ON THE SOLUTION OF AN INITIAL–BOUNDARY VALUE PROBLEM FOR A LOADED FOURTH–ORDER PARTIAL DIFFERENTIAL EQUATION

An initial–boundary value problem is considered for a differential equation with n loads, depending on two variables and involving fourth-order partial derivatives. By introducing a new unknown function, the original problem is reduced to a family of Cauchy problems for a loaded differential equation with first-order partial derivatives. For the unknown function with loads z(t_i,x), a system of functional equations with respect to the variable x is constructed. An algorithm for finding a solution to this system of functional equations is proposed. A theorem on the existence and uniqueness of a solution to the family of Cauchy problems for the loaded differential equation with first-order partial derivatives is proved. Conditions for the existence and uniqueness of a solution to the initial–boundary value problem for the loaded differential equation involving fourth-order partial derivatives and depending on two variables are established. The result is illustrated by an example.