An integral solution for the inverse heat conduction problem after the method of Weber
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.
scientific article; zbMATH DE number 4047012
scientific article