A hybrid analytical-numerical method for solving evolution partial differential equations. 1. The half-line

AMS Citation:
Flyer, N., and A. S. Fokas, 2008: A hybrid analytical-numerical method for solving evolution partial differential equations. 1. The half-line. Proceedings of the Royal Society. Series A, Mathematical Physical & Engineering Sciences, 464, 1823-1849, doi:10.1098/rspa.2008.0041.
Date:2008-04-01
Resource Type:article
Title:A hybrid analytical-numerical method for solving evolution partial differential equations. 1. The half-line
Abstract: A new method, combining complex analysis with numerics, is introduced for solving a large class of linear partial differential equations (PDEs). This includes any linear constant coefficient PDE, as well as a limited class of PDEs with variable coefficients (such as the Laplace and the Helmholtz equations in cylindrical coordinates). The method yields novel integral representations, even for the solution of classical problems that would normally be solved via the Fourier or Laplace transforms. Examples include the heat equation and the first and second versions of the Stokes equation for arbitrary initial and boundary data on the half-line. The new method has advantages in comparison with classical methods, such as avoiding the solution of ordinary differential equations that result from the classical transforms, as well as constructing integral solutions in the complex plane which converge exponentially fast and which are uniformly convergent at the boundaries. As a result, these solutions are well suited for numerics, allowing the solution to be computed at any point in space and time without the need to time step. Simple deformation of the contours of integration followed by mapping the contours from the complex plane to the real line allow for fast and efficient numerical evaluation of the integrals.
Subject(s):Numerical contour integration, Evolution partial differential equations, Integral transforms
Peer Review:Refereed
Copyright Information:An edited version of this paper was published by the Royal Society. Copyright 2008 the Royal Society.
OpenSky citable URL: ark:/85065/d7348mqb
Publisher's Version: 10.1098/rspa.2008.0041
Author(s):
  • Natasha Flyer - NCAR/UCAR
  • A. Fokas
  • Random Profile

    SCIENTIST III

    Recent & Upcoming Visitors