Backward Heat Problem Research Articles

The Levenberg–Marquardt regularization for the backward heat equation with fractional derivative

The backward heat problem with time-fractional derivative in Caputo's sense is studied. The inverse problem is severely ill-posed in the case when the fractional order is close to unity. A Levenberg–Marquardt method with a new a posteriori stopping rule is investigated. We show that optimal order can be obtained for the proposed method under a Hölder-type source condition. Numerical examples for one and two dimensions are provided.

Two-Parameter Regularization Method for a Nonlinear Backward Heat Problem with a Conformable Derivative

In this paper, we consider the nonlinear inverse-time heat problem with a conformable derivative concerning the time variable. This problem is severely ill posed. A new method on the modified integral equation based on two regularization parameters is proposed to regularize this problem. Numerical results are presented to illustrate the efficiency of the proposed method.

A Cauchy problem for the asymmetric Parabolic equation in polar coordinates with the perturbed diffusivity

The inverse problem for the heat equation plays an important area of study and application. Up to now, the backward heat problem (BHP) in Cartesian coordinates has been arisen in many articles, but the BHP in different domains such as polar coordinates, cylindrical one or spherical one is rarely considered. This paper’s purpose is to investigate the BHP on a disk, especially, the problem is associated with the perturbed diffusivity and the space-dependent heat source. In order to solve the problem, the authors apply the separation of variables method, associated with the Bessel’s equation and Bessel’s expansion. Based on the exact solution, the regularized solution is constructed by using the modified quasi-boundary value method. As a result, a Holder type of convergence rate has been obtained. In addition, a numerical experiment is given to illustrate the flexibility and effectiveness of the used method.

On an asymmetric backward heat problem with the space and time-dependent heat source on a disk

Abstract In this article, we investigate the backward heat problem (BHP) which is a classical ill-posed problem. Although there are many papers relating to the BHP in many domains, considering this problem in polar coordinates is still scarce. Therefore, we wish to deal with this problem associated with a space and time-dependent heat source in polar coordinates. By modifying the quasi-boundary value method, we propose the stable solution for the problem. Furthermore, under some initial assumptions, we get the Hölder type of error estimates between the exact solution and the approximated solution. Eventually, a numerical experiment is provided to prove the effectiveness and feasibility of our method.

On a Backward Heat Conduction Problem Associated with Asymmetric Final Data

In this work, we investigate an abstract parabolic problem associated with the final data, with one specific case of the abstract parabolic problem being considered in polar coordinates. Specially, the present paper gives the first treatment which is not only more general but also more applicable in the case the backward heat problem (BHP) is not symmetric and not axisymmetric in polar coordinates. In general, the above problem is severely ill-posed in the sense of Hadamard. Here, we propose the modified quasi-boundary value (MQBV) method in order to obtain the stability of the regularized solution for the problem. To our knowledge, our results are new and they improve the results of two previous papers (see Cheng and Fu (Inverse Probl. Sci. Eng. 17(8), 1085–1093 2009), (Acta Math. Sinica, English Series 26(11), 2157–2164 2010)) while the authors only considered axisymmetric or radially symmetric data. Finally, a numerical example is given to demonstrate the feasibility and efficiency of our method.

A two-dimensional backward heat problem with statistical discrete data

Abstract We focus on the nonhomogeneous backward heat problem of finding the initial temperature θ = θ ⁢ ( x , y ) = u ⁢ ( x , y , 0 ) {\theta=\theta(x,y)=u(x,y,0)} such that { u t - a ⁢ ( t ) ⁢ ( u x ⁢ x + u y ⁢ y ) = f ⁢ ( x , y , t ) , ( x , y , t ) ∈ Ω × ( 0 , T ) , u ⁢ ( x , y , t ) = 0 , ( x , y ) ∈ ∂ ⁡ Ω × ( 0 , T ) , u ⁢ ( x , y , T ) = h ⁢ ( x , y ) , ( x , y ) ∈ Ω ¯ , \left\{\begin{aligned} \displaystyle u_{t}-a(t)(u_{xx}+u_{yy})&\displaystyle=f% (x,y,t),&\hskip 10.0pt(x,y,t)&\displaystyle\in\Omega\times(0,T),\\ \displaystyle u(x,y,t)&\displaystyle=0,&\hskip 10.0pt(x,y)&\displaystyle\in% \partial\Omega\times(0,T),\\ \displaystyle u(x,y,T)&\displaystyle=h(x,y),&\hskip 10.0pt(x,y)&\displaystyle% \in\overline{\Omega},\end{aligned}\right.\vspace*{-0.5mm} where Ω = ( 0 , π ) × ( 0 , π ) {\Omega=(0,\pi)\times(0,\pi)} . In the problem, the source f = f ⁢ ( x , y , t ) {f=f(x,y,t)} and the final data h = h ⁢ ( x , y ) {h=h(x,y)} are determined through random noise data g i ⁢ j ⁢ ( t ) {g_{ij}(t)} and d i ⁢ j {d_{ij}} satisfying the regression models g i ⁢ j ⁢ ( t ) = f ⁢ ( X i , Y j , t ) + ϑ ⁢ ξ i ⁢ j ⁢ ( t ) , \displaystyle g_{ij}(t)=f(X_{i},Y_{j},t)+\vartheta\xi_{ij}(t), d i ⁢ j = h ⁢ ( X i , Y j ) + σ i ⁢ j ⁢ ε i ⁢ j , \displaystyle d_{ij}=h(X_{i},Y_{j})+\sigma_{ij}\varepsilon_{ij}, where ( X i , Y j ) {(X_{i},Y_{j})} are grid points of Ω. The problem is severely ill-posed. To regularize the instable solution of the problem, we use the trigonometric least squares method in nonparametric regression associated with the projection method. In addition, convergence rate is also investigated numerically.

