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Topic: Integral equation


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  Fredholm integral equation - Wikipedia, the free encyclopedia
An inhomogeneous Fredholm equation of the first kind is written:
An inhomogeneous Fredholm equation of the second kind is essentially a form of the eigenvalue problem for the above equation:
Integral Equations at EqWorld: The World of Mathematical Equations.
en.wikipedia.org /wiki/Fredholm_integral_equation   (205 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   (301 words)

  
 Encyclopedia: Integral equation   (Site not responding. Last check: 2007-10-08)
In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables.
Maxwells equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter.
The most basic type of integral equation is a Fredholm equation of the first type: In mathematics, the Fredholm integral equation introduced by Ivar Fredholm gives rises to a Fredholm operator.
www.nationmaster.com /encyclopedia/Integral-equation   (669 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)).
Integral equations are solved by replacing the integral by a numerical integration or quadrature formula.
The integral equation is then reduced to a linear equation with the values of f at the quadrature points being unknown at the outset.
www.numerical-methods.com /inteq.htm   (444 words)

  
 6.3 Integral Equations for the Population Activity
The integral equation (6.44) that we have derived in Section 6.2 via integration of the density equations turns out to be a specific instance of the general framework presented in this section.
Equation (6.77) tells us that in a homogeneous population of neurons the population activity in a stationary state is equal to the firing rate of individual neurons.
The difference between the numerical solution of the Wilson-Cowan integral equation (6.76) and that of the coarse-grained equation (6.86) is shown in Fig.
diwww.epfl.ch /~gerstner/SPNM/node47.html   (2296 words)

  
 [No title]   (Site not responding. Last check: 2007-10-08)
The integral of dI is I. In words those equations mean: The derivative equation: The total current passing through a plane divided by the surface area of that plane (=dA) is called the current density.
The integral equation: The total current through a plane is equal to the current density times the surface area of the plane.
The integral of dV is V. In words those equations mean: For the best interpretation of those equations it is good to realize that the electric potential is a potential energy per unit charge, and that the electric field is a force per unit charge.
uweb.txstate.edu /~WG06/PHYS2425/currentsderivintegr.doc   (582 words)

  
 Equation   (Site not responding. Last check: 2007-10-08)
Although the basic theory of structural equation modeling is an extension of single- equation regression concepts, the matrix language that is typically...
In equations, the values of the variables for which the equation is true are called solutions.
Thus to solve the equation, one must express those values of the variables that are solutions in terms of whatever constants may be included.
hallencyclopedia.com /Equation   (447 words)

  
 Integral Equation Techniques in Computational Electromagnetics   (Site not responding. Last check: 2007-10-08)
The equations are usually discretized using the method of moments in which an unknown physical quantity is expanded in terms of a set of known expansion functions.
The fringe integral equation is not affected by this, since the current peak is treated with analytical techniques and the method of moments is used to solve for the well-behaved fringe current only.
The fringe integral equation technique was recently used to model the radiation pattern of the Danish Rømer satellite and its crossed-slot antenna.
www.emi.dtu.dk /research/afg/research/computional_electromagnetics   (826 words)

  
 Integral Equation Formulation for Ground-Water Flow   (Site not responding. Last check: 2007-10-08)
Advantages of boundary integral equation methods (BIEM) for the analysis of two-dimensional flows through porous media are numerous.
This formulation has the advantage that the equation used for any unknown is a Fredholm equation of the second kind which results in well-conditioned equations and allows nodes to be closely spaced in the region of singularities.
Computational algorithms for solution of the equations by a simple iterative method and by direct solution of the full set of simultaneous linear equations are developed.
www.pubs.asce.org /WWWdisplay.cgi?5016539   (140 words)

  
 Integral equation methods
The current equations (or, rather, the equations for the tangential components of the fields) are to be solved for iteratively.
Then, integral equations are derived for the complementary fields at the surface.
An integral equation  is defined by a Kernel K, and symbolically, it may be described in generality by
europa.ieec.fcr.es /hosted/oppscat/KNMI_slides/node17.html   (280 words)

  
 Section II - The Integral Equations For Free Space
The vector integral in equation (5) can be reduced to a scalar integral equation when the conducting surface S is that of a cylindrical thin wire, thereby making the solution much easier.
Equation (5) then becomes Equation6 where the integration is over the length of the wire.
The thin-wire form of the integral in equation (7) is used over wires while the more general form of equation (5) must be used on surfaces.
www.nec2.org /part_1/section2.html   (827 words)

