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Determination of an additive source in the heat equation

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Inverse Problems

Co-authors: Dinh Nho Hao (Hanoi Institute of Mathematics, Vietnam), Areena Hazanee (University of Leeds, UK), Mikola Ivanchov (Ivan Franko National University of Lviv, Ukraine), Phan Xuan Thanh (Hanoi University of Science and Technology, Vietnam)

Water contaminants arising from distributed or non-point sources deliver pollutants indirectly through environmental changes, e.g. a fertilizer is carried into a river by rain which in turn will affect the aquatic life. Then, in this inverse problem of water pollution, an unknown source in the governing equation needs to be determined from the measurements of the concentration or other projections of the dependent variable of the model. A similar inverse problem, arises in heat transfer.

Inverse source problems for the heat equation, especially in the one-dimensional transient case, have received considerable attention in recent years. In most of the previous studies, in order to ensure a unique solution, the unknown heat source was assumed to depend on only one of the independent variables, namely, space or time, or on the dependent variable, namely, concentration/temperature. It is the puropose of our analysis to investigate an extended case in which the unknown source is assumed to depend on both space and time, but which is additively separated into two unknown coefficient source functions, namely, one component dependent on space and another one dependent on time. The additional overspecified conditions can be a couple of local or nonlocal measurements of the concentration/temperature in space or time.

The unique solvability of this linear inverse problem in classical Holder spaces is proved; however, the problem is still ill-posed since small errors in the input data cause large errors in the output source. In order to obtain a stable reconstruction the Tikhonov regularization or the iterative conjugate gradient method is employed. Numerical results will be presented and discussed.

This talk is part of the Isaac Newton Institute Seminar Series series.

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