BOUNDARY VALUE PROBLEMS FOR A FOURTH-ORDER EQUATION CONTAINING A PARABOLIC-HYPERBOLIC OPERATOR

Abstract

The existence and uniqueness of a solution to a boundary value problem on the plane for a fourth-order equation containing the product of a mixed second-order parabolic-hyperbolic operator and a first-order ordinary differential operator in x with a variation line like y=0 are proved. Boundary data is set on the x=0, x=l and x=-y lines. Using the method of lowering the order of the equation, the problem under consideration for y>0 is reduced to solving the first boundary value problem in a rectangle for the heat equation, and for y<0 in the characteristic triangle, to the problem for the string vibration equation. In a rectangle, the Green's function method is used to obtain a representation of the solution of the problem in an explicit form. Applying the method of general solutions to the equation of string vibrations, an explicit form of the solution of the problem for y<0 is found.