On the behavior of solutions of equations for double waves in the neighborhood of the quiescent region : PMM vol. 39, n 6, 1975, pp. 1043–1050

The structure of solutions of gasdynamic equations is investigated in the case of unsteady double waves in the neighborhood of the quiescent region. A general concept of double waves is presented in the form of special series with logarithmic terms. Results of numerical computations are given.The problem of determining the flow of plane and three-dimensional waves separated from the quiescent region by a weak discontinuity was considered in [1–3], where approximate solutions were derived for that neighborhood, and the formulation of boundary value problems required for solving the equation for the analog of the velocity potential in the hodograph plane was investigated.The more general problem (without the assumption of the degeneration of motion) of arbitrary potential flows of polytropic gas adjacent to the quiescent region and separated by a weak discontinuity was considerd in [4–8]. Solution of that problem was obtained in the form of special series in powers of the mo dulus of the velocity vector r in the space of the time hodograph. The value r = 0 corresponds to the surface of weak discontinuity that separates the perturbed motion region from that at rest. Some applications of derived solutions to problems such as the motion of a convex piston and the propagation of weak shock waves were also investigated in those papers. Convergence in the small of obtained series was proved in [9]. However the attempts of constructing series in powers of r, which were used in [4–8] for the presentation of equations of double waves in the neighborhood of the quiescent region, proved to be unsuccessful.Although parts of expansions in series in powers of r (accurate to within 0 (r²)), were constructed in [1–3], it was found that the coefficient at r⁸ in equations for double waves cannot be determined owing to the insolvability of its equation. This is related to the fact that the surface r = 0in the case of equations for double waves is simultaneously a line of parabolic degeneration and a characteristic.The object of the present note is the formulation of solutions of equations for plane unsteady double waves in the neighborhood of the quiescent region. Parts of the derived series, which generally are nonanalytic functions of r, can be used for defining flows at small r in particular those downstream of two-dimensional normal detonation waves [10] or in problems of angular pistons [11]. The method used for the derivation of series can be also applied in investigations of threedimensional self-similar flows with variables x₁/x₃ and x₂/x₃ (steady flows) or x₁/t, x₂/t and x₃/t (unsteady flows). However it was not possible to obtain in such cases regular series in powers of r.     

Grid approximation of singularly perturbed parabolic convection-diffusion equations subject to a piecewise smooth initial condition

A boundary value problem for a singularly perturbed parabolic convection-diffusion equation on an interval is considered. The higher order derivative in the equation is multiplied by a parameter ? that can take arbitrary values in the half-open interval (0, 1]. The first derivative of the initial function has a discontinuity of the first kind at the point x ₀. For small values of ?, a boundary layer with the typical width of ? appears in a neighborhood of the part of the boundary through which the convective flow leaves the domain; in a neighborhood of the characteristic of the reduced equation outgoing from the point (x ₀, 0), a transient (moving in time) layer with the typical width of ?^1/2 appears. Using the method of special grids that condense in a neighborhood of the boundary layer and the method of additive separation of the singularity of the transient layer, special difference schemes are designed that make it possible to approximate the solution of the boundary value problem ?-uniformly on the entire set $\bar G$ , approximate the diffusion flow (i.e., the product ?(?/?x)u(x, t)) on the set $\bar G^ * = \bar G\backslash \{ (x_0 ,0)\} $ , and approximate the derivative (?/?x)u(x, t) on the same set outside the m-neighborhood of the boundary layer. The approximation of the derivatives ?²(?²/?x ²)u(x, t) and (?/?t)u(x, t) on the set $\bar G^ * $ is also examined.     

