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**The method of fundamental solutions for inverse source problems associated with the steady-state heat conduction.**
*(English)*
Zbl 1194.80101

Summary: This paper presents the use of the method of fundamental solutions (MFS) for recovering the heat source in steady-state heat conduction problems from boundary temperature and heat flux measurements. It is well known that boundary data alone do not determine uniquely a general heat source and hence some a priori knowledge is assumed in order to guarantee the uniqueness of the solution. In the present study, the heat source is assumed to satisfy a second-order partial differential equation on a physical basis, thereby transforming the problem into a fourth-order partial differential equation, which can be conveniently solved using the MFS. Since the matrix arising from the MFS discretization is severely ill-conditioned, a regularized solution is obtained by employing the truncated singular value decomposition, whilst the optimal regularization parameter is determined by the L-curve criterion. Numerical results are presented for several two-dimensional problems with both exact and noisy data. The sensitivity analysis with respect to two solution parameters, i.e. the number of source points and the distance between the fictitious and physical boundaries, and one problem parameter, i.e. the measure of the accessible part of the boundary, is also performed. The stability of the scheme with respect to the amount of noise added into the data is analysed. The numerical results obtained show that the proposed numerical algorithm is accurate, convergent, stable and computationally efficient for solving inverse source problems in steady-state heat conduction.

### MSC:

80A23 | Inverse problems in thermodynamics and heat transfer |

80A20 | Heat and mass transfer, heat flow (MSC2010) |

80M25 | Other numerical methods (thermodynamics) (MSC2010) |

65N80 | Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs |

### Keywords:

method of fundamental solutions; inverse source problem; truncated singular value decomposition; steady-state heat conduction
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\textit{B. Jin} and \textit{L. Marin}, Int. J. Numer. Methods Eng. 69, No. 8, 1570--1589 (2007; Zbl 1194.80101)

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