The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of the mass

Some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications. In this paper, a numerical method which employs the Bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions. The properties of Bernstein polynomials, and the operational matrices for integration, differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new technique.     

On Hölder solutions of the integro-differential Zakai equation

The existence and uniqueness of solutions of the Cauchy problem to a stochastic parabolic integro-differential equation is investigated. The equation considered arises in a nonlinear filtering problem with a jump signal process and continuous observation.     

On the decay of a solution of a nonuniformly parabolic equation

For a uniformly parabolic second-order equation with lower-order terms in an unbounded domain, we obtain an upper bound for the decay rate of the solution of the mixed problem with alternating boundary conditions of the first and third types. We prove that the bound is sharp in the case of an equation without lower-order terms in a wide class of domains of revolution. In addition, we show that a solution of a nonuniformly parabolic equation can decay much more rapidly than a solution of a uniformly parabolic equation.     

Hyperbolic–parabolic singular perturbation for quasilinear equations of Kirchhoff type with weak dissipation

We consider a hyperbolic–parabolic singular perturbation problem for a quasilinear hyperbolic equation of Kirchhoff type with dissipation weak in time. The purpose of this paper is to give time‐decay convergence estimates of the difference between the solutions of the hyperbolic equation above and those of the corresponding parabolic equation, together with the unique existence of the global solutions of the hyperbolic equation above. Copyright © 2009 John Wiley & Sons, Ltd.     

On nonlinear systems consisting of different types of differential equations

We consider a system consisting of a quasilinear parabolic equation and a first order ordinary differential equation where both equations contain functional dependence on the unknown functions. Then we consider a system which consists of a quasilinear parabolic partial differential equation, a first order ordinary differential equation and an elliptic partial differential equation. These systems were motivated by models describing diffusion and transport in porous media with variable porosity. Supported by the Hungarian NFSR under grant OTKA T 049819.     

Reconstruction of the right-hand side of a parabolic equation

A parabolic equation under the influence of external perturbations is considered. It is assumed that the solutions to this equation are measured (possibly with errors). The problem of construction of differential equations for estimation (reconstruction) of perturbations using the measured data is discussed. Differential equations of the auxiliary system whose right-hand sides are approximations of the unknown input are derived.     

单调算子的值域和在混合边值问题中的应用

通过定义几种算子,对其性质加以研究,并利用了Reich的关于值域和的有关结果,研究了非线性椭圆型方程及非线性抛物型方程,推导出这两种方程的解的存在性条件,从而推广了Pascali的相关结果.     

Completeness of the Floquet solutions to the problem of a solid oscillating in a fluid

We prove the completeness of the Floquet solutions to the parabolic equation describing small oscillations of a fluid-solid system. The symmetry axis of the solid is fixed inside a container of an arbitrary shape which is filled with an incompressible viscous fluid. The solid oscillates torsionally under the action of an elastic force with time periodic rigidity.     

非局部边界条件下的抛物型偏微分方程组

本文首先讨论了一个非局部边界条件下的抛物型偏微分方程组,通过一个变量替换,使得在更宽松的边界假设条件下证明了解的存在唯一性;然后讨论了一个完全非线性的抛物型方程组,同样,通过变量替换证明了比较原理.     

Pseudodifferential Multi-Product Representation of the Solution Operator of a Parabolic Equation

By using a time slicing procedure, we represent the solution operator of a second-order parabolic pseudodifferential equation on ? ⁿ as an infinite product of zero-order pseudodifferential operators. A similar representation formula is proven for parabolic differential equations on a compact Riemannian manifold. Each operator in the multi-product is given by a simple explicit Ansatz. The proof is based on an effective use of the Weyl calculus and the Fefferman-Phong inequality.     

