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Topic: Separation of variables

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	Separation of variables - Wikipedia, the free encyclopedia
		In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to re-write an equation so that each of two variables occurs on a different side of the equation.
		At this point we have separated the variables x and y from each other, since x appears only on the right side of the equation and y only on the left.
		Methods of Generalized and Functional Separation of Variables at EqWorld: The World of Mathematical Equations.
	en.wikipedia.org /wiki/Separation_of_variables (644 words)

	Separation of Variables (Site not responding. Last check: 2007-11-06)
		DC MetaData for: On Separation of Variables in the Schrödinger Equation for a Pa...
		Separation of Variables and Reduction of the Equation...
		Separation of variables for the 1-dimensional nonlinear diffusion equation...
	www.scienceoxygen.com /math/502.html (248 words)

	PlanetMath: separation of variables (Site not responding. Last check: 2007-11-06)
		Separation of variables is a valuable tool for solving differential equations of the form
		The solution of this equation is relatively straightforward, we simple separate the variables algebraically and integrate.
		This is version 2 of separation of variables, born on 2002-02-26, modified 2002-02-26.
	www.planetmath.org /encyclopedia/SeparationOfVariables.html (145 words)

	Separation of Variables (Site not responding. Last check: 2007-11-06)
		A reasonable number of variables are kept ``variable'' while all others are assigned a fixed value, the minimization program is run, and the found values are fixed.
		We found that as long as the variables are correlated by tracks, the obtained results are absolutely identical within the uncertainties.
		We tried various combinations: ``tower geometry'' where the positions of strips in the same area of successive planes is determined simultaneously, or ``plane geometry'' where the positions of half of the strips of 1 plane (i.e.
	www.fynu.ucl.ac.be /librairie/theses/gustaaf.brooijmans/node74.html (280 words)

	Separation of variables (Site not responding. Last check: 2007-11-06)
		In mathematics separation of variables is any of several methods of ordinary and partial differential equations.
		Separation of variables for Riemannian spaces of constant curvature (Pitman monographs and surveys in pure and applied mathematics)
		Separation of Variables for Partial Differential Equations: An Eigenfunction Approach
	www.freeglossary.com /Separation_of_variables (595 words)

	Separation of variables: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-11-06)
		(separation of variables is any of several methods of solving ordinary and partial differential equation differential equation quick summary:
		In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables....
		At this point we have separated the variables x and y from each other, EHandler: no quick summary.
	www.absoluteastronomy.com /encyclopedia/s/se/separation_of_variables.htm (1081 words)

	Separation of variables for the 1-dimensional nonlinear diffusion equation (Site not responding. Last check: 2007-11-06)
		Abstract: The class of separable solutions of a 1-dimensional sourceless diffusion equation is stabilized by the action of the generic symmetry group.
		An equation which admits separation of variables in some field coordinate has separable solutions which are not invariant under any subgroup, as in the linear case.
		The class of separable equations significantly extends the class of equations having nongeneric symmetry, that is, those with exponential and power law diffusivities, for which separation of variables is a trivial process.
	wwwmaths.anu.edu.au /research.reports/mrr/97.011 (211 words)

	VI. Notes (played) on the vibrating string
		We shall begin our study of the method of separation of variables with the one-dimensional wave equation, which controls the vibrations of a taut string (see derivation) and some other phenomena.
		Here the spatial variable x might be more than one-dimensional, as for the problem of the vibrating drum, which will be encountered later.
		At this stage, we can exactly repeat the analysis of separation of variables, until the point where we first used the boundary conditions, i.e., (6.1).
	www.mathphysics.com /pde/ch6wr.html (2255 words)

	Separation of variables (Site not responding. Last check: 2007-11-06)
		In mathematics, separation of variables is any of severalmethods of solving ordinary and partial differentialequations.
		At this point we have separated the variables x and y from each other, since xappears only on the right side of the equation and y only on the left.
		Usually, each independent variable creates a separation constant that cannot be determined onlyfrom the equation itself.
	www.therfcc.org /separation-of-variables-164223.html (291 words)

	Solution Procedure - Separation of Variables (Site not responding. Last check: 2007-11-06)
		Solving problems by separation of variables is best learned by practice.
		The general procedure for solving a problem by separation of variables is as follows:
		This is inconsistent with the fact that none of the terms on side 2 depend on the independent variables associated with side 1.
	www.coe.uncc.edu /~rkeanini/funht/not2h/node14.html (296 words)

