The smoothness criterion for the classical solution to inhomogeneous model telegraph equation with the rate a(x,t) on the half-line

Abstract

The smoothness criterion is derived on the right-hand side f for an explicit solution F to u tt (x,t) − a 2 (x,t)u xx (x,t) − a −1 (x,t)a t (x,t)u t (x,t) − a(x,t)a x (x,t)u x (x,t) = f(x,t) (1) with the variable rate a(x,t) in the first quarter of the plane G ∞ =]0,+∞[×]0,+∞[. The smoothness criterion consists of the necessary and sufficient smoothness requirements for the right-hand side f to this model telegraph equation. The necessary smoothness requirements on f are found as derivatives of F along two families of implicit characteristics of the given equation. Hence, by differentiation, we derive their sufficiency for twice continuous differentiability of F. The function F satisfies equation (1) pointwise, since it satisfies its canonical form pointwise. When f depends only on x or on t, then this smoothness criterion is equivalent to continuity f respectively, with respect to x or to t. For the equation, a general integral is constructed under a smoothness criterion of its right-hand side f