Classical solution for the mixed problem for Klein-Gordon-Fock equation with unlocal conditions

Mixed problem for one-dimensional Klein-Gordon-Fock equation is considered with nonlocal conditions in half-strip. Solution of this problem is reduced to the solution of the system of the second type Volterra's equations. Theorems of existence and uniqueness solution in class of the twice continuously differentiable functions were proven for these equations when initial functions are smooth enough. It is proven that fulfillment of the matching conditions on the given functions is necessary and sufficient for the existence of the unique smooth solution when initial functions are smooth enough. The method of characteristics is used for problem analysis. This method is reduced to the splitting original area of the definition to the subdomains. The solution of the subproblem in each subdomain can be constructed with the help of the initial and nonlocal conditions. Then, obtained solutions are glued in common points, and received gluing conditions are the matching conditions. This approach can be used in constructing as analytical solution, in the case when solution of the system of the integral equations can be found in explicit way, so for approximate solution. Moreover, approximate solutions can be constructed in numeric and analytical form. When the numeric solution is constructed then matching conditions are essential and they need to be considered while developing numerical methods.