Exact differential equation

About: Exact differential equation is a research topic. Over the lifetime, 4375 publications have been published within this topic receiving 82226 citations. The topic is also known as: total differential equation.

AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont)

Abstract: Preface This is a guide to the software package AUTO for continuation and bifurcation problems in ordinary differential equations. graphics program PLAUT and the pendula animation program. An earlier graphical user interface for AUTO on SGI machines was written by Taylor & Kevrekidis (1989). Special thanks are due to Sheila Shull, California Institute of Technology, for her cheerful assistance in the distribution of AUTO over a long period of time. Over the years, the development of AUTO has been supported by various agencies through the California Institute of Technology. Work on this updated version was supported by a general research grant from NSERC (Canada). The development of HomCont has much benefitted from various pieces of help and advice from, among others, W. This manual uses the following conventions. command This font is used for commands which you can type in. PAR This font is used for AUTO parameters. filename This font is used for file and directory names. variable This font is used for environment variable. site This font is used for world wide web and ftp sites. function This font is used for function names.

The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods

Abstract: The paper is concerned with the practical determination of the characteristic values and functions of the wave equation of Schrodinger for a non-Coulomb central field, for which the potential is given as a function of the distance r from the nucleus.The method used is to integrate a modification of the equation outwards from initial conditions corresponding to a solution finite at r = 0, and inwards from initial conditions corresponding to a solution zero at r = ∞, with a trial value of the parameter (the energy) whose characteristic values are to be determined; the values of this parameter for which the two solutions fit at some convenient intermediate radius are the characteristic values required, and the solutions which so fit are the characteristic functions (§§ 2, 10).Modifications of the wave equation suitable for numerical work in different parts of the range of r are given (§§ 2, 3, 5), also exact equations for the variation of a solution with a variation in the potential or of the trial value of the energy (§ 4); the use of these variation equations in preference to a complete new integration of the equation for every trial change of field or of the energy parameter avoids a great deal of numerical work.For the range of r where the deviation from a Coulomb field is inappreciable, recurrence relations between different solutions of the wave equations which are zero at r = ∞, and correspond to terms with different values of the effective and subsidiary quantum numbers, are given and can be used to avoid carrying out the integration in each particular case (§§ 6, 7).Formulae for the calculation of first order perturbations due to the relativity variation of mass and to the spinning electron are given (§ 8).The method used for integrating the equations numerically is outlined (§ 9).

Contractive Mapping Theorems in Partially Ordered Sets and Applications to Ordinary Differential Equations

Abstract: We prove the existence and uniqueness of solution for a first-order ordinary differential equation with periodic boundary conditions admitting only the existence of a lower solution. To this aim, we prove an appropriate fixed point theorem in partially ordered sets.