlinear third-order partial differential equations. Then, we apply Kudryashov’s method to look for exact solutions of the nonlinear differential equation. We also determine line soliton solutions of the family of equations in a particular case. Lastly, through the multipliers method, we have constructed low-order conservation laws admitted by the family of equations.

Fully developed entropy-optimized MHD nanofluid flow by a variably thickened rotating surface

Entropy generation analysis for three-dimensional (3D) magnetohydrodynamic (MHD) flow of viscous fluid through a rotating disk is addressed in this article. Entropy generation is explored as a function of temperature and velocity. The modeling of the considered problem is performed through Buongiorno model. Conservation of energy comprises dissipation, convective heat transport and Joule heating. Flow under consideration is because of nonlinear stretching velocity of disk. Transformations used lead to the reduction of partial differential equations into ordinary differential equations. Total entropy generation rate is scrutinized. Non-linear computations have been carried out. Domain of convergence for the obtained solutions is identified. Radial, axial and tangential velocities are interpreted. Entropy equation is studied in the presence of dissipation, Brownian diffusion and thermophoresis effects. Velocity and temperature gradients are discussed graphically. Meaningful results are summed up in the concluding section.

Contact interaction of two rectangular plates made from different materials with an account of physical nonlinearity

A mathematical model of a contact interaction between two plates made from materials with different elasticity modulus is derived taking into account physical and design nonlinearities. In order to study the stress–strain state of this complex mechanical structure, the method of variational iteration has been employed allowing for reduction of partial differential equations to ordinary differential equations (ODEs). The theorem regarding convergence of this method is formulated for the class of similar-like problems. The convergence of the proposed iterational procedure used for obtaining a solution to contact problems of two plates is proved. In the studied case, the physical nonlinearity is introduced with the help of variable parameters associated with plate stiffness. The work is supplemented with a few numerical examples. Both Fourier and Morlet power spectra are employed to detect and analyse regular and chaotic vibrations of two interacting plates.

Isostable reduction with applications to time-dependent partial differential equations

Isostables and isostable reduction, analogous to isochrons and phase reduction for oscillatory systems, are useful in the study of nonlinear equations which asymptotically approach a stationary solution. In this work, we present a general method for isostable reduction of partial differential equations, with the potential power to reduce the dimensionality of a nonlinear system from infinity to 1. We illustrate the utility of this reduction by applying it to two different models with biological relevance. In the first example, isostable reduction of the Fokker-Planck equation provides the necessary framework to design a simple control strategy to desynchronize a population of pathologically synchronized oscillatory neurons, as might be relevant to Parkinson's disease. Another example analyzes a nonlinear reaction-diffusion equation with relevance to action potential propagation in a cardiac system.

The flow of a micropolar fluid through a porous expanding channel: A Lie group analysis

In this paper, the flow of a micropolar fluid through a porous channel with deforming walls is investigated. The Lie group method is applied to determine symmetry reductions of partial differential equations. Compared with previous work, the similar variable for the microrotation velocity also is obtained, which is an extension of the case corresponding to the Newtonian fluids (Boutros et al., 2007). This problem we study here can be considered as an extension of the model discussed in Boutros et al. (2007). The effect of the permeation Reynolds number Re, the micropolar parameter K and the expansion ratio α are studied numerically and the results are plotted. Furthermore, the multiple solutions also can be found for some values of parameters.

Lie Group Method for Studying the Heat Generation Effect on Freeconvection Laminar Boundary-layer Flow Over a Vertical Flat Plate

The nonlinear equations of heat and mass transfer in two-dimensional free-convection, laminar, boundary layer flow of a viscous incompressible fluid over a vertical plate with thermophoresis and heat generation effect have been considered. We apply Lie-group method for determining symmetry reductions of partial differential equations. Liegroup method starts out with a general infinitesimal group of transformations under which the given partial differential equations are invariant. The determining equations are a set of linear differential equations, the solution of which gives the transformation function or the infinitesimals of the dependent and independent variables. After the group has been determined, a solution to the given partial differential equations may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the number of independent variables of the system. The effect of the heat generation parameter He, the Prandtl number Pr, the Schimted number Sc, the thermophoretic parameter τ, the solutal Grashof number Gc and the thermal Grashof number Gr on velocity, concentration and temperature have been studied and the results are plotted.

Magnetohydrodynamic Mixed Convection Flow of Thixotropic Fluid with Thermophoresis and Joule Heating

An analysis has been carried out for the magnetohydrodynamic mixed convection flow of thixotropic fluid over a moving surface. Heat transfer is considered in the presence of thermophoresis, Joule heating, and radiative effects. Similarity transformations are used for the reduction of partial differential equations into the ordinary differential equations. A dimensionless nonlinear problem is computed with the homotopy analysis method. Convergence analysis of homotopic solutions is ensured graphically and numerically. Graphical results are displayed and analyzed. Numerical values of wall shear stress and heat transfer rate are computed and discussed. Comparison with the previous results in a limiting case is made and excellent agreement is noticed.

3-D flow of Jeffery fluid in a channel with stretched wall

This article discusses the generalized three-dimensional channel flow of a Jeffery fluid. The transformation method has been utilized for the reduction of partial differential equations into ordinary differential equations. The solutions of nonlinear systems are presented by employing the homotopy analysis method (HAM). The reported graphical results are analyzed. A comparative study is made with the previous results of viscous fluid.