	Separation of Variables
		Once the equation is in a separated form, finding the solution is a simple matter, if the integrations are simple, that is. As you look at other methods of solving 1st Order Equations notice how, in many cases, you will try to manipulate the equation into this form.
		Both of the last two examples illustrate the fact that whenever you separate the variables in a separable differential equation, you are using a type of Integrating Factor.
		In general an Integrating Factor is a function which, when multiplied on to each side of a differential equation, yields an equation where each side is recognized as a derivative.
	homepage.mac.com /nshoffner/nsh/ODE/Separate.htm (950 words)

	Separation of variables
		Thus, at first sight, it would appear that the method of separation of variables only works for a very special subset of boundary conditions.
		For instance, in spherical geometry, the monotonic functions are power law functions of the radial variable, and the oscillatory functions are Legendre polynomials.
		This implies that the method of separation of variables is of quite general applicability.
	farside.ph.utexas.edu /teaching/em/lectures/node66.html (695 words)

	Amazon.com: Symmetry and Separation of Variables (Encyclopedia of Mathematics and its Applications): Books: Jr, Willard ... (Site not responding. Last check: 2007-11-06)
		This volume is concerned with the relationship between symmetries of a linear second-order partial differential equation of mathematical physics, the coordinate systems in which the equation admits solutions via separation of variables, and the properties of the special functions that arise in this manner.
		Some modern group-theoretic twists in the ancient method of separation of variables that can be used to provide a foundation for much of special function theory are shown.
		In particular, it is shown explicitly that all special functions that arise via separation of variables in the equations of mathematical physics can be studied using group theory.
	www.amazon.com /exec/obidos/tg/detail/-/0521302242?v=glance (489 words)

	Citebase - On Separation of Variables
		Authors: Viazminsky, C. The necessary and sufficient conditions for a function to be totally or partially separable are derived.
		The conditions of separability are expressed neatly in terms of the matrix which has to be diagonal if the function is to be totally separable, and has to assume a diagonal block form in order that the function is partially separable.
		For polynomials, the conditions of separability are shown to hold if and only if the product of the first column and the first row of the coefficients matrix is equal to the matrix itself.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0210167 (313 words)

	Separation of variables in the Kramers (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Abstract: We consider the problem of separation of variables in the Kramers equation admitting a non-trivial symmetry group.
		Provided the external potential V (x) is at most quadratic, a complete solution of the problem of separation of variables is obtained.
		0.6: On separable Schrodinger equations - Renat Zhdanov And
	citeseer.ist.psu.edu /685419.html (158 words)

	Separation Of Variables In Perturbed Cylinders (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		We study the Laplace operator subject to Dirichlet boundary conditions in a two-dimensional domain that is one-toone mapped onto a cylinder (rectangle or infinite strip).
		As a result of this transformation the original eigenvalue problem is reduced to an equivalent problem for an operator with variable coefficients.
		Taking advantage of the simple geometry we separate variables by means of the Fourier decomposition method.
	citeseer.ist.psu.edu /393905.html (415 words)

	Solution by Separation of Variables
		The differential equation (11) above is commonly solved by the technique of separation of variables.
		The basic idea behind this technique is to assume, initially, that the variable of interest can be separated into factors which depend separately on the coordinates.
		If one of the terms was not constant, then there would be no way that the other terms could compensate, since they depend on different variables.
	cow.physics.wisc.edu /~craigm/toroid/toroid/node4.html (156 words)

	Method of separation of variables (Site not responding. Last check: 2007-11-06)
		where each factor is function of only one variable.
		The interesting property of solutions of kind (2) is that all solution of the partial differential equation can be built up out of linear combinations of the members of the family of separated solutions.
		Notice that the right hand side is a constant, therefore the same must be for the left hand side.
	www.uic.edu /classes/eecs/eecs520/week02/SeparationVariables/node1.html (211 words)

	Separable Differential Equations
		In this section, we'd like to show you an application of the method of Change of Variables by introducing you to separable differential equations.
		In case you've forgotten, we'll remind what a differential equation is and why they are so useful in mathematics.
		In other words, what is decaying exponentially is the difference between the terminal velocity and the initial velocity.
	www.ugrad.math.ubc.ca /coursedoc/math101/notes/moreApps/separable.html (1418 words)

	Separation of variables in the nonlinear wave equation
		Separation of variables in the nonlinear wave equation
		We develop a technique making it possible to handle the problem of separation of variables in nonlinear differential equations.
		Using it we obtain a number of new two-dimensional nonlinear wave equations admitting separation of variables and construct their exact solutions.
	stacks.iop.org /0305-4470/27/L291 (188 words)

