11,931 research outputs found
Fractional reaction-diffusion equations
In a series of papers, Saxena, Mathai, and Haubold (2002, 2004a, 2004b)
derived solutions of a number of fractional kinetic equations in terms of
generalized Mittag-Leffler functions which provide the extension of the work of
Haubold and Mathai (1995, 2000). The subject of the present paper is to
investigate the solution of a fractional reaction-diffusion equation. The
results derived are of general nature and include the results reported earlier
by many authors, notably by Jespersen, Metzler, and Fogedby (1999) for
anomalous diffusion and del-Castillo-Negrete, Carreras, and Lynch (2003) for
reaction-diffusion systems with L\'evy flights. The solution has been developed
in terms of the H-function in a compact form with the help of Laplace and
Fourier transforms. Most of the results obtained are in a form suitable for
numerical computation.Comment: LaTeX, 17 pages, corrected typo
Perturbative Linearization of Reaction-Diffusion Equations
We develop perturbative expansions to obtain solutions for the initial-value
problems of two important reaction-diffusion systems, viz., the Fisher equation
and the time-dependent Ginzburg-Landau (TDGL) equation. The starting point of
our expansion is the corresponding singular-perturbation solution. This
approach transforms the solution of nonlinear reaction-diffusion equations into
the solution of a hierarchy of linear equations. Our numerical results
demonstrate that this hierarchy rapidly converges to the exact solution.Comment: 13 pages, 4 figures, latex2
Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations
Computational costs of numerically solving multidimensional partial
differential equations (PDEs) increase significantly when the spatial
dimensions of the PDEs are high, due to large number of spatial grid points.
For multidimensional reaction-diffusion equations, stiffness of the system
provides additional challenges for achieving efficient numerical simulations.
In this paper, we propose a class of Krylov implicit integration factor (IIF)
discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion
equations on high spatial dimensions. The key ingredient of spatial DG
discretization is the multiwavelet bases on nested sparse grids, which can
significantly reduce the numbers of degrees of freedom. To deal with the
stiffness of the DG spatial operator in discretizing reaction-diffusion
equations, we apply the efficient IIF time discretization methods, which are a
class of exponential integrators. Krylov subspace approximations are used to
evaluate the large size matrix exponentials resulting from IIF schemes for
solving PDEs on high spatial dimensions. Stability and error analysis for the
semi-discrete scheme are performed. Numerical examples of both scalar equations
and systems in two and three spatial dimensions are provided to demonstrate the
accuracy and efficiency of the methods. The stiffness of the reaction-diffusion
equations is resolved well and large time step size computations are obtained
Solving reaction-diffusion equations 10 times faster
The most popular numerical method for solving systems of reaction-diffusion equations continues to be a low order finite-difference scheme coupled with low order Euler time stepping. This paper extends previous 1D work and reports experiments that show that with high--order methods one can speed up such simulations for 2D and 3D problems by factors of 10--100. A short MATLAB code (2/3D) that can serve as a template is included.\ud
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This work was supported by the Engineering and Physical Sciences Research Council (UK) and by the MathWorks, Inc
Solution of generalized fractional reaction-diffusion equations
This paper deals with the investigation of a closed form solution of a
generalized fractional reaction-diffusion equation. The solution of the
proposed problem is developed in a compact form in terms of the H-function by
the application of direct and inverse Laplace and Fourier transforms.
Fractional order moments and the asymptotic expansion of the solution are also
obtained.Comment: LaTeX, 18 pages, corrected typo
Voter Model Perturbations and Reaction Diffusion Equations
We consider particle systems that are perturbations of the voter model and
show that when space and time are rescaled the system converges to a solution
of a reaction diffusion equation in dimensions . Combining this result
with properties of the PDE, some methods arising from a low density
super-Brownian limit theorem, and a block construction, we give general, and
often asymptotically sharp, conditions for the existence of non-trivial
stationary distributions, and for extinction of one type. As applications, we
describe the phase diagrams of three systems when the parameters are close to
the voter model: (i) a stochastic spatial Lotka-Volterra model of Neuhauser and
Pacala, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert,
Lieberman, and Nowak, and (iii) a continuous time version of the non-linear
voter model of Molofsky, Durrett, Dushoff, Griffeath, and Levin. The first
application confirms a conjecture of Cox and Perkins and the second confirms a
conjecture of Ohtsuki et al in the context of certain infinite graphs. An
important feature of our general results is that they do not require the
process to be attractive.Comment: 106 pages, 7 figure
Depinning transitions in discrete reaction-diffusion equations
We consider spatially discrete bistable reaction-diffusion equations that
admit wave front solutions. Depending on the parameters involved, such wave
fronts appear to be pinned or to glide at a certain speed. We study the
transition of traveling waves to steady solutions near threshold and give
conditions for front pinning (propagation failure). The critical parameter
values are characterized at the depinning transition and an approximation for
the front speed just beyond threshold is given.Comment: 27 pages, 12 figures, to appear in SIAM J. Appl. Mat
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