(1) PAPER TITLE

Solution of differential equations with Genetic Programming
and the Stochastic Bernstein Interpolation

(2) AUTHORS

Daniel Howard
Biocomputing and Developmental Systems Group
University of Limerick, Ireland
DanielHoward@sunrisemalvern.freeserve.co.uk

Joseph Kolibal
Department of Mathematics
University of Southern Mississippi, USA
Joseph.Kolibal@usm.edu

(3) CORRESPONDING AUTHOR

Daniel Howard

(4) ABSTRACT

This report introduces a new method for the solution of the Convection-Diffusion
equations (CDE) using Genetic Programming and Stochastic Bernstein Interpolation.
Significantly, it is being used to solve a problem
that has resisted analysis for a long time using other methods.
Although the method in this report solves the one-dimensional CDE
which has also been solved analytically and optimally, our
strategy of combining a the Stochastic Bernstein Interpolation
method with GP allows for the method to extend to higher
dimensions, and thus it shows how to construct GP based methods
for solving a range of computational problems in multiple
dimensions which have hitherto resisted numerical solution.


(5) CRITERIA

(A) The result was patented as an invention in the past, is
an improvement over a patented invention, or would qualify today as a
patentable new invention.

(G) The result solves a problem of indisputable difficulty in its field.


(6) STATEMENT

The problem is how to approximate convection-diffusion equations numerically
without resorting to upwinding artificial viscosity and other ad-hoc measures
that can compromise the numerical solution.  The approach taken by GP
circumvents this problem.

The method used for developing the underlying numerical interpolation
of the unknown functions is based on a patent applied for by the University
of Southern Mississippi (Joseph Kolibal).

Attempts to solve the convection-diffusion equation have a long history
and while the equations are solvable, there are still difficulties
associated with constructing methods that iteratively achieve usable
converged solutions as the Peclet number increases. The approach provides
a machine generated solution which is applicable to a range of Peclet numbers
and which is extensible to multiple dimensional problems.

The convection-diffusion equations has generated tremendous activity
in its investigation and analysis. The equations contain significant
mathematical difficulties, and are of significance in several applications
areas in the applied science.  In the preface to Morton's book,

  Numerical Solution of Convection-Diffusion Problems by K.W. Morton,
  Chapman and Hall, 1st edition, 1996.

he begins: "Accurate modelling of the interaction between convective
and diffusive processes is the most ubiquitous and challenging task in the
numerical approximation of partial differential equations."

(7) CITATION
D Howard, K Kolibal "Solution of differential equations with Genetic Programming
and the Stochastic Bernstein Interpolation", Biocoumputing-Developmental
Systems Group, University of Limerick Technical Report No. BDS-TR-2005-001,
Ireland (June, 2005).