Painlevé Classification of All Semilinear Partial Differential Equations of the Second Order. II. Parabolic and Higher Dimensional Equations

This paper extends the work of the previous paper (I) on the Painlevé classification of second-order semilinear partial differential equations to the case of parabolic equations in two independent variables, u_xx = F(x, y, u, u_x, u_y), and irreducible equations in three or more independent variables of the form, Σ_ijR_ij (x₁,…, x_n)u,_ij = F(x₁,…, x_n; u,₁,…, u,_n). In each case, F is assumed to be rational in u and its first derivatives and no other simplifying assumptions are made. In addition to the 22 hyperbolic equations found in paper I, we find 10 equivalence classes of parabolic equations with the Painlevé property, denoted PS-I, PS-I1,…, PS-X, equation PS-II being a generalization of Burgers' equation denoted the Forsyth-Burgers equation, and 13 higher-dimensional Painlevé equations, denoted GS-I, GS-II,…, GS-XIII. The lists are complete up to the equivalence relation of Möbius transformations in u and arbitrary changes of the independent variables. In order to avoid repetition, the proofs are sketched very briefly in cases where they closely resemble those for the corresponding hyperbolic problem. Every equation is solved by transforming to a linear partial differential equation, from which it follows that there are no non trivial soliton equations among the two classes of Painlevé equations treated in this paper.     

Steady compressible Navier–Stokes equations with large potential forces via a method of decomposition

We investigate the steady compressible Navier–Stokes equations near the equilibrium state v = 0, ρ = ρ₀ (v the velocity, ρ the density) corresponding to a large potential force. We introduce a method of decomposition for such equations: the velocity field v is split into a non-homogeneous incompressible part u (div (ρ₀u) = (0) and a compressible (irrotational) part ∇ϕ. In such a way, the original complicated mixed elliptic–hyperbolic system is split into several ‘standard’ equations: a Stokes-type system for u, a Poisson-type equation for ϕ and a transport equation for the perturbation of the density σ = ρ − ρ₀. For ρ₀ = const. (zero potential forces), the method coincides with the decomposition of Novotny and Padula [21]. To underline the advantages of the present approach, we give, as an example, a ‘simple’ proof of the existence of isothermal flows in bounded domains with no-slip boundary conditions. The approach is applicable, with some modifications, to more complicated geometries and to more complicated boundary conditions as we will show in forthcoming papers. © 1998 B.G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Improvement of an estimate of H. Müller involving the order of 2(mod u) II

Let m≧ 1 be an arbitrary fixed integer and let N_m(x) count the number of odd integers u≦ x such that the order of 2 modulo u is not divisible by m. In case m is prime, estimates for N_m (x) were given by Müller that were subsequently sharpened into an asymptotic estimate by the present author. Müller on his turn extended the author’s result to the case where m is a prime power and gave bounds in the case m is not a prime power. Here an asymptotic for N_m (x) is derived that is valid for all integers m. We also generalize to other base numbers than 2. A further analysis of Müller’s method leads us to study and solve a certain Diophantine equation. Received: 23 August 2005     

Some Algebraic Methods for Calculating the Number of Colorings of a Graph

With an arbitrary graph G having n vertices and m edges, and with an arbitrary natural number p, we associate in a natural way a polynomial R(x ₁,...,x _n) with integer coefficients such that the number of colorings of the vertices of the graph G in p colors is equal to p ^m-n R(0,...,0). Also with an arbitrary maximal planar graph G, we associate several polynomials with integer coefficients such that the number of colorings of the edges of the graph G in 3 colors can be calculated in several ways via the coefficients of each of these polynomials. Bibliography: 2 titles.     

On the weak discontinuity surfaces and tapes of equation systems in fluid mechanics

The weak discontinuity surfaces for a system of quasi-linear differential equations of higher order are developed. The classification of equation systems in fluid mechanics is based on the propagative weak discontinuity surfaces. Types of equations for different flow models are discussed. The conclusion is as follows:(a) For incompressible nonviscous flow, incompressible viscous flow and compressible viscous flow, the types of equations are all parabolic in the unsteady case and elliptic in the steady case.(b) For compressible nonviscous flow, the type of equations is hyperbolic in the unsteady case or steady supersonic case, and the type is elliptic in the steady subsonic case.     

