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CivilComp Proceedings
ISSN 17593433 CCP: 84
PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY 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. Usero^{1}, L. Vázquez^{2}^{4} and T. Pierantozzi^{3}
^{1}Departmental Section of Applied Mathematics, Faculty od Chemical Sciences,
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", CivilComp Press, Stirlingshire, UK, Paper 11, 2006. doi:10.4203/ccp.84.11
Keywords: fractional differential equations, RiemannLiouville fractional integrals and derivatives, Caputo fractional derivative, MittagLeffler and Wright functions, Diractype equations.
Summary
In this paper, we present a generalization of the linear
onedimensional 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
KleinGordon 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 evolutiondiffusion
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 kleinGordon 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 KleinGordon 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 sroots 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 spacetime 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. References
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