Identification of a multi-dimensional space-dependent heat source from boundary data

Abstract

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights • Multi-dimensional space-dependent sources are reconstructed from boundary data. • Iterative regularization methods are developed. • Convergence and stability of numerical results are thoroughly investigated.

Recovering Time-Dependent Coefficients in a Two-Dimensional Parabolic Equation Using Nonlocal Overspecified Conditions via ADE Finite Difference Schemes

“…As an illustration, see references [18][19][20][21][22]. Also used in solving in the inverse problems, in references [23,24]. This method is considered unconditionally stable.…”

Identifying the Unknown Source in Linear Parabolic Equation by a Convoluting Equation Method

“…In [24], the authors used the simplified Tikhonov method to identify the unknown source. There are some other the regularization methods to solve this problem, such as the Fourier regularization method [25], the wavelet dual squares method [26], the Tikhonov-type method [27], the mollification regularization method [10,11,28], the optimization method [29], and the Landweber method [30,31].…”

A local meshless method for the numerical solution of space‐dependent inverse heat problems

“…The inverse heat conduction problems (IHCPs) with an unknown space‐dependent heat source are mostly highlighted in transportation, 1,2 conduction processes, 3,4 hydrology, 5,6 and diffusion process in natural materials 7‐9 . Mathematical representation of a class of IHCP with unknown space‐dependent heat source is given as where u ( x , t ) is the state variable, Δ is the Laplace operator, Ω is a bounded domain in R n , and f( x ) is the unknown space‐dependent heat source source term.…”

“…Several methods have been reported for numerical solution of IHCP (), including the boundary element method (BEM), 3 iterative regularization technique, 11 method of fundamental solution (MFS), 12,13 variational iteration method (VIM), 14,15 homotopy perturbation method (HPM), 16 Lie‐group method (LGM), 17 Landweber and conjugate gradient methods, 4 meshless radial basis functions (RBF) method, 18 and Haar wavelets method 19 …”

“…The construction of an accurate and stable numerical solution is difficult due to the ill-conditioned coefficient matrix and error in the extra conditions. Several methods have been reported for numerical solution of IHCP (1), including the boundary element method (BEM), 3 iterative regularization technique, 11 method of fundamental solution (MFS), 12,13 variational iteration method (VIM), 14,15 homotopy perturbation method (HPM), 16 Lie-group method (LGM), 17 Landweber and conjugate gradient methods, 4 meshless radial basis functions (RBF) method, 18 and Haar wavelets method. 19 The horizon of meshless methods is continuously expanding, and their applicability is increasing rapidly due to their ease of implementation in higher dimensions.…”