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Topic: Exact solution

	bakker2000.ht
		The exact solution is obtained with the hodograph method in combination with conformal mapping; the complex specific discharge function and the reference function are used as auxiliary functions.
		The approximate and exact solutions produce results that are identical for at least six significant digits for all cases investigated, which suggests that it may be reasonable to use the approximate formulas outside the range of comparison.
		Solutions for isotropic and anisotropic aquifers obtained with the approximate formulas are compared to results obtained with a Dupuit solution for a set of realistic aquifer parameters; the interface is approximated well with the Dupuit solution.
	www.engr.uga.edu /~mbakker/bakker2000.html (329 words)

	Time and Attendance Exact Solution - Exactsoft (Site not responding. Last check: 2007-11-06)
		Exact Solution is a powerful modular software suite consisting of time and attendance management, shift scheduling, human resources, training (law 90), job costing, access control, clock data collection and other aspects related to time management.
		Exact Solution modules may also be leased at low monthly rates over a period of 1 to 4 years.
		Exact Solution communicates perfectly with other software packages as long as the logical elements required for integration are open to the same integration philosophy.
	www.exactsoft.com /products_exactsolution.htm (1020 words)

	Exact solution -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-06)
		An exact solution of the (Click link for more info and facts about Einstein field equation) Einstein field equation is a (Click link for more info and facts about Lorentz metric) Lorentz metric that corresponds to a physically realizable (Click link for more info and facts about energy-momentum tensor) energy-momentum tensor.
		In the study of exact solutions of the field equations, it is sometimes convenient to decompose the Riemann tensor into its trace and trace-free parts.
		This is one of the major difficulties in finding exact solutions of the field equations, and quite often simplifying assumptions such as linearising the field equations are made.
	www.absoluteastronomy.com /encyclopedia/e/ex/exact_solution.htm (242 words)

	My Personal Reading List
		An asymptotically flat solution of the Einstein field equations representing the exterior gravitational field of a stationary rotating mass and reducting to the Schwarzschild metric in the static limit is presented.
		An exact asymptotically flat solution of the vacuum Einstein equations representing the exterior gravitational field of a stationary axisymmetric mass with an arbitrary mass-multipole structure is presented.
		New exact asymptotically flat solutions of Einstein's vacuum equations for the description of the exterior gravitational field of a static and stationary mass with an arbitrary mass-multipole structure are presented.
	members.localnet.com /~atheneum/bib/staxsymaps.html (4380 words)

	Acid-base equilibria. Exact solution for the pH of a monoprotic acid solution
		Equation 13 is the desired exact relationship, neglecting activity corrections, between the unknown hydrogen ion concentration and the known quantities.
		These plots were generated by solving for the exact roots of the cubic equation over the range of variables shown using the Newton-Raphson method.
		Figures 1A and 1B are concise graphical summaries of the exact solutions to Equation 13 and can be used as a standard against which to judge the adequacy of approximations.
	www.chem.sc.edu /faculty/morgan/resources/acidbase/exact.html (528 words)

	Exact Solution for a Circular Piston Radiating in an Infinite Baffle
		Here we develop an alternate exact solution using a plane wave spectrum, which results in a one-dimensional integral (but over an infinite range), that can be numerically evaluated everywhere on the baffle surface.
		The numerical solution has the great advantage of being applicable to a more general piston shape, and it is applied to typical loudspeaker cone.
		In all of the remaining plots, the exact equation result is shown as a solid line, and the numerical result as dashed.
	www.silcom.com /~aludwig/Physics/Exact_piston/Exact_piston.htm (1093 words)

	General_solution
		The general solution of the wave equation provides a means for computing the sound wave pressure (or velocity) anywhere in a region, in terms of values of pressure and velocity on the boundary of the region.
		Thus given the solution to A, it is only necessary to subtract it from a plane wave to get the solution to B. Note that the integrals in each case yield the total wave inside the volume, equal to the incident plane wave plus the wave scattered by the obstacle.
		Thus the total solution in case B is equal to the scattered wave in case A, and vice-versa.
	www.silcom.com /~aludwig/Physics/Gensol/General_solution.html (2743 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		The solution to this equation is y1 = C1ex + C2e-2x.
		The have of h = 0.002 appears stable in that the solution is bounded.
		The solution for h = 0.0025 is unstable.
	www.csun.edu /~lcaretto/me501a/hw13.doc (2100 words)

	Lab1 - Experimental Results (Site not responding. Last check: 2007-11-06)
		The exact solution is plotted as well for comparison.
		Both Experiments 1 and 2 came "close" to the exact solution within the simulation time frame while the solution calculated by Experiment 3 still did not match the exact solution by the end of the simulation time frame, 75 minutes.
		The solution from Experiment 3 "rings" or oscillates around the exact solution as it apparently "damps down" to the exact solution.
	www.woodrow.org /teachers/esi/1999/princeton/projects/modeling/lab1results.html (448 words)

