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| In [[mathematics]], an '''integral equation''' is an equation in which an unknown [[function (mathematics)|function]] appears under an [[integral]] sign. There is a close connection between [[differential equation|differential]] and integral equations, and some problems may be formulated either way. See, for example, [[Maxwell's equations]].
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| ==Overview==
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| The most basic type of integral equation is called a ''[[Fredholm integral equation|Fredholm equation]] of the first type'':
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| :<math> f(x) = \int \limits_a^b K(x,t)\,\varphi(t)\,dt. </math>
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| The notation follows [[George Arfken|Arfken]].
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| Here φ is an unknown function,
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| ''f'' is a known function,
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| and ''K'' is another known function of two variables,
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| often called the [[Kernel (integral operator)|kernel]] function.
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| Note that the limits of integration are constant; this is what characterizes a Fredholm equation.
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| If the unknown function occurs both inside and outside of the integral, it is known as a ''Fredholm equation of the second type'':
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| :<math> \varphi(x) = f(x)+ \lambda \int \limits_a^b K(x,t)\,\varphi(t)\,dt. </math>
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| The parameter λ is an unknown factor,
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| which plays the same role as the [[eigenvalue]] in [[linear algebra]].
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| If one limit of integration is variable, it is called a [[Volterra integral equation|Volterra equation]]. The following are called ''Volterra equations of the first and second types'', respectively:
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| :<math> f(x) = \int \limits_a^x K(x,t)\,\varphi(t)\,dt </math>
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| :<math> \varphi(x) = f(x) + \lambda \int \limits_a^x K(x,t)\,\varphi(t)\,dt. </math>
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| In all of the above, if the known function ''f'' is identically zero, it is called a ''homogeneous integral equation''. If ''f'' is nonzero, it is called an ''inhomogeneous integral equation''.
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| ==Numerical Solution==
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| It is worth noting that Integral Equations often do not have an analytical solution, and must be solved numerically. An example of this is evaluating the [[EFIE|Electric-Field Integral Equation]] (EFIE) or Magnetic-Field Integral Equation (MFIE) over an arbitrarily shaped object in an electromagnetic scattering problem.
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| One method to solve numerically requires discretizing variables and replacing integral by a quadrature rule
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| :<math> \sum_{j=1}^n w_j K(s_i,t_j)u(t_j)=f(s_i) </math>
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| for <math>i=0,1,..,n</math>. Then we have a <math>n</math> equations and <math>n</math> variables system. By solving it we get the value of the <math>n</math> variables <math>u(t_0),u(t_1),...,u(t_n)</math>. | |
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| ==Classification==
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| Integral equations are classified according to three different dichotomies, creating eight different kinds:
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| ;Limits of integration
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| : '''both fixed:''' [[Fredholm equation]]
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| : '''one variable:''' [[Volterra equation]]
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| ;Placement of unknown function
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| : '''only inside integral:''' first kind
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| : '''both inside and outside integral:''' second kind
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| ;Nature of known function ''f''
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| : '''identically zero:''' homogeneous
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| : '''not identically zero:''' inhomogeneous
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| Integral equations are important in many applications. Problems in which integral equations are encountered include [[radiative energy transfer]] and the [[oscillation]] of a string, membrane, or axle. Oscillation problems may also be solved as [[differential equations]].
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| Both Fredholm and Volterra equations are linear integral equations, due to the linear behaviour of φ(x) under the integral. A nonlinear Volterra integral equation has the general form:
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| :<math> \varphi(x) = f(x) + \lambda \int \limits_a^x K(x,t)\,F(x, t, \varphi(t))\,dt. </math>,
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| where F is a known function.
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| ==Wiener-Hopf integral equations==
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| <math> y(t) =\lambda x(t)+\int^{\infty}_0 k(t-s)x(s)ds,\quad 0\leq t<\infty </math>,
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| Originally, such equations were studied in connection with problems in radiative transfer, and more recently,
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| they have been related to the solution of boundary integral equations for planar problems in which the boundary is only piecewise smooth.
