GOURSAT PROBLEM FOR THE ADJOINT MODEL TELEGRAPH EQUATION WITH THE RATE a(s, t) IN THE UPPER HALF-PLANE

Abstract

In the upper half-plane, the classical solution and correctness criterion of the Goursat problem for a linear inhomogeneous adjoint model telegraph equation with variable rate a(s,t) are found explicitly. An explicit formula is obtained for the classical solution of this Goursat problem, unique and stable with respect to the right hand side of the equation and Goursat data. This formula contains implicit characteristic functions of the equation. In the case of a homogeneous conjugate model telegraph equation, the classical solution of this Goursat problem is the Riemann function in all linear mixed (initial-boundary) problems for an inhomogeneous model telegraph equation with variable rate a(s,t). This Riemann function has been calculated by us. A correctness criterion according to Hadamard (necessary and sufficient conditions) of its unique and stable on the right-hand side of the equation and the Goursat data solvability is found. This criterion consists of smoothness requirements on the righthand side of the equation and two Goursat data. The smoothness requirements on the right side of the equation are the condition of continuity of the right-side and the corresponding integral smoothness conditions on the right side of the equation and on the Goursat data – their twice continuous differentiability in the upper half-plane.