Abstract
The mixed problem for the one-dimensional Klein – Gordon – Fock type equation with oblique derivatives at boundary conditions in the half-strip is considered. 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 twice continuously differentiable func tions were proven for these equations when initial functions are smooth enough. It is proven that fulfilling the matching conditions on the given functions is necessary and sufficient for 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 splitting the ori ginal definition area into subdomains. The solution of the subproblem can be constructed in each subdomain with the help of the initial and boundary conditions. The obtained solutions are then glued in common points, and the obtained glued сonditions are the matching conditions. Intensification of smoothness requirements for source functions is proven when the di rections of the oblique derivatives at boundary conditions are matched with the directions of the characteristics. This approach can be used in constructing both the analytical solution, when the solution of the integral equation can be found explicitly, and the approximate solution. Moreover, approximate solutions can be constructed in numerical and analytical form. When a numerical solution is constructed, the matching conditions are significant and need to be considered while developing numerical methods.
Highlights
The mixed problem for the one-dimensional Klein – Gordon – Fock type equation with oblique derivatives at boundary conditions in the half-strip is considered. 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 twice continuously differentiable functions were proven for these equations when initial functions are smooth enough
It is proven that fulfilling the matching conditions on the given functions is necessary and sufficient for existence of the unique smooth solution, when initial func tions are smooth enough
Summary
Пусть l(t, x) ∈C m+1 Q (k) , тогда решение u(k)(t,x) уравнения (14) существует, ( ) единственно, принадлежит классу C m+2 Q (k) и непрерывно зависит от правой части тогда и только тогда, когда функции p (k) ∈C m+2 ([-(k + 1)l;-(k -1)l]), g (k) ∈C m+2 ([kl;(k + 2)l]), m ∈. Из формулы (16) и того, что функции j(k) ( x), Ψ (k) ( x) ∈ C m+1([0;l]), следует, что p(k) принадлежит классу C m+1 на множестве своего задания. С помощью первого из условий (9) получаем следующее урав нение для нахождения неизвестной функции p(k) в области Q(k,2):. Функция p(k), определенная по формуле (25), принадлежит классу C 2 ([−(k + 1)l;−kl]), если на функции φ(k) (x), ψ(k) (x) будут налагаться более сильные требования на гладкость.
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