Name: Prof. Michael Mascagni Address: Department of Computer ScienceandSchool of Computational Science Florida State University Tallahassee, FL 32306-4530USAOffices: 498 Dirac Science Library/172 Love Building Phone: +1.850.644.3290 FAX: +1.850.644.0098 e-mail: mascagni@fsu.edu

**Title: A
Deterministic Particle Method for One-Dimensional Reaction-Diffusion Equations**

**Abstract:**

**We derive a
deterministic particle method for the solution of nonlinear reaction-diffusion
equations in one spatial dimension. This deterministic method is an analog
of a Monte Carlo method for these problems previously investigated by the
author. The deterministic method uses a uniform discretization of a
dependent variable (the solution) instead of an independent variable (such as
space). This discretization leads to the consideration of a system of
nonlinear ordinary differential equations for the positions of particles defined
by their solution value. These ODEs are nonlinear, with their nonlinearity
corresponding to the reaction-diffusion equation's linear diffusion term, and
are linear corresponding to the reaction-diffusion equation's nonlinear
term. We then study explicit and implicit methods to solve numerically
these equations. These methods are especially attractive when the solution
has a traveling wave solution, as the particles concentrate themselves where the
gradient of the solution is steepest. Thus these methods are naturally
adaptive. Nagumo's equation is known to have traveling wave solutions, and
so we compute the solution to this initial value problem using the new
method. We then compute the wave speed of the numerical solution and
compare it to the wave speed of exact solution. This is done analytically
and is confirmed numerically. We then discuss future research into
adapting these methods to solve non-monotonic problems and systems. The
eventual goal of this research is the rapid solution of nerve equations for
neural simulation.**

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