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  • Open Access

    ARTICLE

    Determination of an Unknown Heat Source Term from Boundary Data

    Y. Hu1, T. Wei1,2

    CMES-Computer Modeling in Engineering & Sciences, Vol.87, No.4, pp. 307-326, 2012, DOI:10.3970/cmes.2012.087.307

    Abstract This paper employ the method of fundamental solutions for determining an unknown heat source term in a heat equation from overspecified boundary measurement data. By a function transformation, the inverse source problem is changed into an inverse initial data problem which is solved by a method of fundamental solutions. The standard Tikhonov regularization technique with the generalized cross-validation criterion for choosing the regularization parameter is adopted for solving the resulting ill-conditioned system of linear algebraic equations. The effectiveness of the algorithm is illustrated by five numerical examples in one-dimensional and two-dimensional cases. More >

  • Open Access

    ARTICLE

    Numerical Reconstruction of a Space-Dependent Heat Source Term in a Multi-Dimensional Heat Equation

    C. Shi1, C. Wang1, T. Wei1,2

    CMES-Computer Modeling in Engineering & Sciences, Vol.86, No.2, pp. 71-92, 2012, DOI:10.3970/cmes.2012.086.071

    Abstract In this paper, we consider a typical ill-posed inverse heat source problem, that is, we determine a space-dependent heat source term in a multi-dimensional heat equation from a pair of Cauchy data on a part of boundary. By a simple transformation, the inverse heat source problem is changed into a Cauchy problem of a homogenous heat conduction equation. We use the method of fundamental solutions (MFS) coupled with the Tikhonov regularization technique to solve the ill-conditioned linear system of equations resulted from the MFS discretization. The generalized cross-validation rule for determining the regularization parameter is More >

  • Open Access

    ARTICLE

    A Three-Point BVP of Time-Dependent Inverse Heat Source Problems and Solving by a TSLGSM

    Weichung Yeih1,2, Chein-Shan Liu3

    CMES-Computer Modeling in Engineering & Sciences, Vol.46, No.2, pp. 107-128, 2009, DOI:10.3970/cmes.2009.046.107

    Abstract We consider an inverse problem for estimating an unknown time dependent heat source H(t) in a heat conduction equation ut(x,t) = uxx(x,t) + H(t). First this inverse problem is formulated as a three-point boundary value problem (BVP) for ODEs discretized from the transformed homogeneous governing equation. To treat this three-point BVP we develop a two-stage Lie-group shooting method (TSLGSM). The novel approach is examined through numerical examples to convince that it is rather accurate and efficient; the estimation error is small even for identifying discontinuous and oscillatory heat sources under noise. More >

  • Open Access

    ARTICLE

    An Accurate Refinement Scheme for Inverse Heat Source Location Identifications

    Leevan Ling1, Tomoya Takeuchi2

    CMES-Computer Modeling in Engineering & Sciences, Vol.20, No.2, pp. 99-110, 2007, DOI:10.3970/cmes.2007.020.099

    Abstract We aim to identify the unknown source locations in a two-dimensional heat equation from scattered measurements. In [Inverse Problems, 22(4):1289--1305, 2006], we proposed a numerical procedure that identifies the unknown source locations of 2D heat equation solely based on three measurement points. Due to the nonlinearity and complexity of the problem, the quality of the resulting estimations is often poor especially when the number of unknown is large. In this paper, we purpose a linear refinement scheme that takes the outputs of the existing nonlinear algorithm as initial guesses and iteratively improves on the accuracy More >

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