IJPAM: Volume 69, No. 3 (2011)
EQUATION ARISING IN NEURONAL VARIABILITY
Department of Mathematics
Chandigarh, 160 014, INDIA
Laboratoire Jacques-Louis Lions
Université Pierre et Marie Curie
Abstract. This paper studies some finite difference approximations to find the numerical solution of first-order hyperbolic partial differential equation of mixed type, , 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 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