We present an overview of recent methods based on semidiscretization (in time) for the inverse ill-posed problems of finding the solutions of evolution equations according to the time-like Cauchy data. Specifically, the values of function and normal derivative are given on a portion of the lateral boundary of a space-time cylinder and the corresponding data should be generated on the remaining lateral part of the cylinder either for the heat equation or for the wave equation. The procedure of semidiscretization in time is based on the application either of the Laguerre transform or of the Rothe method (finite-difference approximation), and has a specific feature that similar sequences of elliptic problems are obtained for the heat and wave equations, and only the values of some parameters are different. The elliptic equations are solved numerically either by the boundary integral approach involving the Nyström method or by the method of fundamental solutions. The theoretical properties are formulated together with discretization strategies in the space. Systems of linear equations are obtained for finding either the values of densities or the coefficients. The Tikhonov regularization is applied for the stable solution of the linear equations. The presented numerical results show that the proposed strategies give good accuracy in combination with economic computational costs.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 2, pp. 274–285, February, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i2.6880.
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Chapko, R., Johansson, B.T. Calculating Heat and Wave Propagation from the Lateral Cauchy Data. Ukr Math J 74, 314–326 (2022). https://doi.org/10.1007/s11253-022-02062-w
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DOI: https://doi.org/10.1007/s11253-022-02062-w