Approximate solution of u′=–A2u using a relation between exp(–tA2) and exp(itA)

Let A be a positive definite operator in a Hilbert space and consider the initial value problem for u _t =–A² u. Using a representation of the semi group exp(–A² t) in terms of the group exp(iAt) we express u in terms of the solution of the standard heat equation w _t = w _yy , with initial values v solving the initial value problem for v _y = iAv. This representation is used to construct a method for approximating u in terms of approximations of v. In the case that A is a 2^nd order elliptic operator the method is combined with finite elements in the spatial variable and then reduces the solution of the 4th order equation for u to that of the 2^nd order equation for v, followed by the solution of the heat equation in one space variable.     

On the Properties of the Solution of a Strongly Degenerate Parabolic Equation

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THE DECAY BEHAVIOR OF SOLUTIONS FOR NONLOCAL BOUNDARY PROBLEMS

A parabolic equation defined on a bounded domain with boundary condition being nonlocal is considered. The existence and the dynamic behavior of the solutions for linear and semilinear equations are investigated in special spaces. One will find that the behavior of the solutions are affected by the boundary conditions. A semigroup approach is employed in this paper.     

LARGE TIME BEHAVIOR OF SOLUTIONS TO SOME DEGENERATE PARABOLIC EQUATIONS

The purpose of this paper is to study the limit in L ¹(Ω), as t → ∞, of solutions of initial-boundary-value problems of the form u_t ? Δw = 0 and u ∈ β(w) in a bounded domain Ω with general boundary conditions ?w/?η + γ(w) ? 0. We prove that a solution stabilizes by converging as t → ∞ to a solution of the associated stationary problem. On the other hand, since in general these solutions are not unique, we characterize the true value of the limit and comment the results on the related concrete situations like the Stefan problem and the filtration equation.     

具有未知源的某些非线性伪抛物型方程的反问题

考 虑具有未 知源项的 某些非 线性伪 抛物 型方程 的反演 问题． 首先 将伪抛 物型 方程初 边值问 题化为非线 性发展方 程 Ｃｏｕｃｈ ｙ 问题,然 后,利用半 群理论,论 证发展 方程反问 题解的存 在唯一 性,最后, 利用不 动点方法得到 伪抛物型方程反 问题的可解性     

关于抛物型方程离散流量的收敛性

关于用差分方法求解具有间断系数的二阶抛型方程的问题, Ａ．Ｈ． ＴＮＸＯＨＯＢ与 Ａ．Ａ．Ｃａｍａｐｃ　从１９６１年山开始曾经作过详尽的研究,他们的结果都总结在专著［２］中,有关的文献也可以在该书中找到．他们指出在系数间断点处附近的网格点上格式的截断误差为Ｏ（１）,但差分格式的解在极大意义下收敛于原微分方程的连续解．他们在构造差分格式,并论证其收敛性时,充分利用了原微分方程中流连续的性质,但是却没有讨论差分格式中的离散流量的收敛性．８０年代 Ｔ．Ａ． Ｍａｎｔｅｎｆｆｅｌ, Ａ．Ｂ． Ｗｈｉｔｅ, Ｊｒ…     

Semigroup approach to a doubly degenerate parabolic equation

1.IntroductionWeareconcernedwiththesemigroupapproachtotheinitialvalueproblemfordoublynonlineardegenerateparabolicequationoftheformwhicharisesfromdifferentphysicalbackgroundssuchasthemodelingofthemotionofnon-Newtonianfluids.Inthepastyears,thenonlinear...     

Evaluation of diffusion coefficient based on multiple forward solutions

This note is concerned with the identification of the unknown diffusion coefficient for a parabolic equation. It introduces an iterative algorithm that can be used to recover the unknown function. The algorithm assumes an initial guess for the unknown function and obtains a background field. It obtains an equation for the error field. It then formulates three forward problems for the error field. These three formulations share the same unknown function which is the correction to the assumed value of the unknown diffusion coefficient. By equating the responses of these three formulations, the algorithm obtains two working equations for the unknown function. A number of numerical examples are also used to study the performance of the algorithm.     

