Abstract:
The paper considers the transverse vibrations of two strings connected to each other at a certain point. A mathematical model of this process, based on the law of conservation of momentum, expressed in the form of an integro-differential equation is constructed. This equation connects the deviations of the strings during vibrations with their characteristics such as density, tension, and sources of external forces. This approach can be considered as a development of the method proposed by A.N. Tikhonov for the equation of string vibrations with nonsmooth data. For the case where the densities of the strings are constants, we set a nonclassical problem for the integro-differential equation. In addition to the Cauchy data, the problem includes necessary matching conditions. A theorem on the existence and uniqueness of a solution to the problem is proved, and explicit formulas are obtained for its solution.
Keywords:wave equation, integral conservation law, Cauchy problem, lattice, discontinuous function.
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