13,310 research outputs found
Second-order Stable Finite Difference Schemes for the Time-fractional Diffusion-wave Equation
We propose two stable and one conditionally stable finite difference schemes
of second-order in both time and space for the time-fractional diffusion-wave
equation. In the first scheme, we apply the fractional trapezoidal rule in time
and the central difference in space. We use the generalized Newton-Gregory
formula in time for the second scheme and its modification for the third
scheme. While the second scheme is conditionally stable, the first and the
third schemes are stable. We apply the methodology to the considered equation
with also linear advection-reaction terms and also obtain second-order schemes
both in time and space. Numerical examples with comparisons among the proposed
schemes and the existing ones verify the theoretical analysis and show that the
present schemes exhibit better performances than the known ones
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
Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview
Over the past few decades, there has been substantial interest in evolution
equations that involving a fractional-order derivative of order
in time, due to their many successful applications in
engineering, physics, biology and finance. Thus, it is of paramount importance
to develop and to analyze efficient and accurate numerical methods for reliably
simulating such models, and the literature on the topic is vast and fast
growing. The present paper gives a concise overview on numerical schemes for
the subdiffusion model with nonsmooth problem data, which are important for the
numerical analysis of many problems arising in optimal control, inverse
problems and stochastic analysis. We focus on the following aspects of the
subdiffusion model: regularity theory, Galerkin finite element discretization
in space, time-stepping schemes (including convolution quadrature and L1 type
schemes), and space-time variational formulations, and compare the results with
that for standard parabolic problems. Further, these aspects are showcased with
illustrative numerical experiments and complemented with perspectives and
pointers to relevant literature.Comment: 24 pages, 3 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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