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

Blended General Linear Methods based on Boundary Value Methods in the GBDF family

Among the methods for solving ODE-IVPs, the class of General Linear Methods (GLMs) is able to encompass most of them, ranging from Linear Multistep Formulae (LMF) to RK formulae. Moreover, it is possible to obtain methods able to overcome typical drawbacks of the previous classes of methods. For example, order barriers for stable LMF and the problem of order reduction for RK methods. Nevertheless, these goals are usually achieved at the price of a higher computational cost. Consequently, many efforts have been made in order to derive GLMs with particular features, to be exploited for their efficient implementation. In recent years, the derivation of GLMs from particular Boundary Value Methods (BVMs), namely the family of Generalized BDF (GBDF), has been proposed for the numerical solution of stiff ODE-IVPs. In particular, this approach has been recently developed, resulting in a new family of L-stable GLMs of arbitrarily high order, whose theory is here completed and fully worked-out. Moreover, for each one of such methods, it is possible to define a corresponding Blended GLM which is equivalent to it from the point of view of the stability and order properties. These blended methods, in turn, allow the definition of efficient nonlinear splittings for solving the generated discrete problems. A few numerical tests, confirming the excellent potential of such blended methods, are also reported.Comment: 22 pages, 8 figure

Numerical Solution of ODEs and the Columbus' Egg: Three Simple Ideas for Three Difficult Problems

On computers, discrete problems are solved instead of continuous ones. One must be sure that the solutions of the former problems, obtained in real time (i.e., when the stepsize h is not infinitesimal) are good approximations of the solutions of the latter ones. However, since the discrete world is much richer than the continuous one (the latter being a limit case of the former), the classical definitions and techniques, devised to analyze the behaviors of continuous problems, are often insufficient to handle the discrete case, and new specific tools are needed. Often, the insistence in following a path already traced in the continuous setting, has caused waste of time and efforts, whereas new specific tools have solved the problems both more easily and elegantly. In this paper we survey three of the main difficulties encountered in the numerical solutions of ODEs, along with the novel solutions proposed.Comment: 25 pages, 4 figures (typos fixed

Tchebychev Polynomial Approximations for mthm^{th} Order Boundary Value Problems

Higher order boundary value problems (BVPs) play an important role modeling various scientific and engineering problems. In this article we develop an efficient numerical scheme for linear $m^{th}$ order BVPs. First we convert the higher order BVP to a first order BVP. Then we use Tchebychev orthogonal polynomials to approximate the solution of the BVP as a weighted sum of polynomials. We collocate at Tchebychev clustered grid points to generate a system of equations to approximate the weights for the polynomials. The excellency of the numerical scheme is illustrated through some examples.Comment: 21 pages, 10 figure