IJPAM: Volume 67, No. 1 (2011)

FINITE DIFFERENCE APPROXIMATIONS FOR
THE FIRST-ORDER HYPERBOLIC PARTIAL
DIFFERENTIAL EQUATION WITH POINT-WISE DELAY

Paramjeet Singh$^1$, Kapil K. Sharma$^2$
$^{1,2}$Department of Mathematics
Panjab University
Sector 14, Chandigarh, U.T. - 160014, INDIA
$^1$e-mail: [email protected]
$^2$e-mail: [email protected]


Abstract.Explicit numerical methods based on Lax-Friedrichs and Leap-Frog finite difference approximations are constructed to find the numerical solution of the first-order hyperbolic partial differential equation with point-wise delay or advance, i.e., shift in space. The differential equation involving point-wise delay and advance models the distribution of the time intervals between successive neuronal firings. In this paper, we continue the numerical study which was initiated in [Sharma and Singh, Appl. Math. Comput., 201 (2008), 229–238]. We construct higher order numerical approximations and discuss their consistency, stability and convergence. The numerical approximations constructed in this paper are consistent, stable under CFL condition, and convergent. We also extend our methods to the higher space dimensions. Some test examples are included to illustrate our approach. These examples verify the theoretical estimates and show the effect of point-wise delay on the solution.

Received: December 2, 2010

AMS Subject Classification: 35L04, 65N06, 39A14

Key Words and Phrases: hyperbolic partial differential equation, differential difference equation, point-wise delay, finite difference method

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2011
Volume: 67
Issue: 1