10,664 research outputs found
Status of the differential transformation method
Further to a recent controversy on whether the differential transformation
method (DTM) for solving a differential equation is purely and solely the
traditional Taylor series method, it is emphasized that the DTM is currently
used, often only, as a technique for (analytically) calculating the power
series of the solution (in terms of the initial value parameters). Sometimes, a
piecewise analytic continuation process is implemented either in a numerical
routine (e.g., within a shooting method) or in a semi-analytical procedure
(e.g., to solve a boundary value problem). Emphasized also is the fact that, at
the time of its invention, the currently-used basic ingredients of the DTM
(that transform a differential equation into a difference equation of same
order that is iteratively solvable) were already known for a long time by the
"traditional"-Taylor-method users (notably in the elaboration of software
packages --numerical routines-- for automatically solving ordinary differential
equations). At now, the defenders of the DTM still ignore the, though much
better developed, studies of the "traditional"-Taylor-method users who, in
turn, seem to ignore similarly the existence of the DTM. The DTM has been given
an apparent strong formalization (set on the same footing as the Fourier,
Laplace or Mellin transformations). Though often used trivially, it is easily
attainable and easily adaptable to different kinds of differentiation
procedures. That has made it very attractive. Hence applications to various
problems of the Taylor method, and more generally of the power series method
(including noninteger powers) has been sketched. It seems that its potential
has not been exploited as it could be. After a discussion on the reasons of the
"misunderstandings" which have caused the controversy, the preceding topics are
concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages,
references and further considerations adde
Multigrid waveform relaxation for the time-fractional heat equation
In this work, we propose an efficient and robust multigrid method for solving
the time-fractional heat equation. Due to the nonlocal property of fractional
differential operators, numerical methods usually generate systems of equations
for which the coefficient matrix is dense. Therefore, the design of efficient
solvers for the numerical simulation of these problems is a difficult task. We
develop a parallel-in-time multigrid algorithm based on the waveform relaxation
approach, whose application to time-fractional problems seems very natural due
to the fact that the fractional derivative at each spatial point depends on the
values of the function at this point at all earlier times. Exploiting the
Toeplitz-like structure of the coefficient matrix, the proposed multigrid
waveform relaxation method has a computational cost of
operations, where is the number of time steps and is the number of
spatial grid points. A semi-algebraic mode analysis is also developed to
theoretically confirm the good results obtained. Several numerical experiments,
including examples with non-smooth solutions and a nonlinear problem with
applications in porous media, are presented
Efficient preconditioning of the method of lines for solving nonlinear two-sided space-fractional diffusion equations
A standard method for the numerical solution of partial differential equations (PDEs) is the method of lines. In this approach the PDE is discretised in space using �finite di�fferences or similar techniques, and the resulting semidiscrete problem in time is integrated using an initial value problem solver. A significant challenge when applying the method of lines to fractional PDEs is that the non-local nature of the fractional derivatives results in a discretised system where each equation involves contributions from many (possibly every) spatial node(s). This has important consequences for the effi�ciency of the numerical solver. First, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. Second, since the Jacobian matrix of the system is dense (partially or fully), methods that avoid the need to form and factorise this matrix are preferred. In this paper, we consider a nonlinear two-sided space-fractional di�ffusion equation in one spatial dimension. A key contribution of this paper is to demonstrate how an eff�ective preconditioner is crucial for improving the effi�ciency of the method of lines for solving this equation. In particular, we show how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach
A comparison between numerical solutions to fractional differential equations: Adams-type predictor-corrector and multi-step generalized differential transform method
In this note, two numerical methods of solving fractional differential
equations (FDEs) are briefly described, namely predictor-corrector approach of
Adams-Bashforth-Moulton type and multi-step generalized differential transform
method (MSGDTM), and then a demonstrating example is given to compare the
results of the methods. It is shown that the MSGDTM, which is an enhancement of
the generalized differential transform method, neglects the effect of non-local
structure of fractional differentiation operators and fails to accurately solve
the FDEs over large domains.Comment: 12 pages, 2 figure
On an explicit finite difference method for fractional diffusion equations
A numerical method to solve the fractional diffusion equation, which could
also be easily extended to many other fractional dynamics equations, is
considered. These fractional equations have been proposed in order to describe
anomalous transport characterized by non-Markovian kinetics and the breakdown
of Fick's law. In this paper we combine the forward time centered space (FTCS)
method, well known for the numerical integration of ordinary diffusion
equations, with the Grunwald-Letnikov definition of the fractional derivative
operator to obtain an explicit fractional FTCS scheme for solving the
fractional diffusion equation. The resulting method is amenable to a stability
analysis a la von Neumann. We show that the analytical stability bounds are in
excellent agreement with numerical tests. Comparison between exact analytical
solutions and numerical predictions are made.Comment: 22 pages, 6 figure
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