Surface Temperature Determination from Borehole Measurements : Regularization and Error Estimates

The authors consider the problem of determining the temperature distribution u(x, t) on the half-line x 0, > 0, from measurements at an interior point, for all > 0. As is well-known, this is an ill-posed problem Using the Tikhonov method, the authors give a regularized solution, and assuming the (unknown) exact solution is in H(R),a > 0 They give an error estimate of the order for e 0, where e > 0 is a bound on the measurement error