共查询到20条相似文献,搜索用时 31 毫秒
1.
For a parabolic equation, we consider inverse problems of reconstructing a coefficient that depends on the space variables
alone. The first problem is to find a lower-order coefficient c(x) multiplying u(x, t), and the second problem is to find the coefficient a(x) multiplying Δu. As additional information, the integral of the solution with respect to time with some weight function is given. The coefficients
of the equation depend both on time and on the space variables. We obtain sufficient conditions for the existence of generalized
solutions of our problems; moreover, for the first problem, we also prove uniqueness and construct an iterative sequence that
converges to the desired coefficient almost everywhere in the domain. We present examples of input data of these problems
for which the assumptions of our theorems are necessarily true. 相似文献
2.
Alemdar Hasanov 《Applied mathematics and computation》2003,140(2-3):501-515
An inverse polynomial method of determining the unknown leading coefficient k=k(x) of the linear Sturm–Liouville operator Au=−(k(x)u′(x))′+q(x)u(x), x(0,1), is presented. As an additional condition only two measured data at the boundary (x=0,x=1) are used. In absence of a singular point (u′(x)≠0,u″(x)≠0,x[0,1]) the inverse problem is classified as a well-conditioned . If there exists at least one singular point, then the inverse problem is classified as moderately ill-conditioned (u′(x0)=0,x0(0,1);u′(x)≠0,x≠x0;u″(x)≠0,x[0,1]) and severely ill-conditioned (u′(x0)=u″(x0)=0,x0(0,1);u′(x)≠0,u″(x)≠0,x≠x0). For each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a Cauchy problem for to the first order differential equation with respect the unknown function k=k(x). An approximate analytical solution of the each Cauchy problems are derived in explicit form. Numerical simulations all the above cases are given for noise free and noisy data. An accuracy of the presented approach is demonstrated on numerical test solutions. 相似文献
3.
Lech Zarȩba 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(4):445-467
In this paper we consider the mixed problem for the equation u
tt
+ A
1
u + A
2(u
t
) + g(u
t
) = f(x, t) in unbounded domain, where A
1 is a linear elliptic operator of the fourth order and A
2 is a nonlinear elliptic operator of the second order. Under natural assumptions on the equation coefficients and f we proof existence of a solution. This result contains, as a special case, some of known before theorems of existence. Essentially,
in difference up to previous results we prove theorems of existence without the additional assumption on behavior of solution
at infinity.
相似文献
4.
In this paper, we study the inverse problem of the reconstruction of the right-hand side of special form for a parabolic equation in u in which the coefficients of u t and u depend on u (x,t) , with overdetermination given by the integral of the solution over time. The Fredholm property for this problem and the existence and uniqueness theorems in Sobolev spaces are established. 相似文献
5.
Yasser Khalili Nematollah Kadkhoda Dumitru Baleanu 《Mathematical Methods in the Applied Sciences》2020,43(12):7143-7151
In the present work, we consider the inverse problem for the impulsive Sturm–Liouville equations with eigenparameter-dependent boundary conditions on the whole interval (0,π) from interior spectral data. We prove two uniqueness theorems on the potential q(x) and boundary conditions for the interior inverse problem, and using the Weyl function technique, we show that if coefficients of the first boundary condition, that is, h1,h2, are known, then the potential function q(x) and coefficients of the second boundary condition, that is, H1,H2, are uniquely determined by information about the eigenfunctions at the midpoint of the interval and one spectrum or partial information on the eigenfunctions at some internal points and some of two spectra. 相似文献
6.
Stephanos Venakides 《纯数学与应用数学通讯》1985,38(2):125-155
The inverse scattering method is used to determine the distribution limit as ? → 0 of the solution u(x, t, ?) of the initial value problem. Ut ? 6uux + ?2uxxx = 0, u(x, 0) = v(x), where v(x) is a positive bump which decays sufficiently fast as x x→±α. The case v(x) ? 0 has been solved by Peter D. Lax and C. David Levermore [8], [9], [10]. The computation of the distribution limit of u(x, t, ?) as ? → 0 is reduced to a quadratic maximization problem, which is then solved. 相似文献
7.
We consider a solution of the Cauchy problem u(t, x), t > 0, x ∈ R
2, for one class of integro-differential equations. These equations have the following specific feature: the matrix of the coefficients of higher derivatives is degenerate for all x. We establish conditions for the existence of the limit lim
t→∞
u(t, x) = v(x) and represent the solution of the Cauchy problem in explicit form in terms of the coefficients of the equation.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1699 – 1706, December, 2004. 相似文献
8.
A. Yu. Shcheglov 《Computational Mathematics and Mathematical Physics》2006,46(5):776-795
The inverse problem of finding the coefficients q(s) and p(s) in the equation u tt = a 2 u xx + q(u)u t ? p(u)u x is investigated. As overdetermination required in the inverse setting, two additional conditions are set: a boundary condition and a condition with a fixed value of the timelike variable. An iteration method for solving the inverse problem is proposed based on an equivalent system of integral equations of the second kind. A uniqueness theorem and an existence theorem in a small domain are proved for the inverse problem to substantiate the convergence of the algorithm. 相似文献
9.
