A numerical method of the Cauchy problem for the heat equation based on integral equations

Abstract In this paper, we consider the Cauchy problem for the heat equation. The inverse problem involves reconstructing the missing data on an inaccessible boundary from the measured data on an accessible boundary, which is severely ill-posed. By Green’s representation theorem, we transform the Cauchy problem into a system of two boundary integral equations. Noticing the ill-posedness of this system, a stable numerical method via a quasi Tikhonov regularization technique is proposed. Then the well-posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proven. The advantages of our proposed method are as follows: on one hand, we can obtain simultaneously the Dirichlet data and the Neumann data by solving a regularized system; on the other hand, compared with the method based on the classical Tikhonov regularization technique for solving the equations, our proposed method is simpler from the numerical point of view, and needs relatively small amount of computations in numerical implementations. Based on the Nyström method and trigonometric approximation for singular integrals, we give an effective way of discretizing the boundary integral equations. Some numerical results, with exact and noisy measurement data, are also presented to show the efficiency and accuracy of our scheme. Furthermore, it is shown that the proposed method remains applicable even when only partial boundary Cauchy data are available.