PROMETHEE technique to select the best radial basis functions for solving the 2-dimensional heat equations based on Hermite interpolation Saeed Kazem a,n , Farhad Hadinejad b a Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave.,15914 Tehran, Iran b Department of Industrial Management, Faculty of Management and Accounting, Allameh Tabataba'i University, Tehran, Iran article info Article history: Received 11 September 2012 Received in revised form 21 December 2013 Accepted 19 June 2014 Keywords: Multiple criteria decision making Radial basis functions PROMETHEE Hermite interpolation Optimal selecting abstract In this work, we have decided to select the best radial basis functions for solving the 2-dimensional heat equations by applying the multiple criteria decision making (MCDM) techniques. Radial basis functions (RBFs) based on the Hermite interpolation have been utilized to approximate the solution of heat equation by using the collocation method. Seven RBFs, Gaussian (GA), Multiquadrics (MQ), Inverse multiquadrics (IMQ), Inverse quadrics (IQ), third power of Multiquadrics (MQ 3 ), Conical splines (CS) and Thin plate Splines (TPS), have been applied as basis functions as well. In addition, by choosing these functions as alternatives and calculating the error, condition number of interpolation matrix, RAM memory and CPU time, obtained by Maple software, as criteria, rating of cases with the help of PROMETHEE technique has been investigated. In the end, the best function has been selected according to the rankings. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction Radial basis functions (RBFs) interpolation is a technique for representing a function starting with data on scattered points. This technique first appears in the literature as a method for scattered data interpolation, and interest in this method exploded after the review of Franke [1], who found it to be the most impressive of the many methods he tested. Later, Kansa [2,3] proposed a scheme for the estimation of partial derivatives using RBFs. The main advan- tage of radial basis function methods is the meshless characteristic of them. The use of radial basis functions as a meshless method for the numerical solution of partial differential equations (PDEs) is based on the collocation method. These methods have recently received a great deal of attention from researchers [4–10]. Recently, RBFs methods were extended to solve various ordin- ary and partial differential equations including the high-order ordinary differential equations [11], the second-order parabolic equation with nonlocal boundary conditions [12,13], the nonlinear Fokker–Planck equation [14], regularized long wave (RLW) equa- tion [15], Hirota–Satsuma coupled KdV equations [16], nonlinear integral equations [17,18], second-order hyperbolic telegraph equation [19], the solution of 2D biharmonic equations [20], the case of heat transfer equations [21] and so on [22–24]. A RBF Ψ ð J x  x i J Þ : R þ ⟶R depends on the separation between a field point x A R d and the data centers x i , for i ¼ 1; 2; …; N, and N data points. The interpolants are classed as radial due to their spherical symmetry around centers x i , where J : J is the Euclidean norm. Some of the infinitely smooth RBFs choices are listed in Table 1. The RBFs can be of various types, for example, Multi- quadrics (MQ), Inverse multiquadrics (IMQ), Gaussian forms (GA) form, etc. In the cases of inverse quadratic, inverse multiquadric (IMQ) and Gaussian (GA), the coefficient matrix of RBFs interpolating is positive definite and, for multiquadric (MQ), it has one positive eigenvalue and the remaining ones are all negative [25]. One of the most powerful interpolation methods with analytic 2-dimensional test function is the RBFs method based on the Multiquadric (MQ) basis function ψ ðrÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 þ c 2 p ; ð1Þ suggested by Hardy [26], where r ¼ J x  x i J and c is a free positive parameter, often referred to as the shape parameter, to be specified by the user. Madych and Nelson [27] showed that interpolation with MQ is exponentially convergent based on the reproducing kernel Hilbert space. Convergence property of the MQ Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/enganabound Engineering Analysis with Boundary Elements http://dx.doi.org/10.1016/j.enganabound.2014.06.009 0955-7997/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author. Tel.: þ98 935 4592405. E-mail addresses: saeedkazem@aut.ac.ir, saeedkazem@gmail.com (S. Kazem), farhad_hdng@yahoo.com (F. Hadinejad). Engineering Analysis with Boundary Elements 50 (2015) 29–38