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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
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]) through the system of fractional evolution-diffusion Dirac like equations with ![]() ![]() ![]()
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
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 References
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