13,673 research outputs found
An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations
In this paper, we consider the initial boundary value problem of the two
dimensional multi-term time fractional mixed diffusion and diffusion-wave
equations. An alternating direction implicit (ADI) spectral method is developed
based on Legendre spectral approximation in space and finite difference
discretization in time. Numerical stability and convergence of the schemes are
proved, the optimal error is , where are the
polynomial degree, time step size and the regularity of the exact solution,
respectively. We also consider the non-smooth solution case by adding some
correction terms. Numerical experiments are presented to confirm our
theoretical analysis. These techniques can be used to model diffusion and
transport of viscoelastic non-Newtonian fluids
The Galerkin Finite Element Method for A Multi-term Time-Fractional Diffusion equation
We consider the initial/boundary value problem for a diffusion equation
involving multiple time-fractional derivatives on a bounded convex polyhedral
domain. We analyze a space semidiscrete scheme based on the standard Galerkin
finite element method using continuous piecewise linear functions. Nearly
optimal error estimates for both cases of initial data and inhomogeneous term
are derived, which cover both smooth and nonsmooth data. Further we develop a
fully discrete scheme based on a finite difference discretization of the
time-fractional derivatives, and discuss its stability and error estimate.
Extensive numerical experiments for one and two-dimension problems confirm the
convergence rates of the theoretical results.Comment: 22 pages, 4 figure
Novel numerical analysis of multi-term time fractional viscoelastic non-Newtonian fluid models for simulating unsteady MHD Couette flow of a generalized Oldroyd-B fluid
In recent years, non-Newtonian fluids have received much attention due to
their numerous applications, such as plastic manufacture and extrusion of
polymer fluids. They are more complex than Newtonian fluids because the
relationship between shear stress and shear rate is nonlinear. One particular
subclass of non-Newtonian fluids is the generalized Oldroyd-B fluid, which is
modelled using terms involving multi-term time fractional diffusion and
reaction. In this paper, we consider the application of the finite difference
method for this class of novel multi-term time fractional viscoelastic
non-Newtonian fluid models. An important contribution of the work is that the
new model not only has a multi-term time derivative, of which the fractional
order indices range from 0 to 2, but also possesses a special time fractional
operator on the spatial derivative that is challenging to approximate. There
appears to be no literature reported on the numerical solution of this type of
equation. We derive two new different finite difference schemes to approximate
the model. Then we establish the stability and convergence analysis of these
schemes based on the discrete norm and prove that their accuracy is of
and ,
respectively. Finally, we verify our methods using two numerical examples and
apply the schemes to simulate an unsteady magnetohydrodynamic (MHD) Couette
flow of a generalized Oldroyd-B fluid model. Our methods are effective and can
be extended to solve other non-Newtonian fluid models such as the generalized
Maxwell fluid model, the generalized second grade fluid model and the
generalized Burgers fluid model.Comment: 19 pages, 8 figures, 3 table
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