INVERSE PROBLEM OF FINDINING THE RIGHT-HAND SIDE OF THE PARTIAL DIFFERENTIAL EQUATION IN FOURTH-ORDER

Abstract

The subject of the research is an inhomogeneous, linear fourth-order partial differential equation with two independent variables. The aim of the research is to find a solution that satisfies both the initial and homogeneous boundary conditions of the first kind. Integral, integro-differential equations can be found in all areas of science, for example, the transfer equation that arises in the processes of slowing down neutrons, which plays an important role in modern physics. We know that the oscillations of a thin wire can be expressed by separate second-order differential equations. If, instead of a wire, we consider a thin solid beam (thin hammer), then the process of its oscillations is expressed by fourth-order differential equations. Such problems arise in the design of heavy equipment. To construct a solution, the Dirichlet formula for the double integral was applied, as a result of which the Volterra integral equations with three unknowns are obtained. The Dirichlet formula was used to solve the Abel problem. In conclusion, the main theorem was proved on the existence of a solution to the inverse problem that satisfies the above conditions.