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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 84
PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 19
Analysis of the Numerical Methods for a Fractional Dirac Equation S. Jiménez1 and Y.F. Tang2
1Department of Applied Mathematics for Information Technology, E.T.S.I. Telecommunication, Polytechnic University of Madrid, Spain
Full Bibliographic Reference for this paper
, "Analysis of the Numerical Methods for a Fractional Dirac Equation", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 19, 2006. doi:10.4203/ccp.84.19
Keywords: fractional equation, Dirac equation, numerical analysis.
Summary
Lately, the fractional Dirac equation has been considered as an interpolation
from the inside between the classical Heat Equation and the Wave
Equation [1,2]. In this work we build a suitable numerical
scheme to
simulate a one-dimensional fractional Dirac equation of the form:
In order to build our numerical scheme, we start using the ideas presented in [3] for a system that depends on a single variable, we will then extend the ideas to the fractional equation. We start considering a fractional equation of the form: where ![]() ![]()
the equation is equivalent to For this equation we have build two first order schemes. Combining both, we can build a second order scheme:
with local truncation error:
and ![]()
and from here
Denoting the spatial index by subscripts and the temporal one by superscripts, we have the numerical scheme The local truncation error can be written as
for some ![]() ![]() ![]() ![]() ![]() ![]() ![]() We show that, under reasonable boundary and initial conditions, the errors at each time-step are bounded by an expression that goes to zero as the mesh sizes go to zero, keeping both a finite fixed time and a spatial length. The scheme is thus convergent. References
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