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Topic: Fredholm equation

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	Fredholm biography
		He solved his operator equation in the particular cases which arise in the study of the physical problem in his thesis (and in the paper which appeared in 1900 based on that thesis) while the general case was solved by Fredholm somewhat later and not published until 1908.
		Fredholm's work on integral equations was met with great interest and boosted the morale and self-respect of Swedish mathematicians who so far had been working under the shadow of the continental cultural empires Germany and France.
		Fredholm received many honours for his mathematical contributions, including the V A Wallmarks Prize for the theory of differential equations in 1903, the Poncelet Prize from the French
	www-history.mcs.st-and.ac.uk /history/Biographies/Fredholm.html (1692 words)

	Springer Online Reference Works
		A linear Fredholm integral equation with a degenerate kernel.
		The conditions for solvability of a degenerate integral equation (1) are given by the Fredholm alternative.
		The importance of degenerate integral equations in the general theory of Fredholm equations is based on the fact that the solution of any Fredholm equation of the second kind can be approximated by solutions of degenerate integral equations in the mean-square (and certain other) metrics to any degree of accuracy.
	eom.springer.de /d/d030810.htm (327 words)

	Integral Equations
		In their simplest form, integral equations are equations in one variable (say t) that involve an integral over a domain of another variable (s) of the product of a kernel function K(s,t) and another (unknown) function (f(s)).
		The integral equation is then reduced to a linear equation with the values of f at the quadrature points being unknown at the outset.
		There is a close analogy between Fredholm equations and linear systems of equations; the functions can be viewed as vectors, the integration over the kernel function as a matrix.
	www.numerical-methods.com /inteq.htm (444 words)

	Springer Online Reference Works
		is a pole of the resolvent (11) of equation (1h) and an eigen value of this latter equation.
		Fredholm extended these theorems to the case of a system of such equations, and also to the case of one class of kernels with a weak singularity (see Integral operator).
		These theorems make it possible to prove the Fredholm theorems for an equation (1) in the case of a variety of concrete classes of integral operators (2), for example if the given and desired functions are square-integrable.
	eom.springer.de /f/f041420.htm (1468 words)

	Springer Online Reference Works
		For a general description of the problems of constructing and investigating numerical methods for solving Fredholm equations of the second kind one uses the language of functional analysis.
		In this case (3) is an integral Fredholm equation with a degenerate kernel (cf.
		A variety of concrete numerical methods for Fredholm equations of the second kind are described in [a2] (with Fortran programs), [a3] and [a4].
	eom.springer.de /f/f041430.htm (1647 words)

	Fredholm integral equation - Wikipedia, the free encyclopedia
		In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators.
		The general theory underlying the Fredholm equations is known as Fredholm theory.
		One of the principal results is that the kernel K is a compact operator, known as the Fredholm operator.
	en.wikipedia.org /wiki/Fredholm_integral_equation (306 words)

	Integral equation - Wikipedia, the free encyclopedia
		In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign.
		There is a close connection between differential and integral equations, and some problems may be formulated either way.
		Problems in which integral equations are encountered include radiative energy transfer and the oscillation of a string, membrane, or axle.
	en.wikipedia.org /wiki/Integral_equation (305 words)

	[No title]
		This dissertation is an examination of the application of the boundary integral equation method to describe axi-symmetric particle motion in Stokes flow.
		The axi-symmetric integral equations are two-dimensional, linear and exhibit a logarithmic singularity via the presence of complete elliptic integrals of the first and second kind.
		The first is a theoretical investigation of Fredholm integral equations of the first kind and the second part is a study of numerical techniques to simulate axi-symmetric particle (rigid and drop) dynamics in Stokes flow.
	rgmia.vu.edu.au /monographs/johnr.html (693 words)

	Fredholm (print-only)
		Fredholm's first publication came in 1890 when he published On a special class of functions in the Royal Swedish Academy of Sciences.
		Fredholm is best remembered for his work on integral equations and spectral theory.
		Fredholm received many honours for his mathematical contributions, including the V A Wallmarks Prize for the theory of differential equations in 1903, the Poncelet Prize from the French Academy of Sciences in 1908, and an honorary doctorate from the University of Leipzig in 1909.
	www-groups.dcs.st-and.ac.uk /~history/Printonly/Fredholm.html (1615 words)

	Springer Online Reference Works
		The set of characteristic numbers of equation (1) is at most countable, with a single possible limit point at infinity.
		When this condition is violated, (3) may turn out to be a non-Fredholm integral equation.
		Instead of the phrases "transposed equation" and "adjoint equation" one sometimes uses "adjoint equation of a Fredholm integral equationadjoint equation" and "conjugate equation of a Fredholm integral equationconjugate equation" (cf.
	eom.springer.de /f/f041470.htm (283 words)

