Lie-group method for unsteady flows in a semi-infinite expanding or contracting pipe with injection or suction through a porous wall

The unsteady incompressible laminar flow in a semi-infinite porous circular pipe with injection or suction through the pipe wall whose radius varies with time is considered. The present analysis simulates the flow field by the burning of inner surface of cylindrical grain in a solid rocket motor, in which the burning surface regresses with time. 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 given partial differential equations are invariant, then, the determining equations are derived [Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, Wiley, New York, 1999; Hydon, Symmetry Methods for Differential Equations, Cambridge University Press, Cambridge, 2000; Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1986; Seshadri, Na, Group invariance in engineering boundary value problems, Springer, New York, 1985; Yi, Fengxiang, Lie symmetries of mechanical systems with unilateral holonomic constraints, Chinese Sci. Bull. 45 (2000) 1354–1358; Moritz, Schwalm, Uherka, Finding Lie groups that reduce the order of discrete dynamical systems, J. Phys. A: Math. 31 (1998) 7379–7402; Nucci, Clarkson, The nonclassical method is more general than the direct method for symmetry reductions. An example of the Fitzhugh–Nagumo equation, Phys. Lett. A 164 (1992) 49–56; Basarab, Lahno, Group classification of nonlinear partial differential equations: a new approach to resolving the problem, Proceedings of Institute of Mathematics of NAS of Ukraine, vol. 43, 2002, pp. 86–92; Burde, Expanded Lie group transformations and similarity reductions of differential equations, Proceedings of Institute of Mathematics of NAS of Ukraine, vol. 43, 2002, pp. 93–101; Gandarias, Bruzon, Classical and nonclassical symmetries of a generalized Boussinesq equation, J. Nonlinear Math. Phys. 5 (1998) 8–12; Hill, Solution of Differential Equations by Means of One-Parameter Groups, Pitman Publishing Co., 1982]. 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 equation may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the number of independent variables in the system. Effect of the cross-flow Reynolds number Re and the dimensionless wall expansion ratio $\alpha$ on velocity, flow streamlines, axial and radial pressure drop, and wall shear stress has been studied both analytically and numerically and the results are plotted.     

Group method analysis of combined heat and mass transfer by MHD non-Darcy non-Newtonian natural convection adjacent to horizontal cylinder in a saturated porous medium

The group theoretic method is applied for solving problem of combined magneto-hydrodynamic heat and mass transfer of non-Darcy natural convection about an impermeable horizontal cylinder in a non-Newtonian power law fluid embedded in porous medium under coupled thermal and mass diffusion, inertia resistance, magnetic field, thermal radiation effects. The application of one-parameter groups reduces the number of independent variables by one and consequently, the system of governing partial differential equations with the boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The ordinary differential equations are solved numerically for the velocity using shooting method. The effects of magnetic parameter M, Ergun number Er, power law (viscosity) index n, buoyancy ratio N, radiation parameter R_d, Prandtl number Pr and Lewis number Le on the velocity, temperature fields within the boundary layer, heat and mass transfer are presented graphically and discussed.     

On some partial integral inequalities in n independent variables

In this paper comparison method will be used to establish a number of fundamental partial integral inequalities in n independent variables. These inequalities can be used as ready and powerful tools in developing the qualitative theory of partial differential and integral equations in n independent variables.     

Group method analysis of magneto-elastico-viscous flow along a semi-infinite flat plate with heat transfer

The group theoretic method is applied for solving problem of the flow of an elastico-viscous liquid past an infinite flat plate in the presence of a magnetic field normal to the plate. The application of one-parameter transformation group reduces the number of independent variables, by one, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate corresponding conditions. Numerical solution of the velocity field and heat transfer have been obtained. The effect of the magnetic parameter M on velocity field, shear stress, temperature fields and heat transfer has been discussed.     

Hypergeometric functions of two variables

Summary The systematic investigation of contour integrals satisfying the system of partial differential equations associated with Appell's hypergeometric functionF ₁ leads to new solutions of that system. Fundamental sets of solutions are given for the vicinity of all singular points of the system of partial differential equations. The transformation theory of the solutions reveals connections between the system under consideration and other hypergeometric systems of partial differential equations. Presently it is discovered that any hypergeometric system of partial differential equations of the second order (with two independent variables) which has only three linearly independent solutions can be transformed into the system ofF ₁ or into a particular or limiting case of this system. There are also other hypergeometric systems (with four linearly independent solutions) the integration of which can be reduced to the integration of the system ofF ₁.     

