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The generalized finite difference method for an inverse time-dependent source problem associated with three-dimensional heat equation. (English) Zbl 1403.65039
Summary: This paper presents a meshless numerical scheme for recovering the time-dependent heat source in general three-dimensional (3D) heat conduction problems. The problem considered is ill-posed and the determination of the unknown heat source is achieved here by using the boundary condition, initial condition and the extra measured data from a fixed point placed inside the domain. The extra measured data are used to guarantee the uniqueness of the solution. The generalized finite difference method (GFDM), a recently-developed meshless method, is then adopted to solve the resulting time-dependent boundary-value problem. In our computations, the second-order Crank-Nicolson scheme is employed for the temporal discretization and the proposed GFDM for the spatial discretization. Several benchmark test problems with both smooth and piecewise smooth geometries have been studied to verify the accuracy and efficiency of the proposed method. No need to apply any well-known regularization strategy, the accurate and stable solution could be obtained with a comparatively large level of noise.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
80A23 Inverse problems in thermodynamics and heat transfer
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