Backward heat equations with locally lipschitz source

Let be in and let , be a locally Lipschitz function with respect to the variable and . In this article, we consider the following backward heat problem with a nonlinear heat source The problem is ill-posed in the sense of Hadamard. Hence a regularization is in order. We use the -dimensional Fourier transform to construct an approximate solution for this problem. We also prove error estimates for our problem and give comments on similar problems.

On an initial inverse problem in nonlinear heat equation associated with time-dependent coefficient

In this paper, a nonlinear backward heat problem with time-dependent coefficient in the unbounded domain is investigated. A modified regularization method is established to solve it. New error estimates for the regularized solution are given under some assumptions on the exact solution.

Inverse Estimates for Nonhomogeneous Backward Heat Problems

We investigate the inverse problem in the nonhomogeneous heat equation involving the recovery of the initial temperature from measurements of the final temperature. This problem is known as the backward heat problem and is severely ill-posed. We show that this problem can be converted into the first Fredholm integral equation, and an algorithm of inversion is given using Tikhonov's regularization method. The genetic algorithm for obtaining the regularization parameter is presented. We also present numerical computations that verify the accuracy of our approximation.

INVERSE ESTIMATION OF THE INITIAL CONDITION FOR THE HEAT EQUATION

In this work,we investigate the inverse problem in the heat equation involving the recovery of the initial temperature from measurements of the final temperature. This problem is known as the backward heat problem and is severely ill-posed. We show that this problem can be converted into the first Fredholm integral equation, and an algorithm of inversion is given using the Tikhonov's regularization method. The Newton root-finding algorithm for obtaining the regularization parameter is presented. We also present numerical computations that verify the accuracy of our approximation.

Some extended results on a nonlinear ill-posed heat equation and remarks on a general case of nonlinear terms

The major object of this paper is to provide a quite convenient regularization method for a nonlinear backward heat problem. Error estimates for this method are provided together with a selection rule for the regularization parameter. Our method improve some results in a previous paper, including the earlier paper [D.D. Trong, N.H. Tuan, Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation, Nonlinear Anal. 71 (9) (2009) 4167–4176] and some other papers. A general case of nonlinear terms for this problem is obtained.

Notes on a new approximate solution of 2-D heat equation backward in time

In this paper, we consider a backward heat problem that appears in many applications. This problem is ill-posed. The solution of the problem as the solution exhibits unstable dependence on the given data functions. Using a new regularization method, we regularize the problem and get some new error estimates. Some numerical tests illustrate that the proposed method is feasible and effective. This work is a generalization of many recent papers, including the earlier paper [A new regularized method for two dimensional nonhomogeneous backward heat problem, Appl. Math. Comput. 215(3) (2009) 873–880] and some other authors such as Chu-Li Fu et al. [1–3], Campbell et al. [4].

Sharp estimates for approximations to a nonlinear backward heat equation

A nonlinear backward heat problem for an infinite strip is considered. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. In this paper, we use the Fourier regularization method to solve the problem. Some sharp estimates of the error between the exact solution and its regularization approximation are given.

Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation

We consider the inverse time problem for the nonlinear heat equation in the form u t − u x x = f ( x , t , u ( x , t ) ) , ( x , t ) ∈ ( 0 , π ) × ( 0 , T ) , u ( 0 , t ) = u ( π , t ) = 0 t ∈ ( 0 , T ) . The nonlinear problem is severely ill-posed. We shall use the method of integral equation to regularize the problem and to get some error estimates. We show that the approximate problems are well-posed and that their solution u ϵ ( x , t ) converges on [ 0 , T ] if and only if the original problem has a unique solution. We obtain several other results, including some explicit convergence rates. Some numerical tests illustrate that the proposed method is feasible and effective.

A quasi Tikhonov regularization for a two-dimensional backward heat problem by a fundamental solution

Consider a two-dimensional backward heat conduction problem for a general domain with a boundary. Based on the fundamental solution to the heat equation, we propose to solve this problem by the boundary integral equation method, which generates a coupled ill-posed integral equation. Then the well posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proven. Our regularizing scheme can be considered a quasi Tikhonov regularization, with the advantage of a relatively small amount of computation compared with the classical Tikhonov regularization. Numerical performances are given to show the validity of our inversion method.

A Nonlinear Case of the 1-D Backward Heat Problem: Regularization and Error Estimate

We consider the problem of finding, from the final data u(x,T)=\varphi(x) , the temperature function u(x,t),\ x\in (0,\pi),\ t\in [0,T] satisfies the following nonlinear system \begin{align*} u_t-u_{xx}&= f(x,t,u(x,t)), &\quad &(x,t)\in (0,\pi)\times (0,T) \\ u(0,t)&= u(\pi,t)=0, &\quad &t\in (0,T). \end{align*} The nonlinear problem is severely ill-posed. We shall improve the quasi-boundary value method to regularize the problem and to get some error estimates. The approximation solution is calculated by the contraction principle. A numerical experiment is given.

On the backward heat problem: evaluation of the norm of

We show in this paper that ‖Δu‖ = ‖ut‖ is bounded ∀t ≤ T(0) < T if one imposes on u (solution of the backward heat equation) the condition ‖u(x, t)‖ ≤ M. A Hölder type of inequality is also given if one supposes ‖ut(x, T)‖ ≤ K.