DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

A DISCRETE EIGENFUNCTIONS METHOD FOR NUMERICAL
SOLUTION OF RANDOM DIFFUSION MODELS
L. JÓDAR, J. C. CORTÉS, AND L. VILLAFUERTE
This paper deals with the construction of numerical solution of random diffusion models
whose coefficients functions and the source term are stochastic processes depending
on a common random variable and an initial condition which depends on a different
one. After discretization, the random difference scheme is solved using a random discrete
eigenfunctions method. Mean-square stability of the numerical solution is studied, and
a procedure for computing the expectation and the variance of the discrete approximate
stochastic process is given.
Copyright © 2006 L. Jódar et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Mathematical models regarding spatial uncertainty are frequent in geostatistic description
of natural variables [4] and modeling hydrology problems [5, 6]. Wave propagation
in random media has been treated in [2] and fishering problems are modeled in [3] using
stochastic processes. In this paper, we consider random diffusion models of the form
u t = [ p(x,β)u x
]x − q(x,β)u + F(x,t,β), 0 0, ∣∣ a 1
∣ ∣ + ∣∣ a 2
∣ ∣ > 0,
b 1 u(1,t)+b 2 u x (1,t) = 0, t>0, ∣∣ a 1
∣ ∣ + ∣∣ a 2
∣ ∣ > 0,
u(x,0)= f (x,γ), 0 ≤ x ≤ 1,
(1.1)
where the unknown u(x,t)aswellascoefficient p(x,β), the initial condition f (x,γ), and
the source term F(x,t,β) are second-order stochastic processes depending on mutually
independent second-order random variables β, andγ defined on a common probability
space (Ω,,P).
Model (1.1) assumes that random variations of the internal influences of the system
undergoing diffusion are stochastic processes depending on the random variable β and
that random external sources to the medium in which the diffusion takes places is also a
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 457–466