Application of Legendre Polynomials in Single-Mode Inverse Problem
In the existing literature on single-mode inverse problems of rods and beams, the correlation functions are set as power series of the independent variables. The excellent properties of Legendre polynomials are noted, and the application of Legendre polynomials to a kind of single mode inverse problem of rod was discussed. The linear density function, displace-ment mode and axial stiffness function of the rod with elastic supports at both ends are firstly set as a linear combination of Legendre polynomials of different orders, and the analytical expression of the second-order vibration equation of the rod was discussed. Specifically, it addresses the determination of the rod's axial stiffness function given its displacement mode and lin-ear density function. The well-posedness of the solution to this inverse problem, including the existence, uniqueness, and sta-bility of the solution was elucidated. At the same time, it is effective to use Legendre polynomials as the basis function is proved.
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