15 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
Approximate Solutions to Fractional Subdiffusion Equations: The heat-balance integral method
The work presents integral solutions of the fractional subdiffusion equation
by an integral method, as an alternative approach to the solutions employing
hypergeometric functions. The integral solution suggests a preliminary defined
profile with unknown coefficients and the concept of penetration (boundary
layer). The prescribed profile satisfies the boundary conditions imposed by the
boundary layer that allows its coefficients to be expressed through its depth
as unique parameter. The integral approach to the fractional subdiffusion
equation suggests a replacement of the real distribution function by the
approximate profile. The solution was performed with Riemann -Liouville
time-fractional derivative since the integral approach avoids the definition of
the initial value of the time-derivative required by the Laplace transformed
equations and leading to a transition to Caputo derivatives. The method is
demonstrated by solutions to two simple fractional subdiffusion equations
(Dirichlet problems): 1) Time-Fractional Diffusion Equation, and 2)
Time-Fractional Drift Equation, both of them having fundamental solutions
expressed through the M-Write function. The solutions demonstrate some basic
issues of the suggested integral approach, among them: a) Choice of the
profile, b) Integration problem emerging when the distribution (profile) is
replaced by a prescribed one with unknown coefficients; c) Optimization of the
profile in view to minimize the average error of approximations; d) Numerical
results allowing comparisons to the known solutions expressed to the M-Write
function and error estimations.Comment: 15 pages, 7 figures, 3 table
Effective Modified Fractional Reduced Differential Transform Method for Solving Multi-Term Time-Fractional Wave-Diffusion Equations
In this work, we suggest a new method for solving linear multi-term time-fractional wave-diffusion equations, which is named the modified fractional reduced differential transform method (m-FRDTM). The importance of this technique is that it suggests a solution for a multi-term time-fractional equation. Very few techniques have been proposed to solve this type of equation, as will be shown in this paper. To show the effectiveness and efficiency of this proposed method, we introduce two different applications in two-term fractional differential equations. The three-dimensional and two-dimensional plots for different values of the fractional derivative are depicted to compare our results with the exact solutions
Solving Linear and Nonlinear Fractional Differential Equations Using Spline Functions
Suitable spline functions of polynomial form are derived and used to solve linear and nonlinear
fractional differential equations. The proposed method is applicable for 0<α≤1 and α≥1, where α denotes the order of the fractional derivative in the Caputo sense. The results obtained
are in good agreement with the exact analytical solutions and the numerical results presented
elsewhere. Results also show that the technique introduced here is robust and easy to apply
Solving a Higher-Dimensional Time-Fractional Diffusion Equation via the Fractional Reduced Differential Transform Method
In this study, exact and approximate solutions of higher-dimensional time-fractional diffusion equations were obtained using a relatively new method, the fractional reduced differential transform method (FRDTM). The exact solutions can be found with the benefit of a special function, and we applied Caputo fractional derivatives in this method. The numerical results and graphical representations specified that the proposed method is very effective for solving fractional diffusion equations in higher dimensions