A splitting positive definite mixed element method for miscible displacement of compressible flow in porous media

A miscible displacement of one compressible fluid by another in a porous medium is governed by a nonlinear parabolic system. A new mixed finite element method, in which the mixed element system is symmetric positive definite and the flux equation is separated from pressure equation, is introduced to solve the pressure equation of parabolic type, and a standard Galerkin method is used to treat the convection‐diffusion equation of concentration of one of the fluids. The convergence of the approximate solution with an optimal accuracy in L²‐norm is proved. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 229–249, 2001     

半线性抛物方程初边值问题解的Blow up

本文考虑了一类半线性抛物型方程的第三类非线性初边值问题,在某些假设条件下,证明了其解在有限时间内Blow up.     

一类非线性伪抛物型方程Cauchy问题的适定性

通过引入算子I-Δ的Bessel势将伪抛物型方程化成抽象的抛物型方程,然后利用算子半群理论讨论了一类非线性伪抛物型方程Cauchy问题的适定性问题.     

Identification of Parameters through the Approximate Periodic Solutions of a Parabolic System

This work is concerned with the identification problem for the perturbation term or error term in a parabolic partial differential equation through its approximate periodic solutions. The observation is made over a subregion of the physical domain. The existence and uniqueness problem of the approximate periodic solutions is studied in the first part of the paper. A solution to the identification problem is given in the second part of the paper. The main ingredients to be used include the classical Galerkin method and the unique continuation property for a parabolic system. This work was supported by the National Natural Science Foundation of China Grant 10571161.     

Partition of Unity for a Class of Nonlinear Parabolic Equation on Overlapping Non-Matching Grids

A class of nonlinear parabolic equation on a polygonal domain Ω  R2 is inves- tigated in this paper. We introduce a finite element method on overlapping non-matching grids for the nonlinear parabolic equation based on the partition of unity method. We give the construction and convergence analysis for the semi-discrete and the fully discrete finite element methods. Moreover, we prove that the error of the discrete variational problem has good approximation properties. Our results are valid for any spatial dimensions. A numerical example to illustrate the theoretical results is also given.     

An efficient approach for the numerical solution of the Monge-Ampère equation

In this paper, we present a new method to compute the numerical solution of the elliptic Monge-Ampère equation. This method is based on solving a parabolic Monge-Ampère equation for the steady state solution. We study the problem of global existence, uniqueness, and convergence of the solution of the fully nonlinear parabolic PDE to the unique solution of the elliptic Monge-Ampère equation. Some numerical experiments are presented to show the convergence and the regularity of the numerical solution.     

Conditions for stabilization of solutions to the first boundary value problem for parabolic equations

We study the stabilization to zero of a solution to the first boundary value problem for a parabolic equation. We obtain necessary and sufficient stabilization conditions in the case of the heat equation and sufficient conditions in the case of a parabolic equation with variable coefficients. Bibliography: 14 titles.     

A model for a finite memory transport in the Fisher equation

A model for a finite memory effect in the Fisher equation had been presented by Cattaneo [Acad. Sci. 247 (1958) 431]. By this model the type of the governing equation is transformed from a parabolic type to a hyperbolic one. But the Cattaneo’s equation does not reduce to the logistic equation in the homogeneous regime. A new model is presented which conserves the parabolic generic equation as well as the reduction property. Memory effects are visualized in the two models through numerical computations of solutions.     

An inverse coefficient problem with nonlinear parabolic equation

The problem related to controlled potential experiments in electrochemistry is studied. Ion transport is regarded as the superposition of diffusion and migration. Modelling of the experiment leads to a problem for a nonlinear parabolic equation with additional condition. Driven by the needs of theoretical analysis, from the point of view a inverse coefficient problem, we analyze the monotonicity of input-output mappings in inverse coefficient and source problems for this parabolic equation. Additionally, we extend the nonlinear parabolic equation to a more general case. Under some proper conditions, we investigate the existence of quasisolution of the generalized nonlinear parabolic equation.     

A posteriori error estimates for optimal distributed control governed by the evolution equations

We describe a technique for a posteriori error estimates suitable to the optimal control problem governed by the evolution equations solved by the method of lines. It is applied to the control problem governed by the parabolic equation, convection-diffusion equation and hyperbolic equation. The error is measured with the aid of the L²-norm in the space-time cylinder combined with a special time weighted energy norm.