7,877 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
A survey on fuzzy fractional differential and optimal control nonlocal evolution equations
We survey some representative results on fuzzy fractional differential
equations, controllability, approximate controllability, optimal control, and
optimal feedback control for several different kinds of fractional evolution
equations. Optimality and relaxation of multiple control problems, described by
nonlinear fractional differential equations with nonlocal control conditions in
Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Computational and Applied Mathematics', ISSN: 0377-0427.
Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication
20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515
On the convergence of two-dimensional fuzzy Volterra-Fredholm integral equations by using Picard method
In this paper we prove convergence of the method of successive approximations used to approximate the solution of nonlinear two-dimensional Volterra-Fredholm integral equations and define the notion of numerical stability of the algorithm with respect to the choice of the first iteration. Also we present an iterative procedure to solve such equations. Finally, the method is illustrated by solving some examples
New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus
This reprint focuses on exploring new developments in both pure and applied mathematics as a result of fractional behaviour. It covers the range of ongoing activities in the context of fractional calculus by offering alternate viewpoints, workable solutions, new derivatives, and methods to solve real-world problems. It is impossible to deny that fractional behaviour exists in nature. Any phenomenon that has a pulse, rhythm, or pattern appears to be a fractal. The 17 papers that were published and are part of this volume provide credence to that claim. A variety of topics illustrate the use of fractional calculus in a range of disciplines and offer sufficient coverage to pique every reader's attention
On the Lp-spaces techniques in the existence and uniqueness of the fuzzy fractional Korteweg-de Vries equation’s solution
In this paper, is proposed the existence and uniqueness of the solution of all fuzzy fractional differential equations, which are equivalent to the fuzzy integral equation. The techniques on LP-spaces are used, defining the LpF F ([0; 1]) for 1≤P≤∞, its properties, and using the functional analysis methods. Also the convergence of the method of successive approximations used to approximate the solution of fuzzy integral equation be proved and an iterative procedure to solve such equations is presented
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Rev. iberoam. bioecon. cambio clim. Vol.1(1) 2015; 95-114Los cambios medioambientales globales hacen pensar en un aumento futuro de la aridez, por ello es necesario buscar alternativas que permitan un uso más eficiente del agua y reducir su consumo, teniendo en cuenta que es un recurso limitado. En la actualidad, aproximadamente el 59,7% del total de agua planificada para todos los usos en Cuba se utiliza en la agricultura, pero no más del 50% de esa agua se convierte directamente en productos agrÃcolas. El estudio de las funciones agua-rendimiento y su uso dentro de la planificación del agua para riego es una vÃa importante para trazar estrategias de manejo que contribuyan al incremento en la producción agrÃcola. Utilizando los datos de agua aplicada por riego y los rendimientos obtenidos en más de 100 experimentos de campo realizados fundamentalmente en suelo FerralÃtico Rojo de la zona sur de La Habana y con ayuda de herramientas de análisis de regresión en este trabajo se estiman las funciones agua aplicada-rendimientos para algunos cultivos agrÃcolas y se analizan las posibles estrategias de optimización del riego a seguir en función de la disponibilidad de agua. Seleccionar una estrategia de máxima eficiencia del riego puede conducir a reducciones de agua a aplicar entre un 21,6 y 46,8%, incrementos de la productividad del agua entre 17 y 32% y de la relación beneficios/costo estimada de hasta un 3,4%. Lo anterior indica la importancia desde el punto de vista económico que puede llegar a alcanzar el uso de esta estrategia en condiciones de déficit hÃdrico. El conocimiento de las funciones agua aplicada por riego-rendimiento y el uso de la productividad del agua, resultan parámetros factibles de introducir como indicadores de eficiencia en el planeamiento del uso del agua en la agricultura, con lo cual es posible reducir los volúmenes de agua a aplicar y elevar la relación beneficio-costo actual.Rev. iberoam. bioecon. cambio clim. Vol.1(1) 2015; 95-11
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