  
 Integral equation   (Site not responding. Last check: 2007-10-08)
In mathematics, an integral equation is an equation inwhich an unknown function appears under an integral sign.
If the unknown function appears only under the integral sign, it is called an integral equation of the first kind.If the unknown function appears both inside and outside the integral sign, it is called an integral equation of the secondkind.
Problems in which integral equations are encountered include radiative energytransfer and oscillation of a string, membrane, or axle.
www.therfcc.org /integral-equation-32832.html   (228 words)

  
 Institute of Mathematics of National Academy of Sciences of Armenia   (Site not responding. Last check: 2007-10-08)
The paper studies the integral equation $f(x,y)=g(x,y)+\iint\limits_GK(x-x',y-y')f(x',y')\;dx'\;dy',$ where $G$ is an unbounded domain in $\R^2$, the kernel $K$ is nonnegative function satisfying the conservativity condition.
We consider renewal equation $f(x)=g(x)+\int_0^x V(x-t)f(t)dt,$ where the kernel $V$ is completely monotone function and prove that if $g\in L_1(0,\y)$ is a completely monotone function of the form $g(x)=\int_a^b e^{-xs}G(s)d\si(s),$ and $G\ge 0$ is a continuous increasing function on $[a,b)$, then the solution is also completely monotone.
For convolution-type integral equations on finite intervals with completely monotone kernels a method for construction of approximate solution is proposed.
math.sci.am /Journal/1997_1.html   (464 words)

  
 AWM's Adsorption page - General/Global integral equation (GIE)
Dilute single-solute adsorption is a special case of binary liquid mixture adsorption where one of the components (solvent) is present in the solution bulk in almost constant concentration.
General Integral Equation of Adsorption (GIE) is a general formulation of adsorption isotherm involving energetical heterogeneity of the adsorption system (i.e.
Another similar in character Integral Equation of Adsorption is the Stoeckli integral equation for adsorption on microporous non-uniform solids.
adsorption.org /awm/ads/GIEc.htm   (704 words)

  
 Integral Equation Methods
In electromagnetic integral equation methods the original boundary value problems for Maxwell's equations are reformulated as integral equations over the boundary interfaces of homogeneous domains.
If the object is inhomogeneous, integral equations over the entire volume of the object has to be considered.
Although the integral equations are in many cases flexible and efficient method for solving various electromagnetic problems, an additional numerical difficulty arises from the singular integral kernels.
www.rni.helsinki.fi /em/EM_int_eq_meth.html   (548 words)

  
 Variational Boundary Integral Equation
The non-regularised variational equation in (3.100) is used as the basis for the Galerkin BIEM.
The variational equation in (3.103) has been considered useful in the standard Galerkin BIEM since the kernel function is regularised to an integrable one.
Actually, FMM formulations with regularised integral equations have been tested in section 3.3 with collocation and were shown to introduce more moments than formulations without regularisation.
gspsun1.gee.kyoto-u.ac.jp /yoshida/doctoral_thesis/node53.html   (274 words)

  
 AWM's Adsorption page - General/Global integral equation (GIE)
General Integral Equation of Adsorption (GIEA) is a general formulation of adsorption isotherm involving energetic heterogeneity of the adsorption system (i.e.
The General Integral Equation of Adsorption may be easily reformulated in a way allowing for use of integral (cumulative) energy distribution function F(E) instead of differential function χ(E).
General Integral Equation of Adsorption (and its special cases for gas and vapor, solute and binary liquid adsorption), see also (in preparation) GIEA for multicomponent systems and its applications in prediction of mutlicomponent adsorption
adsorption.org /awm/ads/En-dist.htm   (1028 words)

  
 integral equation --  Encyclopædia Britannica
The first type approximates the unknown function in the equation by a simpler function, often a polynomial or piecewise...
Using the expression for T above and the divergence theorem of multivariable calculus, which states that integrals over the area of a closed surface S, with integrand nf(x), may be rewritten as integrals over the volume V enclosed by S, with integrand f(x)/x; when f(x) is a differentiable...
He is known for his work with integral equations—mathematical expressions that are used to compute the area beneath a curve.
www.britannica.com /eb/article-9042519?tocId=9042519   (799 words)

