Classical Solutions of Hyperbolic Equation with Translation Operators in Free Terms

Abstract

In this paper, we study the question of constructing explicit solutions in a half-space of a hyperbolic equation containing translation operators in space variables in all coordinate directions. Such equations are a natural generalization of classical equations of hyperbolic type, and the resulting solution relates the value of the desired function at different points of the half-space where the process takes place. To construct solutions, a classical operating scheme is used, namely, the formal application of an integral transformation. A theorem is proved that the constructed solutions are classical if the real part of the symbol of the differential-difference operator in the equation is positive. Classes of equations for which this condition is satisfied are given.