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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 20
A Numerical Study of Fractional EvolutionDiffusion Diraclike Equations T. Pierantozzi^{1} and L. Vázquez^{1}^{2}
^{1}Department of Applied Mathematics, Faculty of Informatics, Universidad Complutense of Madrid, Spain
, "A Numerical Study of Fractional EvolutionDiffusion Diraclike Equations", 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 20, 2006. doi:10.4203/ccp.84.20
Keywords: fractional differential equations, MittagLeffler and Wright functions, Diractype 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 wellknown equation from the KleinGordon equation.
Taking into account that the free Dirac equation is, in some sense, the square root of the KleinGordon 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]) through the system of fractional evolutiondiffusion Dirac like equations with and where A and B are matrices satisfying Pauli's algebra, this is: 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 evolutiondiffusion equations (54) together with certain initialboundary conditions, when the fractional derivative in time is of the Caputo type. Secondly, in order to numerically solve the same initialboundary values problem for each fractional evolutiondiffusion equation, we construct a finite difference scheme employing a convolution quadrature formula for approximating the RiemannLiouville 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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