Classical solution for the first mixed problem for the Klein-Gordon-Fock type equation with inhomogeneous matching conditions

Abstract

The first mixed problem for the Klein-Gordon-Fock type equation in the half strip is considered in the case when inhomogeneous matching conditions are fulfilled. The method of characteristics is used to prove that the fulfillment of the homogeneous matching conditions is not only sufficient but also a necessity for the existence of a unique smooth enough classical solution defined in the whole half strip. The equivalent conjugation problem is formulated when inhomogeneous conditions are fulfilled where conjugation conditions are set on the characteristics. Constructed inhomogeneous conditions uniquely define gaps of the solution or its derivatives on characteristics and given gaps are remained while the time-argument increases.

The solution of the problem is reduced to solving the second-type Volterra-integral equations. Theorems of existence and uniqueness of the solution in the class of the twice continuously differentiable functions were proven for these equations when the initial functions are smooth enough. This approach can be used in constructing as analytical solution, when the solution of the integral equation can be found explicitly, so for the approximate solution. Moreover, approximate solutions can be constructed in numerical and analytical form. When the numerical solution is constructed, the matching conditions are essential and they need to be considered while developing numerical methods.