ABSTRACT
Here we mainly consider the problem of finding the function of two vari ables u(x, y) which gives the minimal value to the double integral
and which assumes some given values on the contour C bounding the domain D. We assume that in the domain D the functions a(x, y), 6(x, y) and c(x, y) are positive. This problem is equivalent, in general, to the Dirichlet problem for the equation of elliptic type:
Among the methods of approximate solution of this problem, two are most often applied: the Ritz method [1] and the Runge method [2] (see also [3]) also known as the finite difference method, consisting in replacing equation (2) by a certain finite difference equation. Here we give a method for solving this problem which differs from the two named above. This method essentially reduces to the following: suppose we must find the func tion u(x, y) which gives the minimal value to the functional J[u(x, y)]. Then choose, in the interval where one of the variables, say y, changes, a certain number of values
and by interpolation construct a function of two variables
which for y — yi , . . . , y = yp would become respectively equal to
f l (x), •••, fp(x)- Substituting the function u constructed into the functional / , we then
obtain the functional I[u(x, y; / 1? / 2, . . . , f p)\ depending on p functions of one variable f i ( x ) , . . . , f p(x). Choose the last variables so that the given functional assumes its minimal value. In this case the function u(x , y) will differ very little from the unknown function u(x,y). Thus the main idea of the method consists in the fact that finding the minimum of a functional dependent on the function of two variables is reduced to a simpler problem on the minimum of the functional depending on functions of one variable. Let us carry out this consideration more precisely.