Long time existence for a slightly perturbed vortex sheet

Consider a flat two-dimensional vortex sheet perturbed initially by a small analytic disturbance. By a formal perturbation analysis, Moore derived an approximate differential equation for the evolution of the vortex sheet. We present a simplified derivation of Moore's approximate equation and analyze errors in the approximation. The result is used to prove existence of smooth solutions for long time. If the initial perturbation is of size ? and is analytic in a strip |??m γ| < ρ, existence of a smooth solution of Birkhoff's equation is shown for time t < k2p, if ? is sufficiently small, with κ → 1 as ? → 0. For the particular case of sinusoidal data of wave length π and amplitude e, Moore's analysis and independent numerical results show singularity development at time t^c = |log ?| + O(log|log ?|. Our results prove existence for t < κ|log ?|, if ? is sufficiently small, with k κ → 1 as ? → 0. Thus our existence results are nearly optimal.     

Unsteady heat and mass transfer over a vertical stretching sheet in a parallel free stream with variable wall temperature and concentration

Our aim in this article is to investigate numerically the unsteady two‐dimensional mixed convection flow along a vertical semi‐infinite stretching sheet in a parallel free stream with a power‐law wall temperature and concentration distributions of the form T _w (x) = T_∞ + Ax^2m?1 and C_w (x) = C_∞ + Bx^2m?1, where A, B and m are constants. The unsteadiness in the flow is caused by the time dependent stretching sheet as well as by the free stream velocity. The governing nonlinear partial differential equations in the velocity, temperature and concentration fields are written in nondimensional form using suitable transformations. The final set of resulting coupled nonlinear partial differential equations is solved using an implicit finite‐difference scheme in combination with a quasi‐linearization technique. The effects of various governing parameters on the velocity, temperature and concentration profiles as well as on the skin friction coefficient, local Nusseltnumber and local Sherwood number are presented and discussed in details. The computed numerically results are compared with previously reported work and are found to be in excellent agreement. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A Green's Function Approach to Wave Propagation and Potential Flow around Obstacles in Infinite Cylindrical Channels

We study (a) acoustic waves generated by a time-harmonic force distribution and (b) the potential flow with prescribed velocity at infinity in an infinite cylinder Ω₀ = Ω′×ℝ with bounded cross-section Ω′⊂ℝ² in the presence of m embedded obstacles B₁,…,B_m. By using Green's function G_κ(x,y) of the Neumann problem for the reduced wave equation ΔU+κ²U = 0 in the unperturbed domain Ω₀, both problems can be reduced to integral equations over the boundaries of the obstacles. The main properties of G_κ(x,y), which are required for this approach, are derived in the first part of this paper.     

Stability of Solution of a Class of Quasilinear Elliptic Equations

In this paper, using capacity theory and extension theorem of Lipschitz functions we first discuss the uniqueness of weak solution of nonhomogeneous quasilinear elliptic equationsin space W(θ,p)(Ω), which is bigger than W1,p(Ω). Next, using revise reverse Holder inequality we prove that if ωc is uniformly p-think, then there exists a neighborhood U of p, such that for all t ∈U, the weak solutions of equation corresponding t are bounded uniformly. Finally, we get the stability of weak solutions on exponent p.     

A direct method for the general solution of a system of linear equations

A computationally stable method for the general solution of a system of linear equations is given. The system isA ^Tx–B=0, where then-vectorx is unknown and then×q matrixA and theq-vectorB are known. It is assumed that the matrixA ^T and the augmented matrix [A ^T,B] are of the same rankm, wheremn, so that the system is consistent and solvable. Whenm<n, the method yields the minimum modulus solutionx _m and a symmetricn ×n matrixH _m of rankn–m, so thatx=x _m+H _my satisfies the system for ally, ann-vector. Whenm=n, the matrixH _m reduces to zero andx _m becomes the unique solution of the system.The method is also suitable for the solution of a determined system ofn linear equations. When then×n coefficient matrix is ill-conditioned, the method can produce a good solution, while the commonly used elimination method fails.This research was supported by the National Science Foundation, Grant No. GP-41158.     

Un Théorème Général de Génération de Semi-Groupes non Linéaires

In this paper, we provide a weak sufficient condition for the existence of solutions to the equation (du/dt)+Au ∋ 0,u(0)=x ₀, whereA is an accretive (possibly nonlinear) operator; a weak criterion form-accretiveness is also given.      