	An Exact Solution of Stikker's Nonlinear Heat Equation
		Exact solutions are derived for a nonlinear heat equation where the conductivity is a linear fractional function of (i) the temperature gradient, or (ii) the product of the radial distance and the radial component of the temperature gradient for problems expressed in cylindrical coordinates.
		The exact solutions are additively separable, isolating the nonlinear component from the remaining independent variables.
		The asymptotic behaviour of these solutions is studied and a boundary value problem is presented which is satisfied by these solutions.
	epubs.siam.org /sam-bin/dbq/article/24917 (157 words)

	Forward and Backward Euler Methods
		Once again, if the true solution is not known a priori, we can choose, depending on the precision required, the solution obtained with a sufficiently small time step as the 'exact' solution to study the convergence characteristics.
		In Figure 4, I have plotted the solutions computed using the BE method for h=0.001, 0.01, 0.1, 0.2 and 0.5 along with the exact solution.
		The accuracy of the computed solution deteriorates as h is increased, and we expect the global error to scale linearly with h.
	web.mit.edu /10.001/Web/Course_Notes/Differential_Equations_Notes/node3.html (914 words)

	The Singular Value Decompostion
		Exact Rational Solutions and Conditioning The first section of this chapter deals with the solution of linear systems of equations Ax = b when the matrix A and the right-hand side b are known exactly as rational numbers.
		Since exact rational solution is the process most familiar to people, it is the one that they most often want --- for example, sometimes people convert a problem with imperfect (say, measured) data into one with rational numbers as entries and then want the exact, rational solution to that problem.
		Note that the elements of the solution are all exact integers, alternate in sign, and are many orders of magnitude larger than the input data.
	www.apmaths.uwo.ca /~rcorless/AM563/NOTES/Jan_31_96/node10.html (1714 words)

	An Exact Solution
		This incompatibility led Einstein, as early as 1907, to the belief that the global invariance of light speed, in the sense of the special theory, could not be maintained.
		To deduce the implications of the field equations for observable phenomena Einstein originally made use of approximate methods, since no exact solutions were known.
		These approximate methods were adequate to demonstrate that the field equations lead in the first approximation to Newton's laws, and in the second approximation to a natural explanation for the anomalous precession of Mercury (see Section 6.2).
	www.mathpages.com /rr/s6-01/6-01.htm (2135 words)

	The Exact Solution of the Gambler's Ruin Problem (Site not responding. Last check: 2007-11-06)
		The solution method is virtually the same as that in the two examples in the previous section.
		In fact the exact solution of the Gambler's Ruin problem turns out to be a function of three variables.
		So we have determined an exact solution for a problem that is a simplification of the problems we first discussed.
	www.math.usu.edu /~koebbe/GR/book/node8.html (897 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		The 'eps' is a small parameter and lends this ODE to an asymptotic solution.
		An adequate asymptotic solution consists of an 'inner,' and 'outer' solution that are pieced together.
		All along the exact solution, either the inner or the outer solution is indistinguishable from the exact solution.
	www.math.sfu.ca /~jrg/apma900.html (464 words)

	Mark Newman: Publications
		Solution for the properties of a clustered network, Juyong Park and M. Newman, Phys.
		Solution of the 2-star model of a network, Juyong Park and M. Newman, Phys.
		Exact solution of site and bond percolation on small-world networks, Cristopher Moore and M. Newman, Phys.
	www-personal.umich.edu /~mejn/pubs.html (1481 words)

	Numerical Techniques
		Since we expect the solution to the differential equation and its tangent line to be close when x is close to -1, we should also expect that the solution to the differential equation at, let's say, x=-0.75 will be close to the tangent line at x=-0.75.
		But, in such a situation you cannot compare the approximation to the exact solution so you have no control over how good your approximation is! If you take a course in Numerical Mathematics you will learn that there are ways to predict the error in Euler's method even if you cannot compute the exact solution.
		The equilibrium is a down node; thus the exact solution with initial condition y(0)=1.1 is decreasing and approaches 1 as t approaches infinity.
	www.sosmath.com /diffeq/first/numerical/numerical.html (722 words)

	Exact solution of the O(n) model on a random lattice (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Exact solution of the O(n) model on a random lattice
		Abstract: We present an exact solution of the O(n) model on a random lattice.
		The coupling constant space of our model is parametrized in terms of a set of moment variables and the same type of universality with respect to the potential as observed for the onematrix model is found.
	citeseer.ist.psu.edu /183931.html (321 words)

	C. Lim (RPI), Exact solution of a three constraints equilibrium statistical mechanics model for 2-D turbulence
		This three constraints model is the third one in a family of few constraints equilibrium statistical mechanics models proposed by Turkington and Majda for the study of 2-D and quasi-geostrophic flows.
		The exact solution of this model is based on a simple but fundamental observation that the enstrophy corresponds exactly to the higher dimen sional spherical constraint introduced by Kac.
		The exact solution is then obtained by extending Kac's and Berlin's solution of the spherical model to a long range logarithmic interaction.
	online.itp.ucsb.edu /online/hydrot00fluid/lim (168 words)

