Classical solution to the mixed problems for the Klein–Gordon–Fock-type equation with curve derivatives in boundary conditions

The mixed problem for one-dimensional Klein–Gordon–Fock-type equation with curve derivatives in boundary conditions is considered in half-strip. The solution of this 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 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 the problem analysis.

This method is reduced to the splitting the original area of the definition to the subdomains. The solution of the subproblem can be constructed in each subdomain with the help of the initial and boundary conditions. Then obtained solutions are glued in common points, and received glued conditions are the matching conditions. This approach can be used in constructing as analytical solution, in case when solution of the integral equation can be found in explicit way, so for approximate solution. Moreover, approximate solutions can be constructed in numerical and analytical form. When numeric solution is constructed, then matching conditions are essential and they need to be considered while developing numerical methods.

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