Asymptotic forms of solutions of certain linear ordinary differential equations

We analyze the asymptotic behavior as x → ∞ of the product integral Π_x0^xe^A(s)ds, where A(s) is a perturbation of a diagonal matrix function by an integrable function on [x₀,∞). Our results give information concerning the asymptotic behavior of solutions of certain linear ordinary differential equations, e.g., the second order equation y″ = a(x)y.     

On the integrability of hyperbolic systems of riccati-type equations

We consider equations of the form U_xy = U * U_x, where U(x, y) is a function taking values in an arbitrary finite-dimensional algebra T over the field ℂ. We show that every such equation can be naturally associated with two characteristic Lie algebras, L_x and L_y. We define the notion of a ℤ-graded Lie algebraB corresponding to a given equation. We prove that for every equation under consideration, the corresponding algebraB can be taken as a direct sum of the vector spaces L_x and L_y if we define the commutators of the elements from L_x and L_y by means of the zero-curvature relations. Assuming that the algebra T has no left ideals, we classify the equations of the specified type associated with the finite-dimensional characteristic Lie algebras L_x and L_y. All of these equations are Darboux-integrable. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 2, pp. 261–275, November, 1997.     

Waiting-Time Behavior in a Nonlinear Diffusion Equation

We study the nonlinear diffusion equation u_t*=(uⁿu_x)_x, which occurs in the study of a number of problems. Using singular-perturbation techniques, we construct approximate solutions of this equation in the limit of small n. These approximate solutions reveal simply the consequences of this variable diffusion coefficient, such as the finite propagation speed of interfaces and waiting-time behavior (when interfaces wait a finite time before beginning to move), and allow us to extend previous results for this equation.     

The presentation of explicit analytical solutions of a class of nonlinear evolution equations

In this paper, we introduce a function set Ω_m. There is a conjecture that an arbitrary explicit travelling-wave analytical solution of a real constant coefficient nonlinear evolution equation is necessarily a linear (or nonlinear) combination of the product of some elements in Ω_m. A widespread applicable approach for solving a class of nonlinear evolution equations is established. The new analytical solutions to two kinds of nonlinear evolution equations are described with the aid of the guess.     

Growth in Poisson algebras

A criterion for polynomial growth of varieties of Poisson algebras is stated in terms of Young diagrams for fields of characteristic zero. We construct a variety of Poisson algebras with almost polynomial growth. It is proved that for the case of a ground field of arbitrary characteristic other than two, there are no varieties of Poisson algebras whose growth would be intermediate between polynomial and exponential. Let V be a variety of Poisson algebras over an arbitrary field whose ideal of identities contains identities {{x ₁, y ₁}, {x ₂, y ₂}, . . . , {x _m , y _m }} = 0 and {x ₁, y ₁} · {x ₂, y ₂} · . . . · {x _m , y _m } = 0, for some m. It is shown that the exponent of V exists and is an integer. For the case of a ground field of characteristic zero, we give growth estimates for multilinear spaces of a special form in varieties of Poisson algebras. Also equivalent conditions are specified for such spaces to have polynomial growth.     

A best approximation for the solution of one‐dimensional variable‐coefficient Burgers' equation

In this article, an iterative method for the approximate solution to one‐dimensional variable‐coefficient Burgers' equation is proposed in the reproducing kernel space W_(3,2). It is proved that the approximation w_n(x,t) converges to the exact solution u(x,t) for any initial function w₀(x,t) ε W_(3,2), and the approximate solution is the best approximation under a complete normal orthogonal system . Moreover the derivatives of w_n(x,t) are also uniformly convergent to the derivatives of u(x,t).© 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Life span of 2‐D irrotational compressible fluids in the halfplane

We consider the Euler equations of barotropic inviscid compressible fluids in the half plane. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In dimension two such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial velocity. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small. We consider smooth irrotational solutions. First, we study the life span, i.e. the largest time interval T(ε) of existence of classical solutions, when the initial data are a small perturbation of size εfrom a constant state. For the proof of this result we use a combination of energy and decay estimates. Then, the estimate of the life span allows to show, by a suitable scaling of variables, the existence of irrotational solutions on any arbitrary time interval, for any small enough Mach number. Copyright © 2002 John Wiley & Sons, Ltd.