IJPAM: Volume 69, No. 3 (2011)

NUMERICAL APPROXIMATIONS TO THE TRANSPORT
EQUATION ARISING IN NEURONAL VARIABILITY

Paramjeet Singh$^1$, Kapil K. Sharma$^2$
$^{1,2}$Department of Mathematics
Panjab University
Chandigarh, 160 014, INDIA
$^1$e-mails: paramjeet.singh@pu.ac.in
$^2$e-mail: kapilks@pu.ac.in
$^1$Laboratoire Jacques-Louis Lions
Université Pierre et Marie Curie
Paris, FRANCE
e-mail: paramjeet@ann.jussieu.fr


Abstract. This paper studies some finite difference approximations to find the numerical solution of first-order hyperbolic partial differential equation of mixed type, $i.e.$, transport equation with point-wise delay and advance. We are interested in the challenging issues in neuronal sciences stemming from the modeling of neuronal variability. The resulting mathematical model is a first-order hyperbolic partial differential equation involving point-wise delay and advance which models the distribution of time intervals between successive neuronal firings. We construct, analyze, and implement explicit numerical schemes for solving such type of initial and boundary-interval problems. Analysis shows that numerical approximations are conditionally stable, consistent and convergent in discrete $L{^\infty}$ norm. Numerical approximations works irrespective the size of point-wise delay and advance. Some numerical tests are reported to validate the computational efficiency of the numerical approximations.

Received: March 9, 2011

AMS Subject Classification: 35L04, 65M06, 92B20

Key Words and Phrases: hyperbolic partial differential difference equation, transport equation, neuronal firing, point-wise delay and advance, finite difference method, upwind scheme, Lax-Friedrichs scheme

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 69
Issue: 3