	XII. Geometry and integral operators
		Such equations are called Fredholm equations of the second kind.
		It is a historical accident that differential equations were understood before integral equations.
		Often an integral equation can be converted into a differential equation or vice versa, so many of the laws of nature which we think of as differential equations might just as well have been developed as integral equations initially.
	www.mathphysics.com /pde/green/g12.html (1565 words)

	[No title]
		This equation occurs in the seperation of Laplace’s equation in spherical polar coordinates (r,(,().
		For orthogonality we require that, for n(m, EMBED Equation.2 and the normalising factor is EMBED Equation.2 Laguerre Polynomials The defining equation is EMBED Equation.2 and occurs in the quantum mechanics of the hydrogen atom, x being proportional to the seperation of the nucleus from the electron in the atom.
		If the equations have some symmetry, or there is some conserved quantity (energy in a mechanical system, for example) then E=constant is a closed trajectory.
	www.maths.ox.ac.uk /~murc/lecture_notes/colman_b5die.doc (5437 words)

	My Personal Reading List
		A matrix Fredholm equation of the second kind that is equivalent to the HHP is also deduced.
		A Fredholm equation that is equivalent to the HHP and that was introduced in the preceding paper is used to effect the proofs.
		For the linear singular integral equation with a scalar (i.e., non-matrix) kernel, which solves the inverse problem of this monodromy transform, an equivalent regularization --- a Fredholm linear integral equation of the second kind is constructed in several convenient forms.
	members.localnet.com /~atheneum/bib/waves.html (2557 words)

	RAND \| Research Memoranda \| The Invariant Imbedding Numerical Method for Fredholm Integral Equations with Displacement ...
		A computer program for the solution of a Fredholm integral equation of the second kind with a displacement kernel is given.
		The Fredholm integral equation is transformed into an initial-value problem by treating the interval length as the independent variable.
		The program was used to solve the basic integral equation of radiative transfer, and results were compared with those obtained by Sobolev, by Viskanta, by Bellman, Kagiwada, and Kalaba, and by Heaslet and Warming.
	www.rand.org /pubs/research_memoranda/RM5919 (291 words)

	RAND \| Research Memoranda \| An Initial-value Method for Fredholm Integral Equations with Degenerate Kernels.
		The final step in the mathematical treatment of many problems in such fields as radiative transfer, neutron transport, and optimal filtering theory involves the solution of a Fredholm integral equation in which the kernel is degenerate or can be closely approximated by a degenerate kernel.
		The standard procedure for solving such an equation is to convert it into an equivalent matrix equation and compute the solution by evaluating a number of integrals and performing a matrix inversion.
		In this study, invariant imbedding techniques are used to convert the Fredholm equation into an initial-value problem, and the troublesome matrix inversion is replaced in this formulation by solving a Riccati system of differential equations.
	www.rand.org /pubs/research_memoranda/RM5516 (315 words)

	Corrections to the Integral (Site not responding. Last check: 2007-10-12)
		Applying these corrections to experimental cross sections, the Born cross section can be expressed as follows (this is an expanded form of Equation 2 in EJ95 [27]):
		It is noted that dropping the last term, which is reasonable by the approximation above, this takes the particular form of an inhomogenous Fredholm equation of the second kind which unfortunately has a non-symmetric kernel and is hence not amenable to Green function methods of solution [159].
		for the Born cross section, and consequently of Equation
	hep.bu.edu /~brown/phd/node106.html (706 words)

	The Worst Case Complexity of the Fredholm Equation with Periodic Free Term and Noisy Information (ResearchIndex)
		Abstract: This paper deals with the complexity of the Fredholm equation Lu = f of the second kind with f 2 H (\Gamma) in a periodic setting.
		The problem elements are free term f and belong to the unit ball of H (\Gamma).
		2 The Worst Case Complexity of the Fredholm Equation of the Se..
	citeseer.ist.psu.edu /693652.html (568 words)

	3 Mathematical Techniques for Global Illumination Calculations
		As a Fredholm equation of the second kind [Atk76], the radiance equation cannot be solved analytically, except for trivial cases.
		In this chapter we discuss the theory of finite element technique for both the radiosity equation (3.36) and the radiance equation (2.10).
		In the ray casting techniques the transfer equation is directly integrated along a sample ray using a suitable quadrature rule for sampling the volume data [Sab88, Krü91, Pat93, Sob94].
	graphics.cs.uni-sb.de /~slusallek/Doc/html/node8.html (4870 words)

	Gravity: Maxwell's Equations (Site not responding. Last check: 2007-10-12)
		The Fredholm equation implies that there exists a form A of the same class (in standard texts, *A is used, a 1-form) such that:
		These are Maxwell's equations, and A is the magnetic vector potential (extended to 4D with the electric potential).
		When the structure of the space (Christoffel symbols) is given and the source field is computed from it, it is not necessary to solve Fredholm's equation or to apply Kodaira's theorem.
	www.math.ucla.edu /~jimc/klein_h/maxwell.html (593 words)