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| ==Power series solution for integral equations==
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| In many cases if the Kernel of the integral equation is of the form K(xt) and the
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| [[Mellin transform]] of K(t) exists we can find the solution of the integral equation
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| <math> g(s)=s \int_{0}^{\infty}dtK(st)f(t) </math> in a form of a power series
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| <math> f(t)= \sum_{n=0}^{\infty}\frac{a_{n}}{M(n+1)}x^{n} </math>
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| with <math> g(s)= \sum_{n=0}^{\infty}a_{n} s^{-n} \qquad M(n+1)=\int_{0}^{\infty}dtK(t)t^{n} </math> are the Z-transform of the function g(s) and M(n+1) is the Mellin transform of the Kernel.
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| ==Integral equations as a generalization of eigenvalue equations==
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| Certain homogeneous linear integral equations can be viewed as the [[continuum limit]] of [[Eigenvalue, eigenvector and eigenspace|eigenvalue equations]]. Using [[index notation]], an eigenvalue equation can be written as
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| :<math> \sum _j M_{i,j} v_j = \lambda v_i^{}</math>, | |
| where <math>\mathbf{M}</math> is a matrix, <math>\mathbf{v}</math> is one of its eigenvectors, and <math>\lambda</math> is the associated eigenvalue.
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| Taking the continuum limit, by replacing the discrete indices <math>i</math> and <math>j</math> with continuous variables <math>x</math> and <math>y</math>, gives
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| :<math> \int \, K(x,y)\varphi(y)\mathrm{d}y = \lambda \varphi(x)</math>, | |
| where the sum over <math>j</math> has been replaced by an integral over <math>y</math> and the matrix <math>M_{i,j}</math> and vector <math>v_i</math> have been replaced by the 'kernel' <math>K(x,y)</math> and the [[eigenfunction]] <math>\varphi(y)</math>. (The limits on the integral are fixed, analogously to the limits on the sum over <math>j</math>.) This gives a linear homogeneous Fredholm equation of the second type.
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| In general, <math>K(x,y)</math> can be a [[Distribution (mathematics)|distribution]], rather than a function in the strict sense. If the distribution <math>K</math> has support only at the point <math>x=y</math>, then the integral equation reduces to a [[Eigenfunction|differential eigenfunction equation]].
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| ==See also==
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| * [[Differential equation]]
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| == References ==
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| * Kendall E. Atkinson ''The Numerical Solution of integral Equations of the Second Kind''. Cambridge Monographs on Applied and Computational Mathematics, 1997.
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| * George Arfken and Hans Weber. ''Mathematical Methods for Physicists''. Harcourt/Academic Press, 2000.
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| * Andrei D. Polyanin and Alexander V. Manzhirov ''Handbook of Integral Equations''. CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4.
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| * [[E. T. Whittaker]] and [[G. N. Watson]]. ''A Course of Modern Analysis'' Cambridge Mathematical Library.
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| * Jose Javier Garcia Moreta "http://www.prespacetime.com/index.php/pst/issue/view/42 Borel Resummation & the Solution of Integral Equations , power series solution for integral equation with Kernel K(st)
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| * M. Krasnov, A. Kiselev, G. Makarenko, ''Problems and Exercises in Integral Equations'', Mir Publishers, Moscow, 1971
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| *{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Chapter 19. Integral Equations and Inverse Theory | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=986}}
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| ==External links==
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| * [http://eqworld.ipmnet.ru/en/solutions/ie.htm Integral Equations: Exact Solutions] at EqWorld: The World of Mathematical Equations.
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| * [http://eqworld.ipmnet.ru/en/solutions/eqindex/eqindex-ie.htm Integral Equations: Index] at EqWorld: The World of Mathematical Equations.
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| * {{springer|title=Integral equation|id=p/i051400}}
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| [[Category:Integral equations| ]]
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