Coverings over lax integrable equations and their nonlocal symmetries

We consider the three-dimensional rdDym equation u_ty = u_xu_xy ?u_yu_xx. Using the known Lax representation with a nonremovable parameter and two hierarchies of nonlocal conservation laws associated with it, we describe the algebras of nonlocal symmetries in the corresponding coverings.     

Symmetries and Conservation Laws of the System: ux= vwx, vy= u wy, uv+wxx+wyy=0

After a short exposition of the theory of local and nonlocal symmetries and conservation laws for systems of PDEs, results on these and the recursion operator are listed for the system of PDEs u_x=vw_x, v_y=uw_y, uv+w_xx+w_yy=0. In between the methods of computation are explained.     

Symmetric marching technique and elliptic equations with variable coefficient involving mixed partial deriavatives

The symmetric marching technique developed in [1,2] has been extended to the elliptic equations with variable coefficients involving mixed partial derivatives. The restricted class of such equations of the form u_xx+2B(x, y)u_xy+c(x,y)u_yy=0 have been considered. Numerical results of model problems solved are presented.     

Optimal Systems and Group Classification of (1+2)-Dimensional Heat Equation

The optimal systems and symmetry breaking interactions for the (1+2)-dimensional heat equation are systematically studied. The equation is invariant under the nine-dimensional symmetry group H ₂. The details of the construction for an one-dimensional optimal system is presented. The optimality of one- and two-dimensional systems is established by finding some algebraic invariants under the adjoint actions of the group H ₂. A list of representatives of all Lie subalgebras of the Lie algebra h ₂ of the Lie group H ₂ is given in the form of tables and many of their properties are established. We derive the most general interactions F(t,x,y,u,u _x ,u _y ) such that the equation u _t =u _xx +u _yy +F(t,x,y,u,u _x ,u _y ) is invariant under each subgroup.     

A Model Equation Illustrating Subcritical Instability to Long Waves in Shear Flows

A model equation somewhat more general than Burger's equation has been employed by Herron [1] to gain insight into the stability characteristics of parallel shear flows. This equation, namely, u_t + uu_y = u_xx + u_yy, has an exact solution U(y) = ?2tanh y. It was shown in [1] that this solution is linearly stable, and more recently, Galdi and Herron [3] have proved conditional stability to finite perturbations of sufficiently small initial amplitude using energy methods. The present study utilizes multiple-scaling methods to derive a nonlinear evolution equation for a long-wave perturbation whose amplitude varies slowly in space and time. A transformation to the heat-conduction equation has been found which enables this amplitude equation to be solved exactly. Although all disturbances ultimately decay due to diffusion, it is found that subcritical instability is possible in that realistic disturbances of finite initial amplitude can amplify substantially before finally decaying. This behavior is probably typical of perturbations to shear flows of practical interest, and the results illustrate deficiencies of the energy method.     

The Instability of Nonmonotonic Wave Solutions of Parabolic Equations

We consider the class of equations u_t=f(u_xx, u_x, u) under the restriction that for all a,b,c. We first consider this equation over the unbounded domain ? ∞ < x < + ∞, and we show that very nearly every bounded nonmonotonic solution of the form u(t, x)=?(x?ct) is unstable to all nonnegative and all nonpositive perturbations. We then extend these results to nonmonotonic plane wave solutions u(t, x, y)=?(x?ct) of u_t = F(u_xx, u_xy, u_x, u_y, u). Finally, we consider the class of equations u_t=f(u_xx, u_x, u) over the bounded domain 0 < x < 1 with the boundary conditions u(t, x)=A at x=0 and u(t, x)=B at x=1, and we find the stability of all steady solutions u(t, x)=?(x).     

A fourth-order difference method for elliptic equations with nonlinear first derivative terms

We present a nine-point fourth-order finite difference method for the nonlinear second-order elliptic differential equation Au_xx + Bu_yy = f(x, y, u, u_x, u_y) on a rectangular region R subject to Dirichlet boundary conditions u(x, y) = g(x, y) on ?R. We establish, under appropriate conditions O(h⁴)-convergence of the finite difference scheme. Numerical examples are given to illustrate the method and its fourth-order convergence.     