	[No title]
		Fredholm integral equations of the first kind arise in the mathematical analysis of many physical problems.
		An important characteristic of such problems is that the information which we seek about a physical quantity A can only be obtained indirectly, by measuring some other quantity B which has some connection with A. Often, this connection can be expressed mathematically in terms of a Fredholm first kind integral equation.
		The first kind Fredholm integral equations is solved by means of the regularization method of Tihonov and Phillips.
	www.cpc.cs.qub.ac.uk /summaries/AABX_v1_0.html (131 words)

	cruise
		The objective of the Ph.D. research is to develop numerical solution techniques for the integral equations of nuclear reactions and scattering:
		Apply this solver to the specific case of the Lippmann-Schwinger integral equation for nuclear reactions/scattering involving large numbers of degrees of freedom.
		Yet, iterative solutions of the Lippmann-Schwinger integral equation generally diverge.
	www.krellinst.org /Fellows/cruise.html (197 words)

	[No title]
		Homework 3 Multidimensional Fredholm integral equation, resolvent operators and eigenvalues of compact integral operators, non-self adjoint compact operator, and solvability of Ku=f.
		Homework 5 Conversion of BVP to integral equations, Fredholm alternative, necessary conditions for solvability, series solutions, eigenfunctions and solvability in systems.
		Separable Fredholm Integral Equations Separable integral equations are equivalent to matrix equations with appropriate definitions.
	www.math.montana.edu /~pernarow/M560/1998/M560.html (663 words)

	The Worst Case Complexity of the Fredholm Equation of the Second Kind with Non-Periodic Free Term and Noise Information ...
		The Worst Case Complexity of the Fredholm Equation with..
		Where Does Smoothness Count the Most for Fredholm Equations of..
		The worst case complexity of the Fredholm equation of the second kind with non-periodic free term and noise information.
	citeseer.ist.psu.edu /jiang98worst.html (636 words)

	The Wavelet Digest :: View topic - Question: Solving fredholm integral equation of the second kind
		The Wavelet Digest :: View topic - Question: Solving fredholm integral equation of the second kind
		Question: Solving fredholm integral equation of the second kind
		Subject: Question: Solving fredholm integral equation of the second kind
	www.wavelet.org /phpBB2/viewtopic.php?t=4351 (182 words)

	DC MetaData for: On the Fredholm ontegral equation for the two-and three-dimensional heat radiation problem. (Site not responding. Last check: 2007-10-12)
		DC MetaData for: On the Fredholm ontegral equation for the two-and three-dimensional heat radiation problem.
		On the Fredholm ontegral equation for the two-and three-dimensional heat radiation problem.
		The author(s) agree, that this abstract may be stored as full text and distributed as such by abstracting services.
	www.math.uni-magdeburg.de /preprints/shadows/03-17report.html (177 words)

	Integral equation of second kind and prime number counting function
		I can not calculate Pi(100000000000000) or any value by myself, because i only know how to solve an integral equation by means of series.
		That is a complete misuse of "solved": you have found an expansion (allegedly) in terms of some other sets of functions (I don't recall you posting here the formlae for the set of orthonormal functions).
		If you're equation is any good, then use it, go on, demonstrate that you can calculate pi(10,000,000,000) in a reasonalbe number of steps with some accuracy.
	www.physicsforums.com /showthread.php?p=239310 (1317 words)

	A fredholm integral equation method for scattering phase shifts
		A method is presented for obtaining exact scattering phase shifts using an integral equation formulation.
		It is similar to recent work by Reinhardt and Szabo but differs in that in the integral equation used, no inherent singularities have to be avoided.
		The method has been applied to both the exponential and screened coulomb potentials, and has proved to be complementary to the method of numerically integrating the radial equation-this Fredholm integral method converges more quickly for non-zero angular momentum and/or high impact energies.
	stacks.iop.org /0022-3700/5/497 (275 words)

	CSIRO PUBLISHING - Marine & Freshwater Research
		Estimation of age composition was considered in the context of non-parametric mixture distribution problems.
		The problem was formulated as a Fredholm first-kind equation with respect to the distribution function of age.
		The solution of the equation was obtained using methods for ‘ill-posed problems’.
	www.publish.csiro.au /?paper=MF04152 (171 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		One of our goals is to connect the Fredholm determinant approach of Froese to the Fourier transform approach of Zworski.
		Another is to prove a result on the number of antibound states --- namely, in a half-line problem there are an odd number of antibound states between any two bound states.
		Let $u(x,\kappa)$ solve the equation $-u'' + qu = -\kappa^2 u$ with $u(0)=0$, $u'(0) =1$.
	www.ma.utexas.edu /mp_arc/papers/00-220 (3439 words)

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