Surfaces with Prescribed Nodes and Minimum Energy Integral of Fractional Order (original) (raw)

This paper presents a method of constructing a continuous surface or a (real-valued) function of two variables z = u(x, y) defined on the square S := [0, 1] 2 , which minimizes an energy integral of fractional order, subject to the condition u(0, y) = u(1, y) = u(x, 0) = u(x, 1) = 0 and u(x i , y j ) = c ij , where 0 < x 1 < · · · < x M < 1, 0 < y 1 < · · · < y N < 1, and c ij ∈ R are given. The function is expressed as a double Fourier sine series, and an iterative procedure to obtain the function will be presented.