Injective module

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. See, for example, Maxwell's equations.

Overview

The most basic type of integral equation is called a Fredholm equation of the first type:

$f(x)=\int \limits _{a}^{b}K(x,t)\,\varphi (t)\,dt.$

The notation follows Arfken. Here φ is an unknown function, f is a known function, and K is another known function of two variables, often called the kernel function. Note that the limits of integration are constant; this is what characterizes a Fredholm equation.

If the unknown function occurs both inside and outside of the integral, it is known as a Fredholm equation of the second type:

$\varphi (x)=f(x)+\lambda \int \limits _{a}^{b}K(x,t)\,\varphi (t)\,dt.$

The parameter λ is an unknown factor, which plays the same role as the eigenvalue in linear algebra.

If one limit of integration is variable, it is called a Volterra equation. The following are called Volterra equations of the first and second types, respectively:

$f(x)=\int \limits _{a}^{x}K(x,t)\,\varphi (t)\,dt$

$\varphi (x)=f(x)+\lambda \int \limits _{a}^{x}K(x,t)\,\varphi (t)\,dt.$

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.

Numerical Solution

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 Electric-Field Integral Equation (EFIE) or Magnetic-Field Integral Equation (MFIE) over an arbitrarily shaped object in an electromagnetic scattering problem.

One method to solve numerically requires discretizing variables and replacing integral by a quadrature rule

$\sum _{j=1}^{n}w_{j}K(s_{i},t_{j})u(t_{j})=f(s_{i})$

for $i=0,1,..,n$. Then we have a $n$ equations and $n$ variables system. By solving it we get the value of the $n$ variables $u(t_{0}),u(t_{1}),...,u(t_{n})$.

Classification

Integral equations are classified according to three different dichotomies, creating eight different kinds:

Limits of integration

both fixed: Fredholm equation

one variable: Volterra equation

Placement of unknown function

only inside integral: first kind

both inside and outside integral: second kind

Nature of known function f

identically zero: homogeneous

not identically zero: inhomogeneous

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.

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:

$\varphi (x)=f(x)+\lambda \int \limits _{a}^{x}K(x,t)\,F(x,t,\varphi (t))\,dt.$,

where F is a known function.

Wiener-Hopf integral equations

$y(t)=\lambda x(t)+\int _{0}^{\infty }k(t-s)x(s)ds,\quad 0\leq t<\infty$,

Originally, such equations were studied in connection with problems in radiative transfer, and more recently, they have been related to the solution of boundary integral equations for planar problems in which the boundary is only piecewise smooth.

Power series solution for integral equations

In many cases if the Kernel of the integral equation is of the form K(xt) and the Mellin transform of K(t) exists we can find the solution of the integral equation

$g(s)=s\int _{0}^{\infty }dtK(st)f(t)$ in a form of a power series

$f(t)=\sum _{n=0}^{\infty }{\frac {a_{n}}{M(n+1)}}x^{n}$

with $g(s)=\sum _{n=0}^{\infty }a_{n}s^{-n}\qquad M(n+1)=\int _{0}^{\infty }dtK(t)t^{n}$ are the Z-transform of the function g(s) and M(n+1) is the Mellin transform of the Kernel.

Integral equations as a generalization of eigenvalue equations

Certain homogeneous linear integral equations can be viewed as the continuum limit of eigenvalue equations. Using index notation, an eigenvalue equation can be written as

$\sum _{j}M_{i,j}v_{j}=\lambda v_{i}^{}$,

where $\mathbf {M}$ is a matrix, $\mathbf {v}$ is one of its eigenvectors, and $\lambda$ is the associated eigenvalue.

Taking the continuum limit, by replacing the discrete indices $i$ and $j$ with continuous variables $x$ and $y$, gives

$\int \,K(x,y)\varphi (y)\mathrm {d} y=\lambda \varphi (x)$,

where the sum over $j$ has been replaced by an integral over $y$ and the matrix $M_{i,j}$ and vector $v_{i}$ have been replaced by the 'kernel' $K(x,y)$ and the eigenfunction $\varphi (y)$. (The limits on the integral are fixed, analogously to the limits on the sum over $j$.) This gives a linear homogeneous Fredholm equation of the second type.

In general, $K(x,y)$ can be a distribution, rather than a function in the strict sense. If the distribution $K$ has support only at the point $x=y$, then the integral equation reduces to a differential eigenfunction equation.

See also

• Differential equation

References

• Kendall E. Atkinson The Numerical Solution of integral Equations of the Second Kind. Cambridge Monographs on Applied and Computational Mathematics, 1997.
• George Arfken and Hans Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000.
• Andrei D. Polyanin and Alexander V. Manzhirov Handbook of Integral Equations. CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4.
• E. T. Whittaker and G. N. Watson. A Course of Modern Analysis Cambridge Mathematical Library.
• 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)
• M. Krasnov, A. Kiselev, G. Makarenko, Problems and Exercises in Integral Equations, Mir Publishers, Moscow, 1971
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External links

• Integral Equations: Exact Solutions at EqWorld: The World of Mathematical Equations.
• Integral Equations: Index at EqWorld: The World of Mathematical Equations.
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