Solving high order ordinary differential equations with radial basis function networks

Abstract

SUMMARYThis paper is concerned with the application of radial basis function networks (RBFNs) for numerical solution of high order ordinary differential equations (ODEs).Two unsymmetric RBF collocation schemes, namely the usual direct approach based on a differentiation process and the proposed indirect approach based on an integration process, are developed to solve high order ODEs directly and the latter is found to be considerably superior to the former. Good accuracy and high rate of convergence are obtained with the proposed indirect method.

An efficient domain‐decomposition pseudo‐spectral method for solving elliptic differential equations

“…A comprehensive review on spectral methods can be found in, for example, [3][4][5]. Previous findings showed that the use of integration to construct the Chebyshev and radialbasis-function expressions provides an effective way to impose the multiple boundary conditions [6][7][8][9][10]. The present study, which is concerned with the case of domain decomposition, will show that the integral collocation formulation allows a C p continuous solution, instead of the usual C p−1 continuity, across the subdomain interfaces.…”

An effective spectral collocation method for the direct solution of high‐order ODEs

“…Recently, in the context of radial-basis-function networks, a new collocation formulation based on integration for solving high-order ODEs was proposed [12]. In this study, this formulation is extended to the case of Chebyshev polynomials.…”

Radial basis function networks for delay differential equation

“…The radial basis function is U. Saeed (B) NUST Institute of Civil Engineering, National University of Sciences and Technology, Sector H-12, Islamabad, Pakistan E-mail: umer.math@gmail.com a mesh-free scheme, which avoids grid generation and the domain of interest can be considered by a set of scattered data points [7]. The multiquadric radial basis functions [8,9] is very important and useful method for the numerical solution of ordinary and partial differential equations. On the basis of numerical and theoretical evidences, it has been shown in [10,11] that the radial basis function collocation method is very accurate even for a small number of collocation points.…”