10.
Identification of a source factor in a control problem for the heat equation with a boundary memory term
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Cecilia Cavaterra Davide Guidetti 《Mathematical Methods in the Applied Sciences》2015,38(18):4818-4839
We consider a material with thermal memory occupying a bounded region Ω with boundary Γ. The evolution of the temperature u(t,x) is described by an integrodifferential parabolic equation containing a heat source of the form f(t)z0(x). We formulate an initial and boundary value control problem based on a feedback device located on Γ and prescribed by means of a quite general memory operator. Assuming both u and the source factor f are unknown, we study the corresponding inverse and control problem on account of an additional information. We prove a result of existence and uniqueness of the solution (u,f). Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
11.
The paper concerns Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. We start with suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Fixing then a solution u
0 such that the linearized at u
0 problem is non-degenerate, we apply the Implicit Function Theorem. As a result we get that for all small perturbations of the coefficients there exists exactly one solution u ≈ u
0 which depends smoothly (in W
2,p
with p larger than the space dimension) on the data. For that, no structure and growth conditions are needed and the perturbations of the coefficients can be general L
∞-functions of the space variable x. Moreover, we show that the Newton Iteration Procedure can be applied in order to obtain a sequence of approximate (in W
2,p
) solutions for u
0. 相似文献
12.
Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: Weak solution approach 总被引:1,自引:0,他引:1
The problem of determining the pair w:={F(x,t);T0(t)} of source terms in the parabolic equation ut=(k(x)ux)x+F(x,t) and Robin boundary condition −k(l)ux(l,t)=v[u(l,t)−T0(t)] from the measured final data μT(x)=u(x,T) is formulated. It is proved that both components of the Fréchet gradient of the cost functional can be found via the same solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is derived. The obtained results permit one to prove existence of a quasi-solution of the considered inverse problem, as well as to construct a monotone iteration scheme based on a gradient method. 相似文献
13.
We establish conditions for the existence and uniqueness of a solution of a problem with multipoint conditions with respect to a selected variable t (in the case of multiple nodes) and periodic conditions with respect to x
1,..., x
p for a nonisotropic partial differential equation with constant complex coefficients. We prove metric theorems on lower bounds for small denominators appearing in the course of the solution of this problem. 相似文献
14.
We investigate the well-posedness of a problem with multipoint conditions with respect to a chosen variable t and periodic conditions with respect to coordinates x
1,..., x
p for a nonisotropic (concerning differentiation with respect to t and x
1,..., x
p) partial differential equation with constant complex coefficients. We establish conditions for the existence and uniqueness of a solution of this problem and prove metric theorems on lower bounds for small denominators appearing in the course of the construction of its solution. 相似文献
15.
This paper presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(ux) in the inhomogenenous quasi‐linear parabolic equation ut(x, t)=(k(ux)ux(x, t))x +F(u), with the Dirichlet boundary conditions u(0, t)=ψ0, u(1, t)=ψ1 and source function F(u). The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[·]:??→C1[0, T], Ψ[·]:??→C1[0, T] via semigroup theory. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
16.
We consider an inverse problem for the determination of a purely time-dependent source in a semilinear parabolic equation with a nonlocal boundary condition. An approximation scheme for the solution together with the well-posedness of the problem with the initial value u0 ∈ H1(Ω) is presented by means of the Rothe time-discretization method. Further approximation scheme via Rothe’s method is constructed for the problem when u0 ∈ L2(Ω) and the integral kernel in the nonlocal boundary condition is symmetric.
相似文献17.
V. L. Kamynin 《Differential Equations》2011,47(1):91-101
We study existence and uniqueness of the solution for the inverse problem of determination of the unknown coefficient ϱ(t) multiplying u
t
in a nondivergence parabolic equation. As additional information, the integral of the solution over the domain of space variables
with some given weight function is specified. The coefficients of the equation depend both on time and on the space variables. 相似文献
18.
We study the problem of existence of periodic and almost periodic solutions of the scalar equation x′ (t) = − δx(t) + pmax
u∈[t − h, t]
x(u) + f(t) where δ, p ∈ R, with a periodic (almost periodic) perturbation f(t). For these solutions, we establish conditions of global exponential stability and prove uniqueness theorems.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 747–754, June, 1998. 相似文献
19.
Arzu Erdem 《Mathematical Methods in the Applied Sciences》2015,38(7):1393-1404
In this paper, we consider an inverse source problem of identification of F(t) function in the linear parabolic equation ut = uxx + F(t) and u0(x) function as the initial condition from the measured final data and local boundary data. Based on the optimal control framework by Green's function, we construct Fréchet derivative of Tikhonov functional. The stability of the minimizer is established from the necessary condition. The CG algorithm based on the Fréchet derivative is applied to the inverse problem, and results are presented for a test example. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
20.
This article presents a semigroup approach to the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(ux) in the quasi‐linear parabolic equation ut(x, t)=(k(ux)ux(x, t))x+F(x, t), with Dirichlet boundary conditions u(0, t)=ψ0, u(1, t)=ψ1 and source function F(x, t). The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[·]: ?? → C1[0, T], Ψ[·]: ?? → C1[0, T] via semigroup theory. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献