Similarity solutions for magneto-forced-unsteady free convective laminar boundary-layer flow

The group theoretic method is applied for solving problem of a unsteady free-convective laminar boundary-layer flow on a non-isothermal vertical plate under the effect of an external velocity and a magnetic field normal to the plate. The application of two-parameter transformation group reduces the number of independent variables, by two, and consequently the system of governing partial differential equations with the boundary and initial conditions reduces to a system of ordinary differential equations with appropriate corresponding conditions. The Runge–Kutta shooting method used to find the numerical solution of the velocity field, shear stress, heat transfer and heat flux has been obtained. The effect of the magnetic field on the velocity field and the Prandtl number on the heat transfer and heat flux has been discussed.     

The geometry of a quasilinear system of two partial differential equations containing the first and the second partial derivatives of two functions in two independent variables

The geometry of the system of two partial differential equations containing the first and second partial derivatives of two functions in two independent variables is studied by using Élie Cartan’s method of invariant forms and the group-theoretic method of extensions and enclosings due to G. F. Laptev (for finite groups) and A. M. Vasil’ev (for infinite groups). Systems of quasilinear equations with the first and second partial derivatives of two functions u and v in two independent variables x and y are classified.     

Convergent series solution of nonlinear equations

The author's decomposition method [1] provides a new, efficient computational procedure for solving large classes of nonlinear (and/or stochastic) equations. These include differential equations containing polynomial, exponential, and trigonometric terms, negative or irrational powers, and product nonlinearities [2]. Also included are partial differential equations [3], delay-differential equations [4], algebraic equations [5], and matrix equations [6] which describe physical systems. Essentially the method provides a systematic computational procedure for equations containing any nonlinear terms of physical significance. The procedure depends on calculation of the author's A_n, a finite set of polynomials [1,13] in terms of which the nonlinearities can be expressed. This paper shows important properties of the A_n which ensure an accurate and computable convergent solution by the author's decomposition method [1]. Since the nonlinearities and/or stochasticity which can be handled are quite general, the results are potentially extremely useful for applications and make a number of common approximations such as linearization, unnecessary.     

Similarity solutions to nonlinear heat conduction and Burgers/Korteweg–deVries fractional equations

We analyze self-similar solutions to a nonlinear fractional diffusion equation and fractional Burgers/Korteweg–deVries equation in one spatial variable. By using Lie-group scaling transformation, we determined the similarity solutions. After the introduction of the similarity variables, both problems are reduced to ordinary nonlinear fractional differential equations. In two special cases exact solutions to the ordinary fractional differential equation, which is derived from the diffusion equation, are presented. In several other cases the ordinary fractional differential equations are solved numerically, for several values of governing parameters. In formulating the numerical procedure, we use special representation of a fractional derivative that is recently obtained.     

A THEOREM OF BÄCKLUND REVISITED

Abstract

Bäcklund's theorem states that the most general contact transformation is an extended point transformation whenever both the number of independent variables and the number of dependent variables exceed one. A partial circumvention of Bäcklund's theorem is obtained by assigning each dependent variable its own distinct manifold of independent variables. This gives rise to extended symplectic product structures. sequences of extended Hamiltonians, and Lie groups of regular maps that satisfy systems of extended Hamilton-Jacobi equations provided the initial data is determined by a regular map. These ideas are applied to the study of systems of nonlinear second order partial differential equations. Lie groups of solutions are shown to be obtained by solving systems of extended Hamilton-Jacobi equations provided the initial data defines a solution.     

Group theoretic method for unsteady free convection flow of a micropolar fluid along a vertical plate in a thermally stratified medium

The group theoretic approach is applied for solving the problem of unsteady natural convection flow of micropolar fluid along a vertical flat plate in a thermally stratified medium. The application of two-parameter transformation group reduces the number of independent variables in the governing system consisting of partial differential equations and a set of auxiliary conditions from three to only one independent variable, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. Numerical solution of the velocity, microrotation and heat transfer have been obtained. The possible forms of the ambient temperature variation with position and time are derived.     

Preconditioners Based on Fundamental Solutions

We consider a new preconditioning technique for the iterative solution of linear systems of equations that arise when discretizing partial differential equations. The method is applied to finite difference discretizations, but the ideas apply to other discretizations too. If E is a fundamental solution of a differential operator P, we have E*(Pu) = u. Inspired by this, we choose the preconditioner to be a discretization of an approximate inverse K, given by a convolution-like operator with E as a kernel. We present analysis showing that if P is a first order differential operator, KP is bounded, and numerical results show grid independent convergence for first order partial differential equations, using fixed point iterations. For the second order convection-diffusion equation convergence is no longer grid independent when using fixed point iterations, a result that is consistent with our theory. However, if the grid is chosen to give a fixed number of grid points within boundary layers, the number of iterations is independent of the physical viscosity parameter. AMS subject classification (2000) 65F10, 65N22     

Backward Error Analysis for Lie-Group Methods

Backward error analysis has proven to be very useful in stability analysis of numerical methods for ordinary differential equations. However the analysis has so far been undertaken in the Euclidean space or closed subsets thereof. In this paper we study differential equations on manifolds. We prove a backward error analysis result for intrinsic numerical methods. Especially we are interested in Lie-group methods. If the Lie algebra is nilpotent a global stability analysis can be done in the Lie algebra. In the general case we must work on the nonlinear Lie group. In order to show that there is a perturbed differential equation on the Lie group with a solution that is exponentially close to the numerical integrator after several steps, we prove a generalised version of Alekseev-Gr: obner's theorem. A major motivation for this result is that it implies many stability properties of Lie-group methods.     

