A regularized integral equation method for the inverse geometry heat conduction problem

This paper addresses a new technique for solving the inverse geometry heat conduction problem of the Laplace equation in a two-dimensional rectangle, which, named regularized integral equation method (RIEM), consists of three parts. First of all, the Fourier series expansion technique is used to calculate the temperature field u(x, y). Second, we consider a Lavrentiev regularization by adding a term αg(x) to obtain a second kind Fredholm integral equation. The termwise separable property of the kernel function allows us to transform the inverse geometry heat conduction problem into a two-point boundary value problem and therefore, an analytical regularized solution is derived in the final part by using orthogonality. Principally, the RIEM possesses the following advantages: it does not need any guess of the initial profile, it does not need any iteration and a regularized closed-form solution can be obtained. The uniform convergence and error estimate of the regularized solution uα(x, y) are proved and a boundary geometry p(x) is solved by half-interval method. Several numerical examples present the effectiveness of our novel approach in providing excellent estimates of unknown boundary shapes from given data.