Determine the source term of a two-dimensional heat equation

Dang Duc Trong (1) , Truong Trung Tuyen (2) , Phan Thanh Nam (1) , Alain Pham Ngoc Dinh (3)

1 UNS-HCMC - University of Natural Sciences HoChiMinh City
2 IU - Department of Mathematics
3 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

Let $\Omega$ be a two-dimensional heat conduction body. We consider the problem of determining the heat source $F(x,t)=\varphi(t)f(x,y)$ with $\varphi$ be given inexactly and $f$ be unknown. The problem is nonlinear and ill-posed. By a specific form of Fourier transforms, we shall show that the heat source is determined uniquely by the minimum boundary condition and the temperature distribution in $\Omega$ at the initial time $t=0$ and at the final time $t=1$. Using the methods of Tikhonov's regularization and truncated integration, we construct the regularized solutions. Numerical part is given.

Mots clés: truncated integration; Error estimate; Fourier transform; ill-posed problem; heat source; Tikhonov's regularization; truncated integration.

Domaines: Equations aux dérivées partielles [math.AP]

Dang Duc Trong, Truong Trung Tuyen, Phan Thanh Nam, Alain Pham Ngoc Dinh. Determine the source term of a two-dimensional heat equation. 2008. ⟨hal-00295103⟩