An integral solution for the inverse heat conduction problem after the method of Weber (Q1101195)

The inverse heat conduction problem (IHCP) amounts to calculating, for a thermally conducting solid, the surface heat flux and/or temperature data from measured temperatures at an interior point. Generally, this is an ill-posed problem. Taking a one-dimensional semi-infinite slab as the body, the IHCP has the following mathematical formulation: The temperature \(u=u(x,t)\) satisfies \(\partial u/\partial t=\partial\) 2u/\(\partial x\) 2, \(0<x<\infty\), \(t>0\), \(u(x,0)=0\), \(0\leq x<\infty\), u(x,t) bounded as \(x\to \infty\), \(t>0\), \(u(1,t)=F(t)\), \(-u_ x(0,t)=q(t)\) for \(t>0\), with F measured, q unknown. Here t stands for time and x for the distance from the heated surface. The flux condition may be replaced by \(u(0,t)=f(t)\), \(t>0.\) This problem is handled through an algorithm based on an integral representation of Weber's hyperbolic approximation, using a filtered version of the noisy data. Error bounds are obtained and numerical experiments presented indicate the stability and accuracy of the method, as well as the proper choice of parameters.