Analysis of a class of potential Korteweg‐de Vries‐like equations

We analyze a class of third‐order evolution equations, i.e. u_t = f(x, u_x, u_xx) u_xxx+g(x, u_x, u_xx) via the method of preliminary group classification. This method is a systematic means of analyzing the equation for symmetries. We find explicit forms of f and g, which allow for a larger dimensional Lie algebra of point symmetries. Copyright © 2009 John Wiley & Sons, Ltd.     

An integrable equation with nonsmooth solitons

We present the bi-Hamiltonian structure and Lax pair of the equation ρ_t = bu_x+(1/2)[(u ² −u_x ² )ρ]_x, where ρ = u − u_xx and b = const, which guarantees its integrability in the Lax pair sense. We study nonsmooth soliton solutions of this equation and show that under the vanishing boundary condition u → 0 at the space and time infinities, the equation has both “W/M-shape” peaked soliton (peakon) and cusped soliton (cuspon) solutions.     

Complete symmetry group and nonlocal symmetries for some two-dimensional evolution equations

The complete symmetry group of an 1+1 evolution equation of maximal symmetry has been demonstrated to be represented by the six-dimensional Lie algebra of point symmetries sl(2,R)s_⊕W, where W is the three-dimensional Heisenberg-Weyl algebra. We construct a complete symmetry group of a 1+2 evolution equation ut=(Fy(u)ux) for some functions F using the point symmetries admitted by the equation. The 1+2 equation is not completely specifiable by point symmetries alone for some specific functions F. We make use of Ansätze already reported by Myeni and Leach [S.M. Myeni, P.G.L. Leach, Nonlocal symmetries and complete symmetry groups of evolution equations, J. Nonlinear Math. Phys. 13 (2006) 377-392] which provide a route to the determination of the required generic nonlocal symmetries necessary to supplement the point symmetries for the complete specification of these 1+2 evolution equations. Further we find that taking some suitable linear combination of Lie point symmetries helps to optimise the procedure of specifying the equation. A general result concerning the number of symmetries required to form a complete symmetry group of evolution is presented in the Conclusion.     

Su un problema esterno per la equazione delle onde

Sunto Si considera il problema diCauchy per la equazione u_tt = u_xx + u_yy con dati iniziali sui due piani caratteristici t = ±y. Si dà un teorema di esistenza ed unicità della soluzione nella ?regione esterna? |t| ≤ |y|, e si prova che la soluzione non dipende in modo continuo dai dati.

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Summary We consider theCauchy problem for the wave equation u_tt = u_xx + u_yy with initial data on the characteristic planes t = ±y. We discuss the problem related to the ?exterior region? |t| ≤ |y|; we prove uniqueness of the solution, its instability with respect to the data, and the existence of the solution if the data are analytic functions of x (this hypothesis is essential, as we prove with an example).

     

Fourth-order finite difference method for 2D parabolic partial differential equations with nonlinear first-derivative terms

We attempt to obtain a two-level implicit finite difference scheme using nine spatial grid points of O(k² + kh² + h⁴) for solving the 2D nonlinear parabolic partial differential equation v₁u_xx + v₂u_yy = f(x, y, t, u, u_x, u_y, u₁) where v₁ and v₂ are positive constants, with Dirichlet boundary conditions. The method, when applied to a linear diffusion-convection problem, is shown to be unconditionally stable. Computational efficiency and the results of numerical experiments are discussed.     

Nonlocal initial-boundary-value problems for a degenerate hyperbolic equation

We consider the equation y ^m u _xx − u _yy − b ² y ^m u = 0 in the rectangular area {(x, y) | 0 < x < 1, 0 < y < T}, where m < 0, b ≥ 0, T > 0 are given real numbers. For this equation we study problems with initial conditions u(x, 0) = τ(x), u _y (x, 0) = ν(x), 0 ≤ x ≤ 1, and nonlocal boundary conditions u(0, y) = u(1, y), u _x (0, y) = 0 or u _x (0, y) = u _x (1, y), u(1, y) = 0 with 0≤y≤T. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems     

Stability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first-order space derivative terms

In this piece of work, we introduce a new idea and obtain stability interval for explicit difference schemes of O(k²+h²) for one, two and three space dimensional second-order hyperbolic equations u_tt=a(x,t)u_xx+α(x,t)u_x-2η²(x,t)u,u_tt=a(x,y,t)u_xx+b(x,y,t)u_yy+α(x,y,t)u_x+β(x,y,t)u_y-2η²(x,y,t)u, and u_tt=a(x,y,z,t)u_xx+b(x,y,z,t)u_yy+c(x,y,z,t)u_zz+α(x,y,z,t)u_x+β(x,y,z,t)u_y+γ(x,y,z,t)u_z-2η²(x,y,z,t)u,0<x,y,z<1,t>0 subject to appropriate initial and Dirichlet boundary conditions, where h>0 and k>0 are grid sizes in space and time coordinates, respectively. A new idea is also introduced to obtain explicit difference schemes of O(k²) in order to obtain numerical solution of u at first time step in a different manner.     

