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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 84
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 11

From a Fractional Mechanical Model to a Fractional Generalization of the Dirac Equation

D. Usero1, L. Vázquez24 and T. Pierantozzi3

1Departmental Section of Applied Mathematics, Faculty od Chemical Sciences,
2Department of Applied Mathematics, Faculty of Informatics
3Department of Applied Mathematics, Faculty of Mathematics
Universidad Complutense of Madrid, Spain
4Centre for Astrobiology (CSIC-INTA), Torrejon de Ardoz, Madrid, Spain

Full Bibliographic Reference for this paper
D. Usero, L. Vázquez, T. Pierantozzi, "From a Fractional Mechanical Model to a Fractional Generalization of the 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 11, 2006. doi:10.4203/ccp.84.11
Keywords: fractional differential equations, Riemann-Liouville fractional integrals and derivatives, Caputo fractional derivative, Mittag-Leffler and Wright functions, Dirac-type equations.

In this paper, we present a generalization of the linear one-dimensional diffusion and wave equations obtained by combining the fractional derivatives and the internal degrees of freedom associated with the system. Taking into account that the free Dirac equation is, in some sense, the square root of the Klein-Gordon equation, in a similar way we can operate a kind of square root of the time fractional diffusion equation in one space dimension through the system of fractional evolution-diffusion equations Dirac like. Solutions of the above system fulfils D'Alambert's formula as we demonstrate.

Following the well known Dirac's approach [2], it is possible to obtain his equation from the classical klein-Gordon equation. The free Dirac's equation is


where is a spinor, an structure that describes the state of the system and A and B are matrices satisfying Pauli's algebra. In fact this equation can be considered as a square root of the Klein-Gordon wave equation


where u is a scalar field.

As can be suspected, this relationship between both equations results in highly interesting to define roots of known operators. In the above context, Vázquez et al. recently considered in [1,3,4] the fractional diffusion equations with internal degrees of freedom. They can be obtained as the s-roots of the standard scalar linear diffusion equation. Thus, a possible definition of the square root of the standard diffusion equation (SDE) in one space dimension


is the following:


where A and B are matrices satisfying Pauli's algebra


being I the identity operator.

Here is a multicomponent function with at least two scalar space-time components. Also, each scalar component satisfies the SDE. Such solutions can be interpreted as probability distributions with internal structures associated with internal degrees of freedom of the system. They are named diffunors in analogy with the spinors in quantum mechanics.

We deal with a further generalization of Dirac's method. Concretely, we study the system of fractional evolution equations


with and where A and B are the same matrices as before satisfying Pauli's algebra.

We also make a mechanical formulation of the problem and we are able to obtain a conserved quantity, a fractional analogous to the classic hamiltonian.

T. Pierantozzi, 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
S. S. Schweber, "An Introduction to Relativistic Quantum Field Theory", Harper Int. Ed. New York, 1966.
L. Vázquez, "Fractional diffusion equation with internal degrees of freedom", J. Comp. Math., Vol. 21, n. 4, 491-494, 2003. doi:10.1016/S0096-3003(02)00326-0
L. Vázquez, R. Vilela Mendes, "Fractionally coupled solutions of the diffusion equation", Applied Mathematics and Computation, 141, 125-130, 2003. doi:10.1016/S0096-3003(02)00326-0

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