CLASSICAL SOLUTION of THE MIXED PROBLEM FOR THE KLEIN–GORDON–FOCK EQUATION WITH NONLOCAL CONDITIONS

The mixed problem for the one-dimensional Klein–Gordon–Fock equation with nonlocal conditions in a halfstrip is considered. Solving this problem reduces to solving the systems of the second-type Volterra equations. The theorems of existence and uniqueness of a solution in the class of twice continuously differentiable functions were proved for these equations, when initial functions are smooth enough. It is proved that fulfillment of the matching conditions for given functions is necessary and sufficient for the existence of a unique smooth solution when initial functions are smooth enough. The method of characteristics is used for the problem analysis. This method reduces to splitting the original area of the definition into subdomains. The solution of the subproblem can be constructed with in each subdomain, the help of the initial and nonlocal conditions. The obtained solutions are then glued at common points, and these gluing conditions are the matching conditions.

This approach can be used in constructing both an analytical solution, when the solution of the systems of integral equations can be found explicitly, and an approximate solution. Moreover, approximate solutions can be constructed numerically and analytically. When the numerical solution is constructed, matching conditions are essential and need to be considered while developing numerical methods. 

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