The Reduction of Chazy Classes and Other Third-Order Differential Equations Related to Boundary Layer Flow Models

We study the symmetries, conservation laws and reduction of third-order equations that evolve from a prior reduction of models that arise in fluid phenomena. These could be the ordinary differential equations (ODEs) that are reductions of partial differential equations (PDEs) or, alternatively, PDEs related to given ODEs. In this class, the analysis includes the well-known Blasius, Chazy, and other associated third-order ODEs.

Hidden and Not So Hidden Symmetries

Hidden symmetries entered the literature in the late Eighties when it was observed that there could be gain of Lie point symmetry in the reduction of order of an ordinary differential equation. Subsequently the reverse process was also observed. Such symmetries were termed “hidden”. In each case the source of the “new” symmetry was a contact symmetry or a nonlocal symmetry, that is, a symmetry with one or more of the coefficient functions containing an integral. Recent work by Abraham‐Shrauner and Govinder (2006) on the reduction of partial differential equations demonstrates that it is possible for these “hidden” symmetries to have a point origin. In this paper we show that the same phenomenon can be observed in the reduction of ordinary differential equations and in a sense loosen the interpretation of hidden symmetries.

Three-dimensional flow of a Jeffery fluid over a linearly stretching sheet

This investigation reports the three-dimensional flow of Jeffrey fluid over a linearly stretching surface. Transformation method has been utilized for the reduction of partial differential equations into the ordinary differential equations. The solutions of the nonlinear systems are presented by a homotopy analysis method (HAM). The reported graphical results are analyzed. A comparative study with the previous results of viscous fluid in the literature is made.

A similarity reduction of the Grad–Shafranov equation

A direct method for finding similarity reductions of partial differential equations is applied to a specific case of the Grad–Shafranov equation. As an illustration of the method, the frequently used Solov’ev equilibrium is derived. The method is employed to obtain a new family of exact analytical solutions, which contain both the classical and nonclassical group-invariant solutions of the Grad–Shafranov equation and thus greatly extends the range of the available analytical solutions. All the group-invariant solutions based on the classical Lie symmetries are shown to be particular cases in the new family of solutions.

Exact solutions of a generalized Boussinesq equation

We analyze a generalized Boussinesq equation using the theory of symmetry reductions of partial differential equations. The Lie symmetry group analysis of this equation shows that the equation has only a two-parameter point symmetry group corresponding to traveling-wave solutions. To obtain exact solutions, we use two procedures: a direct method and the G′/G-expansion method. We express the traveling-wave solutions in terms of hyperbolic, trigonometric, and rational functions.

Travelling wave solutions for a generalized double dispersion equation

We analyze the generalization of a double dispersion equation by means of the theory of symmetry reductions of partial differential equations. We consider travelling wave reductions depending on the form of an arbitrary function. We present some explicit solutions: soliton solutions, kinks and anti-kinks.

On a preliminary group classification of the nonlinear heat conduction equation

We apply the method of preliminary group classification to a particular form of the nonlinear heat equation,viz. u t = f(u)u xx + g(u)u x 2. This results in an optimal system of one-dimensional subalgebras required for the systematic reduction of partial differential equations (PDEs). This system of subalgebras yields different forms of f(u) and g(u) that allow for an additional symmetry. We also apply a preliminary group classification with respect to potential symmetries to a related system with g(u) = f′(u).

Exact solutions using symmetry methods and conservation laws for the viscous flow through expanding–contracting channels

We present a range of definitions and methods dealing with the reduction of partial differential equations on the basis of the underlying symmetry structure, conservation laws and a combination of these. The method is used to reduce a complex system to an easy-to-handle second-order ordinary differential equation system independent of restrictions on any physical parameters. In particular, we construct exact solutions of a system modelling viscous flow between slowly expanding and contracting walls.

Lie-group method of solution for steady two-dimensional boundary-layer stagnation-point flow towards a heated stretching sheet placed in a porous medium

The boundary-layer equations for two-dimensional steady flow of an incompressible, viscous fluid near a stagnation point at a heated stretching sheet placed in a porous medium are considered. We apply Lie-group method for determining symmetry reductions of partial differential equations. Lie-group method starts out with a general infinitesimal group of transformations under which the given partial differential equations are invariant. The determining equations are a set of linear differential equations, the solution of which gives the transformation function or the infinitesimals of the dependent and independent variables. After the group has been determined, a solution to the given partial differential equations may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the number of independent variables of the system. The effect of the velocity parameter λ, which is the ratio of the external free stream velocity to the stretching surface velocity, permeability parameter of the porous medium k1, and Prandtl number Pr on the horizontal and transverse velocities, temperature profiles, surface heat flux and the wall shear stress, has been studied.

Double reduction of PDEs from the association of symmetries with conservation laws with applications

We use the definition of the association of symmetries of partial differential equations with conservation laws [A. Kara, F. Mahomed, The relationship between symmetries and conservation laws, Int. J. Theor. Phys. 39 (1) (2000) 23–40] to find a double reduction of partial differential equations with two independent variables. Applications to the linear heat equation, the BBM equation, the sine-Gordon equation and a system of differential equations from one dimensional gas dynamics illustrate the new theory.