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1.
The paper is concerned with the non-linear backward heat equation in the rectangle domain. The problem is severely ill-posed. We shall use a modified integral equation method to regularize the nonlinear problem. The error estimates of Hölder type of the regularized solutions are obtained. Numerical results are presented to illustrate the accuracy and efficiency of the method. This work is a generalization of many earlier papers, including the recent 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].  相似文献   

2.
We consider the problem of finding, from the final data u(x,y,T)=g(x,y), the initial data u(x,y,0) of the temperature function u(x,y,t),(x,y)I=(0,π)×(0,π),t[0,T] satisfying the following system
The problem is severely ill-posed. In this paper a simple and convenient new regularization method for solving this problem is considered. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. A numerical example also shows that the method works effectively.  相似文献   

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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. ,  and , Campbell et al. [4].  相似文献   

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In this paper, we consider the Cauchy problem for the Helmholtz equation in a rectangle, where the Cauchy data is given for y=0 and boundary data are for x=0 and x=π. The solution is sought in the interval 0<y≤1. A quasi-reversibility method is applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.  相似文献   

8.
In this paper, a Cauchy problem for the Laplace equation is investigated. Based on the fundamental solution to the elliptic equation, we propose to solve this problem by the truncation method, which generates well-posed problem. Then the well posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proved. Error estimates for this method are provided together with a selection rule for the regularization parameter. The numerical results show that our proposed numerical methods work effectively. This work extends to earlier results in Qian et al. (2008) [14] and Hao et al. (2009) [5].  相似文献   

9.
This work is devoted to solving the radially symmetric backward heat conduction problem, starting from the final temperature distribution. The problem is ill-posed: the solution (if it exists) does not depend continuously on the given data. A modified Tikhonov regularization method is proposed for solving this inverse problem. A quite sharp estimate of the error between the approximate solution and the exact solution is obtained with a suitable choice of regularization parameter. A numerical example is presented to verify the efficiency and accuracy of the method.  相似文献   

10.
ONTHECAUCHYPROBLEMOFNONLINEARDEGENERATEPARABOLICEQUATION¥YANGJINSHUNAbstract:Inthispaper,weprovetheexistenceofsolutionoftheCa...  相似文献   

11.
In this paper a simple and convenient new regularization method for solving backward heat equation—Fourier regularization method is given. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. A numerical example also shows that the method works effectively.  相似文献   

12.
This work studies the inverse problem of reconstructing an initial value function in the degenerate parabolic equation using the final measurement data. Problems of this type have important applications in the field of financial engineering. Being different from other inverse backward parabolic problems, the mathematical model in our article may be allowed to degenerate at some part of boundaries, which may lead to the corresponding boundary conditions missing. The conditional stability of the solution is obtained using the logarithmic convexity method. A finite difference scheme is constructed to solve the direct problem and the corresponding stability and convergence are proved. The Landweber iteration algorithm is applied to the inverse problem and some typical numerical experiments are also performed in the paper. The numerical results show that the proposed method is stable and the unknown initial value is recovered very well.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1900–1923, 2017  相似文献   

13.
We study the backward heat conduction problem in an unbounded region. The problem is ill-posed, in the sense that the solution if it exists, does not depend continuously on the data. Continuous dependence of the data is restored by cutting-off high frequencies in Fourier domain. The cut-off parameter acts as a regularization parameter. The discrepancy principle, for choosing the regularization parameter and double exponential transformation methods for numerical implementation of regularization method have been used. An example is presented to illustrate applicability and accuracy of the proposed method.  相似文献   

14.
We study Neumann problem for a class of nonlinear degenerate parabolic PDE. A typical nonlinearity we have in mind is, for instance, β(u)=−1/u(u>0). We establish a necessary and sufficient condition on given data for existence of solution.  相似文献   

15.
In this paper, we investigate the periodic homogenization of nonlinear parabolic equation arising from heat exchange in composite material problems. This problem, defined in periodical domain, is nonlinear at the interface. This nonlinearity models the heat radiation on the interface, which constitutes the transmission boundary conditions, between the two components of the material. The main challenge is, first, to show the well-posedness of the microscopic problem using the topological degree of Leray–Schauder tools. Then, we apply the two scale convergence to identify the equivalent macroscopic model using homogenization techniques. Finally, in order to confirm the efficiency of the homogenization process, we present some numerical results obtained via finite element approximation.  相似文献   

16.
We have recently developed two quasi-reversibility techniques in combination with Euler and Crank–Nicolson schemes and applied successfully to solve for smooth solutions of backward heat equation. In this paper, we test the viability of using these techniques to recover non-smooth solutions of backward heat equation. In particular, we numerically integrate the backward heat equation with smooth initial data up to a time of singularity (corners and discontinuities) formation. Using three examples, it is shown that the numerical solutions are very good smooth approximations to these singular exact solutions. The errors are shown using pseudo-L- and U-curves and compared where available with existing works. The limitations of these methods in terms of time of simulation and accuracy with emphasis on the precise set of numerical parameters suitable for producing smooth approximations are discussed. This paper also provides an opportunity to gain some insight into developing more sophisticated filtering techniques that can produce the fine-scale features (singularities) of the final solutions. Techniques are general and can be applied to many problems of scientific and technological interests.  相似文献   

17.
We find conditions on data guaranteeing global nonexistence of solutions to an inverse source problem for a class of nonlinear parabolic equations. We also establish a stability result on a bounded domain for a problem with the opposite sign on the power type nonlinearity.  相似文献   

18.
We consider a special sideways parabolic equation which appears in some applied subjects. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. In this paper some wavelet and spectral regularization methods for this problem are given. The error estimates are also established respectively.  相似文献   

19.
In this study, we consider a coefficient problem of a quasi-linear two-dimensional parabolic inverse problem with periodic boundary and integral over determination conditions. We prove the existence, uniqueness and continuously dependence upon the data of the solution by iteration method. Also, we consider numerical solution for this inverse problem by using linearization and the implicit finite-difference scheme.  相似文献   

20.
This article deals with iterative algorithms for domain decomposition applied to the solution of a singularly perturbed parabolic problem. These algorithms are based on finite difference domain decomposition methods and are suitable for parallel computing. Convergence properties of the algorithms are established. Numerical results for test problems are presented. © 1999 Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 389–405, 1999  相似文献   

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