## List of issues > Series «Mathematics». 2017. Vol. 19

##
The Identification of External Force Dynamics in The Modeling of Vibration

The linear homogeneous wave equations with initial and boundary conditions are considered. It is assumed that the non-uniform terms describing an external force, are expanded into Fourier series, and the objective is to determine its N time depending coefficients. In order to determine these coefficients uniquely, N non-local boundary conditions are introduced in accordance with the required averaged dynamics of oscillations. The sufficient conditions are given when the formulated problem enjoy unique classical solution, which can be found by solving the system of Volterra integral equations, explicitly built in this work. Kernels of such integral equations enable resolvent construction using Laplace transform. This problem statement and method can be generalized and applied for system of inhomogeneous wave equations. These results can be useful in the formulation and solution of some problems arising in the optimization of boundary controls of string vibrations.

1. Verlan A.F., Sizikov V.S. Methods for the Solution of Integral Equations using Computer Programs. Nauk. Dumka, Kiev, 1978.

2. Doetsch G. Anleitung zum praktischen Gebrauch der Laplace-Transformation under der Z-Transformation. R. Oldenburg Verlag, Munchen, 1981.

3. Il’in V.A., Moiseev E.I. Optimization of boundary controls of string vibrations, Russian Mathematical Surveys, 2005, vol. 60, no 6, pp. 1093-1119.

4. Kamynin V.L. The inverse problem of determining the lower-order coefficient in parabolic equations with integral observation. Math. Notes, 2013, vol. 94, no 1, pp. 205-213.

5. Kamynin V.L., Kostin A.B. Two inverse problems of finding a coefficient in a parabolic equation. Differ. Eq., 2010, vol. 46, no 3, pp. 375-386.

6. Kozhanov A. I. A nonlinear loaded parabolic equation and a related inverse problem. Math. Notes, 2004, vol. 76, no 5-6, pp. 784-795.

7. Muftahov I.R., Sidorov D.N., Dreglea A.I. On the role of Volterra equations in the inverse heat conduction problem with a nonlocal boundary condition. In: “Mathematical and computer modeling of natural-scientific and social issues”, XIntern. scientific. conf. of young professionals and students (Russia, Penza, 23-27 May 2016), Editor I.V. Boikov. Penza, Publishing House of Penza State University, 2016, pp. 15-19.

8. Muftahov I.R., Sidorov D.N., Sidorov N.A. The Lavrentiev regularization of integral equations of the first kind in the space of continuous functions. Izv. Irkutsk. Gos. Univ. Ser. Mat., 2016, vol. 16, pp. 62–77.(in Russian)

9. Prilepko A.I., Kostin A.B. Inverse problems of the determination of the coefficient in parabolic equations. I, Siberian Math. J., 1992, vol. 33, no 3, pp. 489–496.

10. Prilepko A.I., Kostin A.B. On inverse problems of determining a coefficient in a parabolic equation. II, Siberian Math. J., 1993, vol. 34, no 5, pp. 923–937.

11. Tikhonov A.N., Samarsky A.A. The equations of mathematical physics. Moscow, Nauka Publ., 1977.