101,033 research outputs found
Reduced Differential Transform Method for (2+1) Dimensional type of the Zakharov-Kuznetsov ZK(n,n) Equations
In this paper, reduced differential transform method (RDTM) is employed to
approximate the solutions of (2+1) dimensional type of the Zakharov-Kuznetsov
partial differential equations. We apply these method to two examples. Thus, we
have obtained numerical solution partial differential equations of
Zakharov-Kuznetsov. These examples are prepared to show the efficiency and
simplicity of the method
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
Modeling an Aquifer: Numerical Solution to the Groundwater Flow Equation
We present a model of groundwater dynamics under stationary flow and governed
by Darcy's Law of water motion through porous media, we apply it to study a 2D
aquifer with water table of constant slope comprised of an homogeneous and
isotropic media, the more realistic case of an homogeneous anisotropic soil is
also considered. Taking into account some geophysical parameters we develop a
computational routine, in the Finite Difference Method, that solves the
resulting elliptic partial equation, both in a homogeneous isotropic and
homogeneous anisotropic media. After calibration of the numerical model, this
routine is used to begin a study of the Ayamonte-Huelva aquifer in Spain, a
modest analysis of the system is given, we compute the average discharge vector
as well as its root mean square as a first predictive approximation of the flux
in this system, providing us a signal of the location of best exploitation;
long term goal is to develop a complete computational tool for the analysis of
groundwater dynamics.Comment: 13 pages and 12 figure
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