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

APPLIED MATHEMATICS AND COMPUTATION

In the present paper, a Taylor method is developed to find the approximate solution of high-order linear Volterra-Fredholm integro-differential equations under the mixed conditions in terms of Taylor polynomials about any point, In addition, examples that illustrate the pertinent features of the method are presented, and the results of study are discussed. (C) 2000 Elsevier Science Inc, All rights reserved

APPLIED MATHEMATICS AND COMPUTATION

In this study, a Legendre collocation matrix method is presented to solve high-order Linear Fredholm integro-differential equations under the mixed conditions in terms of Legendre polynomials. The proposed method converts the equation and conditions to matrix equations, by means of collocation points on the interval [-1,1], which corresponding to systems of linear algebraic equations with Legendre coefficients. Thus, by solving the matrix equation, Legendre coefficients and polynomial approach are obtained. Also examples that illustrate the pertinent features of the method are presented and by using the error analysis, the results are discussed. (c) 2009 Elsevier Inc. All rights reserved

INTERNATIONAL JOURNAL OF MODERN PHYSICS B

The purpose of this study is to implement a new approximate method for solving system of nonlinear Volterra integral equations. The technique is based on, first, differentiating both sides of integral equations n times and then substituting the Taylor series the unknown functions in the resulting equation and later, transforming to a matrix equation. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Taylor coefficients of the solution function. Some numerical results are also given to illustrate the efficiency of the method

JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS

In this paper, a numerical method based on polynomial approximation, using Hermite polynomial basis, to obtain the approximate solution of generalized pantograph equations with variable coefficients is presented. The technique we have used is an improved collocation method. Some numerical examples, which consist of initial conditions, are given to illustrate the reality and efficiency of the method. In addition, some numerical examples are presented to show the properties of the given method; the present method has been compared with other methods and the results are discussed. (C) 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved

Orbitofrontal cortex projecting mediodorsal thalamic population’s contributions to instrumental and incentive learning and performance

Cognitive control processes do not solely rely on the prefrontal cortex (PFC) proper. Mediodorsal thalamus (MD), the higher-order thalamic region known to be prominently connected with the PFC, has been recognized as an important node in the cortico-striatal-thalamic-cortical loops mediating flexible goal-directed behavior, both in clinical and basic research. Previously, studies trying to understand MD activity and function have largely examined or manipulated the structure as a whole. Recently, there has been an increasing appreciation of, and methods to target, the subpopulations within the MD as they relate to the functions of their respective PFC targets; but most in vivo studies focus on MD projections into medial and dorsal PFC. The lateral orbitofrontal cortex (lOFC) also receives input from the MD, and has been implicated in various aspects of goal-directed decision-making such as outcome valuation and maintaining an up-to-date internal representation of tasks and the contingencies therein; yet MD’s contributions to lOFC functions have remained unclear.In this dissertation, I sought to image and manipulate the endogenous activity pattern of the MD projection population into lOFC during the learning and performance of a self-initiated goal-directed task, which to our knowledge is the first time this MD-PFC subcircuit has been examined in this manner in vivo. We found that the activity of the MD terminal population in lOFC was differentially sensitive to trials based on the probabilistic outcome of instrumental actions. In concert, we found that animals’ expectation built across learning and changing task requirements, with expectation-modulated instrumental performance affected by optogenetically inhibiting the activity of lOFC projecting MD somas. We did not find evidence of motivational state-induced outcome value representation in the MD-lOFC terminal population; however, attenuating the activity of MD-lOFC projection neurons during outcome revaluation and, in particular, the use of updated outcome value in extinction did compromise adaptive instrumental behavior. These findings suggest that MD input population into lOFC provides prospective information that modulates instrumental actions within the overall cognitive control framework of monitoring interactions with the world, comparing expectations with actual experiences, and adapting an internal model of the world in order to optimize goal-directed behaviors