  
 Integral equation -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-08)
There is a close connection between (A quality that differentiates between similar things) differential and integral equations, and some problems may be formulated either way.
If the limits of integration are fixed, it is called a (Click link for more info and facts about Fredholm equation) Fredholm equation.
Oscillation problems may also be solved as (An equation containing differentials of a function) differential equations.
www.absoluteastronomy.com /encyclopedia/i/in/integral_equation.htm   (297 words)

  
 DIE
Each paper requires a separate page containing an abbreviated version of the title for the running head no more than 40 characters long, and the name and mailing address of the author to whom proofs should be sent.
Equations should be centered with the number of the equation placed in parentheses at the right margin.
Each figure is to be drafted in high contrast on a separate piece of white paper in a form suitable for photographic reproduction and reduction.
www.aftabi.com /DIE/die.html   (449 words)

  
 The Numerical Solution Of A Nonlinear Boundary Integral Equation On Smooth Surfaces - Atkinson (ResearchIndex)   (Site not responding. Last check: 2007-10-08)
This nonlinear boundary value problem is reformulated as a nonlinear boundary integral equation, with u on the boundary as the solution being sought.
The integral equation is solved numerically by using the collocation method, with piecewise quadratic functions used as approximations to u.
26 Piecewise polynomial collocation for boundary integral equat..
citeseer.ist.psu.edu /atkinson94numerical.html   (658 words)

  
 Prof. Weile's Research
Differential equation methods solve for the actual fields, whereas integral equation methods solve for electric currents.
Moreover, if the object under study is homogeneous, the unknown current in the integral equation exists only on the surface, as opposed to filling the whole volume of the structure.
Differential equation methods, on the other hand, lead to sparse systems of equations (i.e., most of the coefficients in the equations are zero) but they always require volumetric meshing and special approximate boundary conditions to terminate the simulation.
www.eecis.udel.edu /~weile/research.html   (827 words)

  
 Boundary Element Methods
In general, a method can be derived by replacing the integrals in an integral equation by a quadrature formula or by a weighted residual method such as the Galerkin method.
There are two fundamental approaches to the derivation of an integral equation formulation of a partial differential equation.
The first is often termed the direct method and the integral equations are derived through the application of Green's second theorem.
www.personal.psu.edu /users/y/u/yuh104/bem.html   (459 words)

  
 Integrals in Kinetic Molecular Theory, erf, and etc.
The integral in Equation 6c can be done by elementary methods.
This integral can be calculated, as was done in Equations 6 and 7, by changing variables.
These integrals can be easily obtained in the same manner that we obtained the integrals in Equations 11 and 12.
www.chem.arizona.edu /~salzmanr/480a/480ants/integral/integral.html   (853 words)

  
 On An Integral Equation For Center Manifolds: A Direct Proof For Nonautonomous Differential Equations On Banach Spaces ...   (Site not responding. Last check: 2007-10-08)
We study a nonlinear integral equation for a center manifold of a semilinear nonautonomous differential equation having mild solutions.
We prove that this integral operator can be factorized as a composition of a nonlinear substitution operator and a...
21 Dichotomies for linear evolutionary equations in Banach spac..
citeseer.ist.psu.edu /1796.html   (629 words)

  
 Local Estimation for an Integral Equation of First Kind with Analytic Kernel - Cheng, Proessdorf, Yamamoto ...
Abstract: In this paper, an integral equation of the ørst kind with Riesz kernel is discussed.
Since the kernel of this integral equation is analytic, this problem is severe ill-posed.
We prove that, for solutions of the integral equation, a local conditional pointwise estimate holds at a point if the solution has some additional smoothness properties in a neighbourhood of this point.
citeseer.ist.psu.edu /cheng98local.html   (453 words)

  
 IP6 Fast Integral Equation Methods for Composite Materials   (Site not responding. Last check: 2007-10-08)
Integral equation methods are natural candidates for the numerical solution of such problems, since they discretize the boundary alone and can achieve a high order of accuracy.
Recent fast algorithms, such as the Fast Multipole Method, allow for the dense N by N matrices which arise from integral equation discretizations to be applied in O(N) operations.
The speaker will describe several fast integral equation methods which allow for large-scale calculations (hundreds of thousands of interface points) to be carried out with ease on modest scientific workstations.
www.siam.org /meetings/archives/ms97/ip6.htm   (153 words)

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