Solutions Of Quasilinear Parabolic Equations Research Articles

On weak solutions of quasilinear parabolic equations

Sur un domaine borne Ω⊂R u N on considere le probleme aux valeurs limites et initiales parabolique suivant: u'+A(u)=f dans Q T =(O,T)×Ω, D x (u) (o,T)×∂Ω =0 pour |α|≤m-1, u(o)=φ dans Ω, ou A(u) est un operateur elliptique quasilineaire d'ordre 2m. On trouve des conditions suffisantes pour etablir l'existence de solutions faibles

Relations between travelling wave solutions of quasilinear parabolic equations

A transformation between travelling wave solutions of v t = v x x + f ( v ) {v_t} = {v_{xx}} + f\left ( v \right ) and u t = ( D ( u ) u x ) x + g ( u ) {u_t} = {\left ( {D\left ( u \right ){u_x}} \right )_x} + g\left ( u \right ) is given.

Properties of solutions unbounded in time of diffusion equations (in Russian)

We survey results on unbounded solutions of quasilinear parabolic equations, obtained by the team directed by A. A. Samarskij and S. P. Kurdjumov. Considered are similarity solutions and comparison theorems. The investigated solutions exhibit a number of interesting and, for linear diffusion equations, unusual properties.

Asymptotic form of solution of quasilinear parabolic equation in neighbourhood of front

Asymptotically smooth forms of the solution of a quasilinear parabolic equation are constructed. The connection between these forms and the metasable states of the solution of the first boundary value problem where they can replace one another in the neighbourhood of the front, is shown.

Iterative solution of quasilinear parabolic equations by parabolic equations with constant coefficients

AbstractA method for solving quasilinear parabolic equations of the typesthat differs radically from previously known methods is proposed. For each initial-boundary-value problem of one of these types that has boundary conditions of the first kind (second kind), a conjugate initial-boundary-value problem of the other type that has boundary conditions of the second kind (first kind) is defined. Based on the relations connecting the solutions of a pair of conjugate problems, a series of parabolic equations with constant coefficients that do not change step to step is constructed. The method proposed consists in calculating the solutions of the equations of this series. It is shown to have linear convergence. Results of a series of numerical experiments in a finite-difference setting show that one particular implementation of the proposed method has a smaller domain of convergence than Newton's method but that it sometimes converges faster within that domain.

The linearization principle for the stability of solutions of quasilinear parabolic equations, I

The linearization principle for the stability of solutions of quasilinear parabolic equations, I

Behavior at the boundary of solutions of quasilinear parabolic equations

Behavior at the boundary of solutions of quasilinear parabolic equations

Some quadrature rules for Galerkin methods using B-spline basis functions

Abstract Quadrature rules are obtained for the evaluation of integrals resulting from the solution of quasilinear parabolic equations using the Galerkin method with B - spline basis functions.

Periodic and stationary solutions of quasilinear parabolic equations

Periodic and stationary solutions of quasilinear parabolic equations

Regularity of solutions for quasi-linear parabolic equations

Let Ω be a bounded domain in n-dimensional Euclidian space En (n ≧ 2), and consider the space-time cylinder Q = Ω × (0, T] for some fixed T > 0. In this paper we deal with the Cauchy and Dirichlet problem for a second order quasi-linear equation(1.1) ut — div A(x, t, u, ux) + B(x, t, u, ux) = 0 for (x, t) ∈ Q,(1.2) u(x, 0) = (ϕ)(x) in Ω and u(x, t) = tψ(x, t) for (x, t) ∈ Γ = ∂Ω × (0, T] ,where ∂Ω is a boundary of Ω which satisfies the following condition (A) : Condition (A). There exist constants ρ0 and »0 both in (0,1) such that, for any sphere K(ρ) with center on ∂Ω and radius ρ ≦ ρ0, the inequality meas [K(ρ) ∩ Ω] ≦ (1 — λ0) × meas E(ρ) holds, where meas E means the measure of a measurable set E.

Net schemes for the navier—stokes equations associated with projection

DIFFERENCE schemes of the projection type for the Navier Stokes equations are considered. In the construction of schemes of the alternating directions type the continuity equation is replaced by a regularized equation. For all the difference schemes considered the convergence of the approximate to the exact solution is proved and estimates of the rate of convergence are given. In [1–4] schemes were considered which reduce at each time step the solution of a Navier- Stokes equation to the solution of quasilinear parabolic equations for the velocity components and to a subsequent projection of the velocity vector obtained in this way onto a space of solenoid functions. This requires a Poisson equation to be solved at each time step. In this paper schemes are proposed combining the projection with the method of alternating directions. Their construction uses methods of regularizing the continuity equation (see [5–7]) and the limitation of non-linearity (see [8, 9]). For all the schemes considered in this paper convergence is proved and estimates are given of the rate of convergence without restrictions on the nature of the dependence of r on h and v, where T is the time step, h is the step along each of the spatial variables, and v is the value of the kinematic viscosity.

Continuous branches of periodic solutions of quasilinear parabolic equations

Continuous branches of periodic solutions of quasilinear parabolic equations

Regularity Properties of Flows Through Porous Media: A Counterexample

Regularity Properties of Flows Through Porous Media: A Counterexample

Removable Singularities of Solutions of Quasilinear Parabolic Equations

Removable Singularities of Solutions of Quasilinear Parabolic Equations

Local behavior of solutions of quasilinear parabolic equations

Local behavior of solutions of quasilinear parabolic equations

Local solutions for quasi-linear parabolic equations

Local solutions for quasi-linear parabolic equations

QUASI-LINEAR SECOND-ORDER PARABOLIC EQUATIONS WITH MANY INDEPENDENT VARIABLES

CONTENTS § 1. Introduction § 2. A priori estimates for the solutions of linear parabolic equations § 3. A priori estimates for the solutions of quasi-linear parabolic equations § 4. Existence theorems for solutions of the first boundary value problem and Cauchy's problem in the large for quasi-linear parabolic equations § 5. An approximate solution of the boundary value problem and Cauchy's problem for quasi-linear parabolic equations. Generalized solutions of these problems § 6. Theorems on a priori estimates and the solvability of the Dirichlet problem for quasi-linear elliptic equations with many independent variables References