Determination of a Time-Dependent Term in the Right-Hand Side of Linear Parabolic Equations
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Determination of a Time-Dependent Term in the Right-Hand Side of Linear Parabolic Equations Nguyen Thi Ngoc Oanh1 · Bui Viet Huong1
Received: 10 December 2014 / Revised: 23 December 2014 / Accepted: 26 December 2014 / Published online: 9 July 2015 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015
Abstract The problem of determining a time-dependent term from integral observations in the right-hand side of parabolic equations is studied. It is reformulated into a variational problem, and a formula for the gradient of the functional to be minimized is derived via an adjoint problem. The variational problem is discretized by the splitting method based on finite differences. A formula for the gradient of the discretized functional is given and the conjugate gradient method is suggested for numerically solving the problem. Several numerical examples are presented for illustrating the efficiency of the algorithm. Keywords Determination of the right-hand side · Inverse problems · Ill-posed problems · Integral observations · Splitting methods · Conjugate gradient method Mathematics Subject Classification (2010) 65M32 · 65N20 · 65J20
1 Introduction In many practical contexts, the sources in diffusion or heat transfer processes are not known and required to be determined from several additional conditions, say, observations or measurements [1, 2, 6, 9, 15, 16]. These are inverse problems of determining a term in the right-hand side of parabolic equations. Due to their importance in practice, a great number of researchers took part in studying them and a lot of theoretical and numerical methods for them have been developed [3–5, 11–13, 17–19, 25–28]. As there is a vast literature on
Nguyen Thi Ngoc Oanh
[email protected] Bui Viet Huong [email protected] 1
College of Science, Thai Nguyen University, Thai Nguyen, Vietnam
314
N. T. N. Oanh, B. V. Huong
these inverse problems, we do not attempt to give a review on them, but refer to the excellent survey by Prilepko and Tkachenko [26] and the recent paper by Hasanov [12] and the references therein, as well as in the above cited references. In this paper, we concentrate ourselves on a particular case: determine a time-dependent term from an integral observation in the right-hand side of parabolic equations in the time direction. We note that the problem of determining a time-dependent term in the righthand side of parabolic equations has been mainly investigated for one-dimensional case (see above cited references) with pointwise observations, except for [19, 24] where some multidimensional problems with integral observations are studied. Our motivation for such a formulation of this problem is that in practice any instrument has a width, and so any measurement is average. Therefore, integral observations are more reasonable than pointwise ones. Following this viewpoint, the pointwise observations can be regarded as an average process or the limit of average processes. However, when so
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