An integral equation method to recover non-additive and non-separable heat source without initial temperature

When the unknown heat source is a non-additive and non-separable function of space-time it is very difficult to recover it. The paper proposes a more difficult mixed type inverse heat conduction problem by simultaneously recovering heat source term and unknown initial temperature, under extra measured data of boundary heat fluxes and a final time condition. We derive a domain/boundary integral equation method (DBIEM), and two different types adjoint Trefftz test functions are used to develop two different numerical algorithms, namely the DBIEM1 and the DBIEM2, which are effective and robust to recover the unknown heat source and initial value. In the adjoint Trefftz test functions we introduce a scaling factor and the multiple-scales are introduced in the trial functions, which play key roles to stabilize the resulting linear system. When we add a large absolute and relative noise up to 10% on the supplementary data, the present numerical methods are quite stable against large noise, and with a fast convergent solution of the linear system to determine the expansion coefficients, typically from five to twenty steps. It is remarkable that we can recover the general heat source function and unknown initial temperature without needing of internal measurement of temperature.