TAILIEUCHUNG - Regularization of a Cauchy problem for the heat equation

In this paper, we study a Cauchy problem for the heat equation with linear source in the. This problem is ill-posed in the sense of Hadamard. To regularize the problem, the truncation method is proposed to solve the problem in the presence of noisy Cauchy data and satisfying We give some error estimates between the regularized solution and the exact solution under some different a-priori conditions of exact solution. | Science Technology Development Vol 5 2017 Regularization of a Cauchy problem for the heat equation Vo Van Au University of Science VNU-HCM Can Tho University of Technology Nguyen Hoang Tuan University of Education Ho Chi Minh Received on 5th December 2016 accepted on 28th November 2017 ABSTRACT In this paper we study a Cauchy problem for the heat equation with linear source in the form ut x t Ux x t f x t u L t p t u L t ự t x t e 0 L X 0 2 . This problem is ill-posed in the sense of Hadamard. To regularize the problem the truncation method is proposed to solve the problem in the presence of noisy Key words elliptic equation ill-posed problem cauchy problem regularization method truncation method Cauchy data p8 and ự8 satisfying p8 -p ự8 -ự 8 and that f8 satisfying If x - f x . 8. We give some error estimates between the regularized solution and the exact solution under some different a-priori conditions of exact solution. INTRODUCTION In this paper the temperature u x t for x t e 0 L X 0 2 1 is sought from known boundary temperature u L t p t and heat flux u L t ự t measurements satisfying the following problem ut x t uxx x t f x t 0 x L 0 t In u L t p t 0 t In WheeKtp ự are 0 tgiven functions usually in L2 0 2 and f is a given linear heat source which may depend on the independent variables x t . Note that we have no initial condition prescribed at t 0 and moreover the Cauchy data p and ự are perturbed so as to contain measurement errors in the form of the input noisy Cauchy data p8 and ự8 also in L2 0 2 satisfying kM k-d e 2 where Il-Il denotes the L2 0 2 7 -norm and S Q is a small positive number representing the level of noise. Trang 184 It is well-known that at least in the linear case the problem 1 has at most one solution using classical analytical sideways continuation for the parabolic heat equation. The existence of solution also holds in the case f 0 . However the problem is still ill-posed in the sense that the solution if it exists does not .

• Vật lý
• Regularization of a Cauchy problem for the heat equation
• Cauchy problem for the heat equation
• A priori conditions
• Different a priori conditions
• Quasi reversibility method