Heat Conduction Equation In The Inverse Problem

In this paper, some inverse problems for the heat equation are considered.First, we discuss the noncharacteristic Cauchy problem for 1-D heat equation. By using fundamental solution, the problem is transformed into a boundary integral equations. We prove the uniqueness for the noncharacteristic Cauchy problem and propose the numerical algorithm. The numerical results show that our method is effective.For the noncharacteristic Cauchy problem for 2-D and 3-D heat equation, we give the explicit Holder type stability estimates by using weighted energy method. Especially, Holder type stability estimate for discrete solutions of the heat equation is obtained through a fully discrete method. The numerical results are presented.We also discuss the inverse source problem for the heat equation, where the function of the source term has the form f(t)Φ(x). If the function Φ(x) is given, we prove the Lipschitz stability results under the assumption that f(t) is a piecewise constant function. The new algorithm is proposed to reconstruct the function f(t). Some numerical results are presented.