The inverse problem for determining the right part of the pseudo-parabolic equation

Abstract

In this pap er the inverse problem of determining a solution and an unknown right-hand side that
dep ends only on spatial variable for the linear pseudo-parab olic equation of the third order is
investigated. In inverse problems, together with the initial and b oundary conditions also consider
an additional information, the need for which is due to the presence of unknown co efficients or the
right side of the equation. In this pap er, as additional information the integral overdetermination
condition is considered. Inverse problems of determining the right-hand side of a differential equation arise in the mathematical mo deling of some physical pro cesses in the case when, in addition
to solving the equation, it is necessary to restore the action of external sources. To day, studies
of direct and inverse problems for pseudo-parab olic equations are rapidly developing due to the
needs of mo deling and pro cess control in thermophysics, hydro dynamics and continuum mechanics. Similar pseudo-parab olic equations to considered in this pap er arise in the description of heat
and mass transfer pro cesses, pro cesses of motion of non-Newtonian fluids, wave pro cesses, and
in many other areas. Using series expansion, the existence and uniqueness theorems of classical
solutions to this problem are proved. The result of this work is a solution presented in the series
form, which allows the necessary numerical calculations to b e p erformed with a given accuracy.