	Separation of Variables (Site not responding. Last check: 2007-11-06)
		Using the technique of separation of variables, let u(r,t) = v(r)w(t).
		Notice that we can factor w[t] out of the right hand side and separate the functions of r from the functions of t.
		Since one side of the equation is a function of t and the other is a function of r, both must be a constant.
	www.ma.iup.edu /projects/CalcDEMma/drum/drum3.html (90 words)

	Separation of variables (Site not responding. Last check: 2007-11-06)
		of n variables, it is sometimes useful to guess solution of the form :
		The ForCES working group is defining a protocol to allow a Control Element (CE) to control the behavior of a Forwarding Element (FE).
		It explains the rationale for choosing the transport protocols and also describes how this TML addresses all the requirements described in the Forces [3] requirements and ForCES protocol [5] document.
	www.serebella.com /encyclopedia/article-Separation_of_variables.html (1942 words)

	Use of separation of variables and graphical analysis in solving a special case of Torricelli’s Law
		Use of separation of variables and graphical analysis in solving a special case of Torricelli’s Law
		In this problem we are presented with a situation where we have a can full of water with a small hole in the bottom.
		We now need to find a linear relationship between the variables in order to produce a graph of the data.
	www.etsu.edu /math/kerley/de/s2000/chal/pres.htm (427 words)

	Amazon.com: Separation of Variables for Partial Differential Equations: An Eigenfunction Approach: Books: George Cain (Site not responding. Last check: 2007-11-06)
		Separation of Variables for Partial Differential Equations: An Eigenfunction Approach (Hardcover)
		This book shows the power of the separation of variables technique by solving a variety of problems, many of which go well beyond the usual textbook problems.
		With a focus on the potential, heat, and wave equations throughout the book, the authors take a spectral exapnsion view of the technique and explore it in detail.
	www.amazon.com /exec/obidos/tg/detail/-/1584884207?v=glance (429 words)

	Separation of Variables
		In physics, many PDEs' equation can be reduced to a number of ODEs' equations by "Separation of Variables".
		A PDE that is "separable" in Cartesian coordinates may not be "separable" in polar coordinates.
		The separation of variables usually comes second.First the equation (assumed linear) must be brought to canonical form.Just then,u have to find ways to integrate it.Separation of variables is an excellent method,when u know that the solution of the eq.is unique.U can use separation of variables and Fourier series to find that solution.
	www.physicsforums.com /showthread.php?t=59508 (340 words)

	Separation of variables (Site not responding. Last check: 2007-11-06)
		Read about separation of variables in the free online encyclopedia and dictionary.
		Find separation of variables at one of the best sites the Internet has to offer!
		Find results for separation of variables and anything else you are looking for instantly!
	www.logicjungle.com /wiki/Separation_of_variables (77 words)

	SIAP Volume 61 Issue 4
		In the 2-D case the Maxwell operator splits into two scalar operators corresponding to so-called TE and TM modes of propagating electromagnetic wave.
		In the present paper we show that both of these operators can be slightly modified in such a way that separation of variables can be used.
		The eigenfunctions and the spectra for the modified operators are then used to obtain Rayleigh--Ritz-type estimates of the spectrum of the Maxwell operator.
	epubs.siam.org /sam-bin/dbq/article/32089 (209 words)

	P.P. Cook's Tangent Space: Knots, Lattices and then Separation of Variables
		Infrequent and random comments on maths and theoretical physics as seen from the point of view of a PhD student.
		There were very many other good books available too (including paperback copies of Polchinski vols 1 and 2) and one I hadn't heard about before, but which I will look up in the library, called "Gravity and Strings" by Tomas Ortin.
		This afternoon's first talk was about lattice gauge theory and was given by Fedor Smirnov, his title was "Correlation functions for lattice exactly solvable models", and the second was by Evgeni Sklyanin and entitled "The Q operator, Bäcklund transformation and separation of variables".
	ppcook.blogspot.com /2005/07/knots-lattices-and-then-separation-of.html (364 words)

	Find the error! Separation of Variables (Site not responding. Last check: 2007-11-06)
		You have just been to a calculus class, and are feeling very good about yourself.
		You are especially proud that you have just learned how to solve separable differential equations, and are reasonably confident that you understand why the method works.
		As you leave the building, a wild-eyed, hungry-looking stranger stops you on the street.
	www.dougshaw.com /findtheerror/FTEsep.html (253 words)

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