Multigrid method for solving convection-diffusion problems with dominant convection

A modification of the multigrid method for the solution of linear algebraic equation systems with a strongly nonsymmetric matrix obtained after difference approximation of the convection-diffusion equation with dominant convection is proposed. Specially created triangular iterative methods have been used as the smoothers of the multigrid method. Some theoretical and numerical results are presented.     

On a class of sublinear singular elliptic problems with convection term

We establish several results related to existence, nonexistence or bifurcation of positive solutions for the boundary value problem −Δu+K(x)g(u)+a|∇u|=λf(x,u) in Ω, u=0 on ∂Ω, where Ω⊂RN(N?2) is a smooth bounded domain, 0<a?2, λ is a positive parameter, and f is smooth and has a sublinear growth. The main feature of this paper consists in the presence of the singular nonlinearity g combined with the convection term a|∇u|. Our approach takes into account both the sign of the potential K and the decay rate around the origin of the singular nonlinearity g. The proofs are based on various techniques related to the maximum principle for elliptic equations.     

Solutions of a class of converse problems

Mathematical Notes -     

Nonsimilarity solutions of a class of free convection problems

The nonsimilarity solutions of laminar boundary layer free convection flows along a rotating isothermal plate under nonuniform gravity and along a non isothermal vertical plate have been obtained. The governing nonlinear partial differential equations are reduced to a set of ordinary differential equations and then solved numerically using the method of ‘shooting’ with least square convergence criterion. The results for large values of axial distance are found to be more accurate than those obtained by series and integral methods, but for small values of axial distance they are in good agreement.     

Bifurcation for a class of singular elliptic problems with quadratic convection term

We study the bifurcation problem ?Δu=g(u)+λ|?u|²+μ in $\text{Ω,}\text{u=0}$ on $\text{?Ω}$, where λ,μ?0 and $\text{Ω}$ is a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$. The singular character of the problem is given by the nonlinearity g which is assumed to be decreasing and unbounded around the origin. In this Note we prove that the above problem has a positive classical solution (which is unique) if and only if λ(a+μ)<λ₁, where a=lim_t→+∞g(t) and λ₁ is the first eigenvalue of the Laplace operator in $\text{H}{}^1{}_0\text{(Ω)}$. We also describe the decay rate of this solution, as well as a blow-up result around the bifurcation parameter. To cite this article: M. Ghergu, V. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Natural convection on a vertical radially stretching sheet

The natural convection on a vertical radially stretching surface is studied. The resulting flow is a new exact similarity solution of the Navier-Stokes and energy equations. Asymptotic properties and uniqueness are shown.     

Solutions for a class of optimal control problems with time delay,part 1

The optimal control of a system whose state is governed by a nonlinear autonomous Volterra integrodifferential equation with unbounded time interval is considered. Specifically, it is assumed that the delay occurs only in the state variable. We are concerned with the existence of an overtaking optimal trajectory over an infinite horizon. The existence result that we obtain extends the result of Carlson (Ref. 1) to a situation where the trajectories are not necessary bounded. Also, we study the structure of approximate solutions for the problem on a finite interval.The author thanks A. Leizarowitz for fruitful discussions.     

Solutions for a class of optimal control problems with time delay,part 2

In this work, we analyze the structure of optimal solutions for a class of optimal control problems with time delay whose state is governed by a nonlinear autonomous Volterra integrodifferential equation. We establish that an almost optimal finite trajectory defined on an interval [0, ] is contained in a small neighborhood of the optimal steady state for all , where is a constant which depends on the neighborhood and does not depend on .The author thanks A. Leizarowitz for fruitful discussions.     

A dominant subset of V-shaped sequences for a class of single machine sequencing problems

In this note we define a subset of V-shaped sequences, ‘V-shaped about T’, which generalize ‘V-shaped about d’ sequences. We derive a condition under which this subset contains an optimal sequence for a class of single machine sequencing problems. Cost functions from the literature are used to illustrate our results.     

Solutions to boundary Cauchy problems with infinite delay

In this paper, we investigate the existence, uniqueness of solutions to boundary Cauchy equations with infinite delay, which are more general than the previous studies. The conclusions are applied to an age dependent population with delay in the birth process.     

On the number of radially symmetric solutions to Dirichlet problems with jumping nonlinearities of superlinear order

This paper is concerned with the multiplicity of radially symmetric solutions to the Dirichlet problem

on the unit ball with boundary condition on . Here is a positive function and is a function that is superlinear (but of subcritical growth) for large positive , while for large negative we have that , where is the smallest positive eigenvalue for in with on . It is shown that, given any integer , the value may be chosen so large that there are solutions with or less interior nodes. Existence of positive solutions is excluded for large enough values of .

     

Stefan problems with nonlinear diffusion and convection

We consider a class of Stefan-type problems having a convection term and a pseudomonotone nonlinear diffusion operator. Assuming data in L¹, we prove existence, uniqueness and stability in the framework of renormalized solutions. Existence is established from compactness and monotonicity arguments which yield stability of solutions with respect to L¹ convergence of the data. Uniqueness is proved through a classical L¹-contraction principle, obtained by a refinement of the doubling variable technique which allows us to extend previous results to a more general class of nonlinear possibly degenerate operators.     

A mathematical model of a class of problems for the transport of a mixture in air

An analytic solution of a class of boundary-value problems of mathematical physics describing the transport of a mixture in the atmosphere is considered. To solve these problems we apply the substitution method and the Fourier method. The solution of a boundary problem describing the process of contamination of the atmosphere by various substances is presented in the form of a series. The result obtained is useful for the solution of problems concerning the protection of the atmosphere.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 87–90, 1989.     

Solutions to a class of Schrödinger equations

We establish existence and multiplicity of solutions to a class of nonlinear Schrödinger equations with, e.g., ``atomic' Hamiltonians, via critical point theory.

     

一类二阶积分型发展方程反周期解问题

本文中,我们研究一类带有非单调扰动算子的二阶非线性发展方程的反周期问题,证明方程中的非单调扰动算子为极大单调的,并用极大单调算子的微单调扰动理论来证明此类方程的反周期解的存在性。     

Explicit solutions to a class of nonlinear filtering problems

In this paper we obtain the solution of a class of nonlinear filtering problems in the form of a series expansion in terms of multiple Wiener integrals. The solution is explicit in the sense that the kernels of the integrals in the expansion are explicitly determine.