1 A 2-D INTERPOLATION METHOD THAT MINIMIZES AN ENERGY INTEGRAL 1 E. Rusyaman, 2 H. Gunawan, 1 A.K. Supriatna, 3 R.E. Siregar Mathematics Dept 1 Padjadjaran University, 2 ITB, 3 Physics Dept Padjadjaran University e-mail: 1 e_rusyaman@unpad.ac.id, 2 hgunawan@math.itb.ac.id Abstract This paper will present a method of interpolation which is used to construct a surface -expressed as a continuous function of two variables- passing some arbitrary points on a square domain. The function must minimize an energy integral of fractional order. To construct such a function, the double Fourier sine series as well as the functional analysis arguments are used. An iterative procedure to obtain the solution is also presented. Keywords : interpolation, energy integral, double Fourier sine series. Presented at IICMA 2009 in Yogyakarta, October 12 th - 13 th , 2009 . 1. INTRODUCTION Newton (1675), Lagrange (1795) and Hermite (1870) pioneered and developed the theory and methods of interpolation. In this decade, interpolation methods continue to grow rapidly. Some of them including the problem of energy minimizing interpolation can be seen in [1, 5, 10] and other references. In [7], a method to construct a surface u(x , y) on a square domain D = { (x , y)⎮0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 }, passing the MN points (x i , y j , c ij ), where 0 < x 1 < . . . < x M < 1, 0 < y 1 < . . . < y N < 1 and u(0, y) = u(1 , y) = u(x , 0) = u(x , 1) = 0 has been proposed. The surface must minimize the energy integral with fractional order: ∫∫ ∆ − = 1 0 1 0 2 2 / ) ( : ) ( dy dx u u E β β . For β = 2, the integral is nothing but the total curvature of u on D. In reality, the available data of (x i , y j , c ij ) points in the domain, are not always homogeneous, but may scatter randomly. This paper will develop an interpolation method in which the number of known points is K and scatter randomly in region D.