INVERSE PROBLEM OF DETERMINING THE KERNEL IN THE INTEGRO-DIFFERENTIAL BEAM VIBRATION EQUATION

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This paper addresses the problem of identifying the kernel that represents the memory of the medium in the integro-differential equation governing the forced vibrations of a beam. The problem is reduced to integral equations of the second kind of Volterra type with respect to the solution of the direct problem and the unknown kernel of the inverse problem. To solve these equations, we apply the method of compressive mappings in the space of continuous functions with exponential weight norm. The global solvability of the inverse problem and the conditional stability of the solution are established. In addition, the paper introduces an efficient numerical approach for solving the inverse problem associated with the beam equation. The method leverages the finite difference method to obtain solutions for both the displacement field u(x, t) and the media viscosity coefficient k(t), with the latter expressed in terms of an integral. Rigorous algorithmic development ensures accurate numerical solutions, which are validated through comparison with analytical solutions, demonstrating a high level of agreement. Furthermore, the proposed scheme is evaluated under noisy conditions, where it exhibits robustness in reducing numerical fluctuations over time. This comprehensive study highlights the reliability and effectiveness of the developed numerical approach, positioning it as a valuable tool for tackling inverse problems in structural mechanics and related fields.