一类强退化抛物方程弱解的惟一性

研究了如下形式的强退化抛物方程(C)(u)/(t)=(2A(u,x,t))/(x2)+(B(u,x,t))/(x),基于Holm gren方法,证明了弱解的惟一性.     

变系数时滞抛物型微分方程解的振动的充要条件

建立了一类变系数时滞抛物型微分方程解的振动的若干充要条件 .     

Regularity Results for a Strongly Degenerate Parabolic Equation

M. Bertsch & R. Dal Passo proved the existence and uniqueness of the Cauchy problem for u_t = (φ(u),ψ(u_x))_x, where φ > 0, ψ is a strictly increasing function with lim_{s → ∞}ψ(s) = ψ_∞ < ∞. The regularity of the solution has been obtained under the condition φ" < 0 or φ = const. In the present paper, under the condition φ" ≤ 0, we give some regularity results. We show that the solution can be classical after a finite time. Further, under the condition φ" ≤ -α_0 (where -α_0 is a constant), we prove the gradient of the solution converges to zero uniformly with respect to x as t → +∞.     

Non-linear Semigroup of a Class of Abstract Semilinear Functional Differential Equations with a Non-Dense Domain

In this work, we are concerned with a general class of abstract semilinear autonomous functional differential equations with a non-dense domain on a Banach space. Our objective is to study, using the Crandall-Liggett approach, the solutions as a semigroup of non-linear operators.     

Analysis of a semigroup approach in the inverse problem of identifying an unknown parameters

This article presents a semigroup approach to the mathematical analysis of the inverse parameter problems of identifying the unknown parameters p(t) and q in the linear parabolic equation u_t(x, t)  = u_xx + qu_x(x, t) + p(t)u(x, t), with Dirichlet boundary conditions u(0, t) = ψ₀, u(1, t) = ψ₁. The main purpose of this paper is to investigate the distinguishability of the input-output mapping Φ[·]:P→H^1,2[0,T], via semigroup theory. In this paper, it is shown that if the nullspace of the semigroup T(t) consists of only zero function, then the input-output mapping Φ[·] has the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of the mapping. Moreover, under the light of the measured output data u_x(0, t) = f(t) the unknown parameter p(t) at (x, t) = (0, 0) and the unknown coefficient q are determined via the input data. Furthermore, it is shown that measured output data f(t) can be determined analytically by an integral representation. Hence the input-output mapping Φ[·]:P→H^1,2[0,T] is given explicitly interms of the semigroup.     

Identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach

In this article, we study the semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(x) in the linear parabolic equation ut(x,t)=(k(x)uxx(x,t)), with Dirichlet boundary conditions u(0,t)=ψ₀, u(1,t)=ψ₁. Main goal of this study is to investigate the distinguishability of the input-output mappings Φ[⋅]:K→C¹[0,T], Ψ[⋅]:K→C¹[0,T] via semigroup theory. In this paper, we show that if the null space of the semigroup T(t) consists of only zero function, then the input-output mappings Φ[⋅] and Ψ[⋅] have the distinguishability property. Moreover, the values k(0) and k(1) of the unknown diffusion coefficient k(x) at x=0 and x=1, respectively, can be determined explicitly by making use of measured output data (boundary observations) f(t):=k(0)ux(0,t) or/and h(t):=k(1)ux(1,t). In addition to these, the values k^′(0) and k^′(1) of the unknown coefficient k(x) at x=0 and x=1, respectively, are also determined via the input data. Furthermore, it is shown that measured output dataf(t) and h(t) can be determined analytically, by an integral representation. Hence the input-output mappings Φ[⋅]:K→C¹[0,T], Ψ[⋅]:K→C¹[0,T] are given explicitly in terms of the semigroup. Finally by using all these results, we construct the local representations of the unknown coefficient k(x) at the end points x=0 and x=1.