Regularization of a Cauchy problem for the heat equation

Abstract

In this paper, we study a Cauchy problem for the heat equation with linear source in the form u_t(x,t)= u_xx(x,t)+f(x,t), u(L,t)=  φ(t), u(L,t)= Ψ (t), (x,t) ∈ (0,L) ×(0, 2π). 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 φ^ε and Ψ^ε satisfying ‖ φ^ε - φ ‖+‖ Ψ^ε - Ψ ‖ ≤ ε and that f^ε satisfying ‖ f^ε(x,^. ) - f(x,^.) ‖ ≤ ε .  We give some error estimates between the regularized solution and the exact solution under some different a-priori conditions of exact solution.