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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 20

A Numerical Study of Fractional Evolution-Diffusion Dirac-like Equations

T. Pierantozzi1 and L. Vázquez12

1Department of Applied Mathematics, Faculty of Informatics, Universidad Complutense of Madrid, Spain
2Centre for Astrobiology (CSIC-INTA), Torrejón de Ardoz, Madrid, Spain

Full Bibliographic Reference for this paper
, "A Numerical Study of Fractional Evolution-Diffusion Dirac-like Equations", 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 20, 2006. doi:10.4203/ccp.84.20
Keywords: fractional differential equations, Mittag-Leffler and Wright functions, Dirac-type equations, finite difference methods, stability analysis.

Summary
A possible interpolation between the Dirac and the diffusion equations in one space dimension can be derived through the fractional calculus and following the method used by Dirac to obtain his well-known equation from the Klein-Gordon equation.

Taking into account that the free Dirac equation is, in some sense, the square root of the Klein-Gordon equation [1], in a similar way we can operate a kind of square root of the time fractional diffusion equation in one space dimension (see for example [2,3,4,5,6])

 (53)

through the system of fractional evolution-diffusion Dirac like equations

 (54)

with and where A and B are matrices satisfying Pauli's algebra, this is:

being I the identity operator.

System (54) was introduced in the previous works [7,8,9,10] and it represents a fractional generalization of the diffusion and wave equations for , as, if we calculate its square, they are returned for and , respectively. Solutions of the above system could model the diffusion of particles whose behavior depends not only on the space and time coordinates, but also on their internal structures.

Firstly we find the analytical solution of each component of the system of fractional evolution-diffusion equations (54) together with certain initial-boundary conditions, when the fractional derivative in time is of the Caputo type.

Secondly, in order to numerically solve the same initial-boundary values problem for each fractional evolution-diffusion equation, we construct a finite difference scheme employing a convolution quadrature formula for approximating the Riemann-Liouville fractional derivative and the classical forward Euler formula for the first order derivative. The stability bounds of this scheme, resulting from a previous discrete von Neumann type analysis, are checked in some representative examples when we know the underlying exact analytical results. The wide number of simulations we performed for different values of and different space and time steps, indicates that a necessary and sufficient condition for the stability of the difference scheme to be ensured should be almost stronger than the pure necessary condition we obtained with the von Neumann type analysis.

References
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A.A. Kilbas, T. Pierantozzi, J.J. Trujillo and L. Vázquez, "On the solution of fractional evolution equation", J. Phys. A: Math. Gen., 37, 9, 3271-3283, 2004. doi:10.1088/0305-4470/37/9/015
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T. Pierantozzi and L. Vázquez, "An interpolation between the wave and diffusion equations through the fractional evolution equations Dirac like", Journal of Mathematical Physics, 46, 113512, 2005. doi:10.1063/1.2121167

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