Symmetry Classification of Third-Order Nonlinear Evolution Equations. Part I: Semi-simple Algebras

We give a complete point-symmetry classification of all third-order evolution equations of the form u _t =F(t,x,u,u _x ,u _xx )u _xxx +G(t,x,u,u _x ,u _xx ) which admit semi-simple symmetry algebras and extensions of these semi-simple Lie algebras by solvable Lie algebras. The methods we employ are extensions and refinements of previous techniques which have been used in such classifications.     

Numerical similarity reductions of the (1+3)-dimensional Burgers equation

We consider the (1+3)-dimensional Burgers equation u_t = u_xx + u_yy + u_zz + uu_x which has considerable interest in mathematical physics. Lie symmetries are used to reduce it to certain ordinary differential equations. We employ numerical methods to solve a number of these ordinary differential equations.     

Nonlocal Dezin’s problem for Lavrent’ev–Bitsadze equation

Using the method of spectral analysis, for the mixed type equation u_xx + (sgny)u_yy = 0 in a rectangular domain we establish a criterion of uniqueness of its solution satisfying periodicity conditions by the variable x, a nonlocal condition, and a boundary condition. The solution is constructed as the sum of a series in eigenfunctions for the corresponding one-dimensional spectral problem. At the investigation of convergence of the series, the problem of small denominators occurs. Under certain restrictions on the parameters of the problem and the functions, included in the boundary conditions, we prove uniform convergence of the constructed series and stability of the solution under perturbations of these functions.     

Problem with analogs of the Frankl’ condition on a characteristic and the degeneration segment for an equation of mixed type with a singular coefficient

For the equation (sign y)|y| ^m u _xx +u _yy ?m(2y)^?1 u _y = 0, where m > 0, considered in some mixed domain, we prove existence and uniqueness theorems for the solution of the boundary value problem with an analog of the Frankl’ condition on a characteristic and on the degeneration segment of the equation.     

A generalized Cole-Hopf transformation for a two-dimensional burgers equation with a variable coefficient

The direct method is applied to the two dimensional Burgers equation with a variable coefficient (u _t + uu _x ? u _xx ) _x + s(t)u _yy = 0 is transformed into the Riccati equation $H' - \tfrac{1} {2}H^2 + \left( {\tfrac{\rho } {2} - 1} \right)H = 0$ via the ansatz $u\left( {x,y,t} \right) = \tfrac{1} {{\sqrt t }}H(\rho ) + \tfrac{y} {{2\sqrt t }}\rho \left( {x,y,t} \right) = \tfrac{x} {{\sqrt t }} - y$ , provided that s(t) = t ^?3/2. Further, a generalized Cole-Hopf transformations $u\left( {x,y,t} \right) = \tfrac{y} {{2\sqrt t }} - \tfrac{2} {{\sqrt t }}\tfrac{{U_\rho (\rho ,r)}} {{U(\rho ,r)}}$ , $\rho \left( {x,y,t} \right) = \tfrac{x} {{\sqrt t }} - y$ , r(t) = log t is derived to linearize (u _t + uu _x ? u _xx ) _x + t ^?3/2 u _yy to the parabolic equation $U_r = U_{\rho \rho } + \left( {\tfrac{\rho } {2} - 1} \right)U_\rho$ .     

On central difference Approximations to general second order elliptic equations

Consider the second order elliptic equation Lu ≔ − au_xx − 2bu_xy − cu_yy + d(x, y)u = f(x, y) in [0, 1] × [0, 1], with periodic boundary conditions, and b² < ac, a > 0, c > 0, d(x, y) ⩾ 0. Finite difference discretizations require a much stronger condition than ellipticity to give a scheme of positive type. In this paper, it is shown that the standard central difference discretization of (1) is of monotone type although it is not positive type. Specifically, the inverse matrix arising from it has one sign.