39 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
A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations
In this paper, we develop regularized discrete least squares collocation and
finite volume methods for solving two-dimensional nonlinear time-dependent
partial differential equations on irregular domains. The solution is
approximated using tensor product cubic spline basis functions defined on a
background rectangular (interpolation) mesh, which leads to high spatial
accuracy and straightforward implementation, and establishes a solid base for
extending the computational framework to three-dimensional problems. A
semi-implicit time-stepping method is employed to transform the nonlinear
partial differential equation into a linear boundary value problem. A key
finding of our study is that the newly proposed mesh-free finite volume method
based on circular control volumes reduces to the collocation method as the
radius limits to zero. Both methods produce a large constrained least-squares
problem that must be solved at each time step in the advancement of the
solution. We have found that regularization yields a relatively
well-conditioned system that can be solved accurately using QR factorization.
An extensive numerical investigation is performed to illustrate the
effectiveness of the present methods, including the application of the new
method to a coupled system of time-fractional partial differential equations
having different fractional indices in different (irregularly shaped) regions
of the solution domain
On the analysis of mixed-index time fractional differential equation systems
In this paper we study the class of mixed-index time fractional differential
equations in which different components of the problem have different time
fractional derivatives on the left hand side. We prove a theorem on the
solution of the linear system of equations, which collapses to the well-known
Mittag-Leffler solution in the case the indices are the same, and also
generalises the solution of the so-called linear sequential class of time
fractional problems. We also investigate the asymptotic stability properties of
this class of problems using Laplace transforms and show how Laplace transforms
can be used to write solutions as linear combinations of generalised
Mittag-Leffler functions in some cases. Finally we illustrate our results with
some numerical simulations.Comment: 21 pages, 6 figures (some are made up of sub-figures - there are 15
figures or sub-figures
Splitting physics-informed neural networks for inferring the dynamics of integer- and fractional-order neuron models
We introduce a new approach for solving forward systems of differential
equations using a combination of splitting methods and physics-informed neural
networks (PINNs). The proposed method, splitting PINN, effectively addresses
the challenge of applying PINNs to forward dynamical systems and demonstrates
improved accuracy through its application to neuron models. Specifically, we
apply operator splitting to decompose the original neuron model into
sub-problems that are then solved using PINNs. Moreover, we develop an
scheme for discretizing fractional derivatives in fractional neuron models,
leading to improved accuracy and efficiency. The results of this study
highlight the potential of splitting PINNs in solving both integer- and
fractional-order neuron models, as well as other similar systems in
computational science and engineering
Efficient multistep methods for tempered fractional calculus: Algorithms and Simulations
In this work, we extend the fractional linear multistep methods in [C.
Lubich, SIAM J. Math. Anal., 17 (1986), pp.704--719] to the tempered fractional
integral and derivative operators in the sense that the tempered fractional
derivative operator is interpreted in terms of the Hadamard finite-part
integral. We develop two fast methods, Fast Method I and Fast Method II, with
linear complexity to calculate the discrete convolution for the approximation
of the (tempered) fractional operator. Fast Method I is based on a local
approximation for the contour integral that represents the convolution weight.
Fast Method II is based on a globally uniform approximation of the trapezoidal
rule for the integral on the real line. Both methods are efficient, but
numerical experimentation reveals that Fast Method II outperforms Fast Method I
in terms of accuracy, efficiency, and coding simplicity. The memory requirement
and computational cost of Fast Method II are and ,
respectively, where is the number of the final time steps and is the
number of quadrature points used in the trapezoidal rule. The effectiveness of
the fast methods is verified through a series of numerical examples for
long-time integration, including a numerical study of a fractional
reaction-diffusion model