General exact solutions for linear and nonlinear waves in a Thirring model

In the present paper, we construct exact solutions to a system of partial differential equations iu_x + v + u | v | ² = 0, iv_t + u + v | u | ² = 0 related to the Thirring model. First, we introduce a transform of variables, which puts the governing equations into a more useful form. Because of symmetries inherent in the governing equations, we are able to successively obtain solutions for the phase of each nonlinear wave in terms of the amplitudes of both waves. The exact solutions can be described as belonging to two classes, namely, those that are essentially linear waves and those which are nonlinear waves. The linear wave solutions correspond to waves propagating with constant amplitude, whereas the nonlinear waves evolve in space and time with variable amplitudes. In the traveling wave case, these nonlinear waves can take the form of solitons, or solitary waves, given appropriate initial conditions. Once the general solution method is outlined, we focus on a number of more specific examples in order to show the variety of physical solutions possible. We find that radiation naturally emerges in the solution method: if we assume one of u or v with zero background, the second wave will naturally include both a solitary wave and radiation terms. The solution method is rather elegant and can be applied to related partial differential systems. Copyright © 2014 John Wiley & Sons, Ltd.     

The use of second degree normalized implicit conjugate gradient methods for solving large sparse systems of linear equations

Second degree normalized implicit conjugate gradient methods for the numerical solution of self-adjoint elliptic partial differential equations are developed. A proposal for the selection of certain values of the iteration parameters ?_i, γ_i involved in solving two and three-dimensional elliptic boundary-value problems leading to substantial savings in computational work is presented. Experimental results for model problems are given.     

The asymptotic estimates of solutions of difference equations

The asymptotic estimate of the solution y(t) of linear difference equations with almost constant coefficients and the condition of asymptotic equivalence between y(t) and y₀(t) are given, where y₀(t) is the corresponding solution of linear equations with constant coefficients. A brief and elementary proof of the results is presented. The method of proof carries over to differential equations in which some similar results are stated.     

Group method analysis of melting effect on MHD mixed convection flow from radiate vertical plate embedded in a saturated porous media

The linear transformation group approach is developed to simulate problem of hydromagnetic heat transfer by mixed convection along vertical plate in a liquid saturated porous medium in the presence of melting and thermal radiation effects for opposing external flow. The application of a one-parameter transformation group reduces the number of independent variables by one so that the governing partial differential equations with the boundary conditions reduce to an ordinary differential equations with appropriate corresponding conditions. The Runge-Kutta shooting method is used to solve the determining equations of the set of nonlinear ordinary differential equations. are presented in the form of the temperature and flow fields in the melting region within the boundary layer for different parameters entering into the analysis. Also the effects of the pertinent parameters on the rate of the heat transfer in terms of the local Nusselt number at the solid–liquid interface are also discussed.     

Exact solution ofP n time-dependent equations with time-dependent cross-sections for slab geometry

Monoenergetic, time-dependentP _n equations with time-dependent cross-sections are solved using the method of the generalized Lie series. The variables are separated in the solution obtained. This solution is given in a closed form for the case of time-dependent cross-sections. a few numerical results are given for the case ofP ₃ approximation and compared with those obtained by the time-dependentS ₄ code.     

On the Partial Equiasymptotic Stability in Functional Differential Equations

A system of functional differential equations with delay dz/dt = Z(t, z_t), where Z is the vector-valued functional is considered. It is supposed that this system has a zero solution z = 0. Definitions of its partial stability, partial asymptotical stability, and partial equiasymptotical stability are given. Theorems on the partial equiasymptotical stability are formulated and proved.     

Initial boundary value problems for the nonlinear Schrödinger equation

A new spectral method for solving initial boundary value problems for linear and integrable nonlinear partial differential equations in two independent variables is applied to the nonlinear Schrödinger equation and to its linearized version in the domain {x≥l(t), t≥0}. We show that there exist two cases: (a) if l″(t)<0, then the solution of the linear or nonlinear equations can be obtained by solving the respective scalar or matrix Riemann-Hilbert problem, which is defined on a time-dependent contour; (b) if l″(t)>0, then the Riemann-Hilbert problem is replaced by a respective scalar or matrix $\bar \partial $ problem on a time-independent domain. In both cases, the solution is expressed in a spectrally decomposed form.