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Abstract
RECENTLY the straight-line method has been attracting more and more attention both among theoreticians and among those using it for practical purposes (see [1]). For one particular class of boundary value problems it possesses obvious advantages over the method of nets with respect to the accuracy of the results and the number of arithmetical operations. Retaining the derivative with respect to one of the variables not only permits us to solve the problem, but also to investigate it. Moreover this method may also be used for the effective solution of non-linear boundary-value problems. To solve equations of the elliptic type in a rectangular region of Integration, the method of straight lines was first used In [2]. Later, many authors tried to use this method of solving problems in trapezoidal regions too. However, as was shown in [3], the method of straight lines was not completely correct for these regions. Very simple regions exist in which the boundary value problem for the set of differential equations of the method of straight lines is insoluble for certain n. In the present paper a solution is obtained, for regions with curvilinear boundaries, of linear and non-linear boundary-value problems for the two-dimensional Laplace equation for a one-dimensional family of curves.
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