	[No title]
		In addition, the solution obtain from this method is compared with the steady state exact and the sub cases analytical solutions and with the finite difference solution.
		Solution for Case 2: (Pe is very large) The analytical solution for this case in in Apendix A. EMBED Equation.3 Eq.
		Comparison between Galerkin method and exact solution for Pe=10 Comparison between Galerkin Method and Exact Solution for set 2 of BC’s For this set of boundary conditions the steady state concentration always is going to be 1 for any Pe used.
	www.wam.umd.edu /~wassali/finalproject.doc (2246 words)

	ELECTRIC FIELD DUE TO A ROD OF CHARGE
		Approximate solution: Divide the rod into N equal sections, and approximate each section as a point charge located at its center.
		Exact Solution: As shown in the book, the exact solution is
		A good approach is to plot E versus N. For comparison, plot the exact solution (which doesn’t depend on N, of course) on the same graph.
	bama.ua.edu /~stjones/erod.htm (335 words)

	CSC4.html (Site not responding. Last check: 2007-11-06)
		In this case the exact solution is known (what is it?), so we are in a position to compare the approximate solution generated by Euler's method with the exact value.
		Explain why all the approximate solution curves fall short of this value.
		Hint: Use Maple's fsolve function, applied to the exact solution you found for this problem.
	www.dartmouth.edu /~math3f98/csc_archive/CSC4part1/CSC48.html (134 words)

	Comparisons of Solutions at R=8M_0 (Site not responding. Last check: 2007-11-06)
		We compare the wavetrain from the QE approximation with the exact integrated solution.
		We also compare the exact integrated solution to the "monopole formula." The monopole approximation is given by the equation in the top panel.
		Above we compare the monopole approximation with the exact, integrated solution at three different times.
	www.physics.uiuc.edu /research/cta/movies/scalar/8m (178 words)

	4
		As can be seen from this figure the numerical solution reach the same steady state calculated by the analytical solution (red).
		As can be seen from this figure both solutions reach the same steady state.
		As can be seen from figures 6, 7 and 8 the Galerkin method reach the same steady state as the exact solution.
	www.wam.umd.edu /~wassali/galerkin.htm (487 words)

	Solution of Exact D. E.'s
		We wish to solve the exact D. with a given I. it is necessary that
		The general solution is a family of curves f[x,y] = c, and the
		Now form the general solution to the exact D. Plot several curves in the family of solutions with Mathematica's generic command.
	math.fullerton.edu /mathews/N310/projects/p3.htm (268 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		To # investigate the behavior of the solutions in these circumstances, we # can either construct the direction field and sketch in solution curves # or use numerical methods to approximate the solutions.
		For the differential equation > dy/dx = f(x,y)#, the slope of the line tangent to the solution* at # (xi, yi) is simply f(xi, yi), and the recurrence relation becomes: # yi+1 = yi + f(xi, yi)Dx # so that the change in y is # f(xi, yi)Dx = Dy.
		We shall compare # this exact solution to the solution found using Euler's method.
	www.math.hmc.edu /maple-odes/text/euler.mwt (637 words)

	Exact solution to Richards' equation (Site not responding. Last check: 2007-11-06)
		Many of the current solutions to Richards' equation depend upon special mathematical forms of the diffusivity or conductivity, or are limited to special boundary conditions.
		Fortran programs for the exact solution and appropriate finite difference models used to confirm the results are available for download.
		That means that the "exact" solution grid points with stick with the wetting front, no matter how sharp, at every time and depth.
	www.aquarien.com /gsreq (361 words)

	ipedia.com: Quantum harmonic oscillator Article (Site not responding. Last check: 2007-11-06)
		It is one of the most important problems in quantum mechanics, because a simple exact solution ex...
		It is one of the most important problems in quantum mechanics, because (i) a simple exact solution exists, and (ii) a wide variety of physical situations can be reduced to this.
		This is consistent with the classical harmonic oscillator, in which the particle spends most of its time (and is therefore most likely to be found) at the turning points, where it is the slowest.
	www.ipedia.com /quantum_harmonic_oscillator.html (1595 words)

	Numerical Relaxation
		If we knew the exact answer U(i,j) to the numerical equations (Q) then the error vector e = U-u would be a good measure of how close we were, and we could decide to terminate iterations if the error was small enough.
		The boundary values are of course exact too: g(i,j) = V(i,j) on boundary = U(x(i),y(j))2 by definition of V. So in fact it is corect to use U(x,y) as the exact solution**.
		The Jacobi method may often be improved by mixing the new solution with a fraction of the old one.
	www.cs.colorado.edu /~mcbryan/3656.04/mail/81.htm (3212 words)

	Exact solutions of the Ising model in 1 and 2 dimensions
		Exact solutions of the Ising model in 1 and 2 dimensions
		Exact solutions of the Ising model are possible in 1 and 2 dimensions and can be used to calculate the exact critical exponents for the two corresponding universality classes.
		which are a set of exact exponents for the d=2, n=1 universality class.
	www.nyu.edu /classes/tuckerman/stat.mech/lectures/lecture_26/node2.html (423 words)

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