Recovering Heat Source from Fourth-Order Inverse Problems by Weighted Gradient Collocation

Judy P. Yang*, Hsiang Ming Li

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


The weighted gradient reproducing kernel collocation method is introduced to recover the heat source described by Poisson’s equation. As it is commonly known that there is no unique solution to the inverse heat source problem, the weak solution based on a priori assumptions is considered herein. In view of the fourth-order partial differential equation (PDE) in the mathematical model, the high-order gradient reproducing kernel approximation is introduced to efficiently untangle the problem without calculating the high-order derivatives of reproducing kernel shape functions. The weights of the weighted collocation method for high-order inverse analysis are first determined. In the benchmark analysis, the unclear illustration in the literature is clarified, and the correct interpretation of numerical results is given particularly. Two mathematical formulations with four examples are provided to demonstrate the viability of the method, including the extreme cases of the limited accessible boundary.

Original languageEnglish
Article number241
Issue number2
StatePublished - 1 Jan 2022


  • Fourth-order PDE
  • Gradient approximation
  • Heat source
  • Inverse problems
  • Reproducing kernel approximation
  • Weighted collocation method


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