30,259 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
An implicit algorithm for validated enclosures of the solutions to variational equations for ODEs
We propose a new algorithm for computing validated bounds for the solutions
to the first order variational equations associated to ODEs. These validated
solutions are the kernel of numerics computer-assisted proofs in dynamical
systems literature. The method uses a high-order Taylor method as a predictor
step and an implicit method based on the Hermite-Obreshkov interpolation as a
corrector step. The proposed algorithm is an improvement of the -Lohner
algorithm proposed by Zgliczy\'nski and it provides sharper bounds.
As an application of the algorithm, we give a computer-assisted proof of the
existence of an attractor set in the R\"ossler system, and we show that the
attractor contains an invariant and uniformly hyperbolic subset on which the
dynamics is chaotic, that is, conjugated to subshift of finite type with
positive topological entropy.Comment: 33 pages, 11 figure
A steady flow of MHD Maxwell viscoelastic fluid on a flat porous plate with the outcome of radiation and heat generation
Maxwell fluids display viscous flow on a long timescale but exhibit additional elastic resistance during rapid deformations. Among various types of rate-type fluids, the Maxwell fluid has achieved prominence in numerous study fields. This viscoelastic fluid has viscous and elastic properties. Due to their reduced complexity, this Maxwell fluid is utilized used in the polymeric industries. We have established a mathematical model based on the applications. This article examines the mathematical and graphical analysis for steady-state magnetohydrodynamic flow in a horizontal flat plate of Maxwell viscoelastic fluid for a permeable medium with heat and thermal radiation. The non-dimensional and similarity transformation used to frame the partial differential equations with restored ordinary differential equations. The shooting technique is originated to find solutions to nonlinear boundary value problems with the help of MATLAB software via the Runge-Kutta Fehlberg method. The primary idea behind this strategy is to change the boundary conditions of boundary value problems into initial value problems. Several plots illustrate the leading parameters such as Prandtl number (Pr), Deborah number (De), Eckert number (Ec), heat generation (Q), radiation (Rd), Lewis number (Le), magnetic parameter (M), and thermal slip condition (β) on the velocity profile and energy transfer behaviour. We validated our results with published work. The most significant impact of this study is that the Nusselt number drops as the Eckert number rises and climbs when heat radiation increases. The skin friction coefficient increases as Deborah number increases
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