Interest in inverse dynamical, spectral and scattering problems for differential equations on graphs is motivated by possible applications to nano-electronics and quantum waveguides and by a variety of other classical and quantum applications. Recently a new effective leaf peeling method has been proposed by S. Avdonin and P. Kurasov [3] for solving inverse problems on trees (graphs without cycles). It allows recalculating efficiently the inverse data from the original tree to the smaller trees, 'removing' leaves step by step up to the rooted edge. In this paper we describe the main step of the spectral and dynamical versions of the peeling algorithm -recalculating the inverse data for the 'peeled tree'.
Introduction.Metric graphs with defined on them differential operators or differential equations are called quantum graphs (or differential equation networks). There are two groups of uniqueness results concerning boundary inverse problems for quantum trees. Brown and Weikard [10], Yurko [12], and Freiling and Yurko [11] proved uniqueness results for trees with a priori known topology (connectivity) and lengths of edges using the Titchmarsh-Weyl matrix function (TW-function) as the inverse data.The second group concerns inverse problems with unknown topology, lengths of edges and potentials. Belishev [8] and Belishev and Vakulenko [9] used the spectral data, i.e. eigenvalues and the traces of the derivatives of eigenfunctions at the boundary vertices (that is equivalent to the knowledge of the TW-function).The paper by Avdonin and Kurasov [3] contains the most complete results in this direction. It proves that a quantum tree is uniquely determined by the reduced TW-function associated to all except one boundary vertices. Moreover, Key words and phrases. wave equation, boundary control method, Titchmarsh-Weyl function, leaf peeling method.
Avdonin and Kurasov proposed a leaf peeling method based on the boundary control to recover a potential for the wave equation on a tree. Avdonin and Nicaise considered a source identification problem for the wave equation on a tree. This paper extends the methodology to the wave equation with unknown potential and source distributed parameters defined on a general tree graph.
О решении обратной задачи на графе для уравнения теплопереноса с памятью Аннотация:Представлена новая постановка обратной задачи для уравнения теплопереноса, учитывающая эффект переноса теплоты с памятью. Получены условия корректной разрешимости прямой задачи и достаточные условия восстановления функции влияния на тепловой процесс (функции источника теплоты) на примере вырожденного графа. Указаны пути переноса полученных результатов на произвольный носительпроизвольный граф.Ключевые слова: уравнение теплопереноса с памятью, восстановления источника теплоты на сетевых носителях, метод граничного управления.On solving the inverse graph problem for the heat transfer equation with memory Abstract. A new formulation of the inverse problem for the heat transfer equation is presented, taking into account the effect of heat transfer with memory. The conditions for correct solvability of the direct problem and sufficient conditions for recovering the function of influence on the heat process (the function of the heat source) on the example of a degenerate graph are obtained. The ways of transferring the obtained results to an arbitrary carrier-an arbitrary graph-are specified.
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