A procedure for determining a spacewise dependent heat source and the initial temperature

Bjorn Johansson, D Lesnic

Research output: Contribution to journalArticle

63 Citations (Scopus)


The inverse problem of determining a spacewise dependent heat source, together with the initial temperature for the parabolic heat equation, using the usual conditions of the direct problem and information from two supplementary temperature measurements at different instants of time is studied. These spacewise dependent temperature measurements ensure that this inverse problem has a unique solution, despite the solution being unstable, hence the problem is ill-posed. We propose an iterative algorithm for the stable reconstruction of both the initial data and the source based on a sequence of well-posed direct problems for the parabolic heat equation, which are solved at each iteration step using the boundary element method. The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for a typical benchmark test example, which has the input measured data perturbed by increasing amounts of random noise. The numerical results show that the proposed procedure gives accurate numerical approximations in relatively few iterations.
Original languageEnglish
Pages (from-to)265-276
Number of pages12
JournalApplicable Analysis
Issue number3
Publication statusPublished - 1 Mar 2008


  • heat source
  • parabolic heat
  • equation
  • boundary element method
  • iterative regularization
  • inverse problem
  • initial condition
  • discrepancy principle


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