Surfaces with Prescribed Nodes and Minimum Energy Integral of Fractional Order

Abstract

This paper presents a method of finding a continuous, 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(xi ,yj)=c𝑖𝑗 , where 0<x1<...<xM,<1, 0<y1<...<yN<1, and c𝑖𝑗 ∈ ℝ are given. The function is expressed as a double Fourier sine series, and an iterative procedure to obtain the function will be presented.