Engineering Analysis with Boundary Elements 91 (2018) 73–81 Contents lists available at ScienceDirect Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound The generalized finite difference method for an inverse time-dependent source problem associated with three-dimensional heat equation Yan Gu a,∗ , Jun Lei b,∗ , Chia-Ming Fan c , Xiao-Qiao He d a School of Mathematics and Statistics, Qingdao University, Qingdao 266071, PR China b Department of Engineering Mechanics, Beijing University of Technology, Beijing 100124, PR China c Department of Harbor and River Engineering & Computation and Simulation Center, National Taiwan Ocean University, Keelung 20224, Taiwan d Department of Civil and Architectural Engineering, City University of Hong Kong, Hong Kong a r t i c l e i n f o Keywords: Generalized finite difference method Meshless method Time-dependent heat source Inverse problems Three-dimensional problems a b s t r a c t 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 unique- ness 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 regular- ization strategy, the accurate and stable solution could be obtained with a comparatively large level of noise. 1. Introduction The problem of recovering the unknown heat sources arising in time- dependent heat conduction equations presents an interesting challenge in many areas of science and engineering. Specific applications can be found, for example, in aerospace, chemical, mechanical and nuclear en- gineering [1,2]. The problem belongs to the broad class of inverse source problems which are usually ill-posed because small random errors in measurement may result in arbitrarily large errors in the numerical so- lutions [3–5]. The existence and uniqueness of solutions for this class of inverse problems have been discussed by Savateev in Ref. [6], when some priori information is available on the functional form of the un- known sources. Some numerical techniques for determining the unknown sources in a parabolic equation have been considered by many authors. In Refs. [7–19], the identification of unknown sources in steady-state heat con- duction problems was considered. In Refs. [20–28], several numerical schemes have been proposed to recover the unknown heat sources in transient heat conduction problems in which the heat source is taken to be time-dependent only. The problems of recovering a heat source de- pendent only on space were considered in Refs. [29–31]. In Ref. [32] the method of fundamental solutions (MFS) coupled with method of radial basis functions (RBFs) has been employed for an inverse heat source ∗ Corresponding authors. E-mail address: guyan1913@163.com (Y. Gu). problem, without any restriction for the form of unknown sources. In some more recent studies [33,34], inverse source problems for frac- tional diffusion equations have been considered. Impressive results have been obtained from aforementioned techniques, however only a limited number of papers devoted to 3D transient heat conduction problems are available in the literature. The recovery of heat sources in this subject is one of the purposes of this paper. In this paper, we investigate a numerical scheme based on the gen- eralized finite difference method (GFDM), a relatively new meshless method, for the recovery of the time-dependent unknown heat source in 3D heat conduction problems. The basis of the GFDM was proposed in the 80s by Lizska and Orkisz [35,36] and were later essentially extended and improved by Benito, Urena and Gavete [37–41]. The main idea of the method is to combine the Taylor series expansions and the moving- least squares (MLS) approximation to derive explicit formulae for the required partial derivatives of unknown variables. In Refs. [37,39], Ben- ito et al. proposed GFDM formulae for second-order partial differential equations in two dimensions. The influence of key parameters, which in- volved criterions of point generation, weighting function and the shape of the domain, has been well-studied, which can be viewed as a good guidance for using the GFDM. An h-adaptive algorithm for GFDM were described in Refs. [38,42] for 2D and 3D cases, respectively. Ureña et al. [43] studied the GFDM solution for advection-diffusion equations and https://doi.org/10.1016/j.enganabound.2018.03.013 Received 3 December 2017; Received in revised form 13 February 2018; Accepted 16 March 2018 0955-7997/© 2018 Elsevier Ltd. All rights reserved.