| Abstrakt: |
The problem of minimizing the functional(A) $${}_a\smallint ^b \varphi (x,y,y',y'')dx$$ under the conditions(B) $$y(a) = a_0 ,y'(a) = a_1 ,y(b) = b_0 ,y'(b) = b_1$$ is replaced by the problem of finding the vector (y1,y2,...,yn-1) on which the sum(C) $$\sum\limits_{\kappa = 0}^n {C_\kappa \varphi (x_\kappa ,y_\kappa ,\left. {\frac{{y_{\kappa + 1} - y_\kappa }}{h},\frac{{y_{\kappa + 1} - 2y_\kappa + y_{\kappa + 1} )}}{{h^2 }}} \right)}$$ takes a minimal value. Under certain conditions on ? andCk it is proved that a solution exists for the difference scheme constructed. The method of differentiation with respect to a parameter is used for the proof. |
Full text from SpringerLink