Integral equation

An equation is used in the mathematical integral equation , called when the desired function at an integral occurs. Integral equations can be used in science and technology to describe various phenomena . A well-known example of an integral equation with some applications is Abel's integral equation , which historically is one of the first integral equations examined.

The branch of mathematics that deals with integral equations and the compact operators mentioned below is functional analysis .

definition

Linear integral equation

A linear integral equation is an equation for an unknown function and has for the form ${\displaystyle u}$ ${\displaystyle x\in \Omega }$

${\displaystyle \lambda (x)u(x)+\int _{\Omega }k(x,y)u(y)\,\mathrm {d} y=f(x),}$

wherein , , given functions and compact are. The function is called the core . ${\displaystyle \lambda }$ ${\displaystyle f}$ ${\displaystyle k}$ ${\displaystyle \Omega \subseteq \mathbb {R} ^{n}}$ ${\displaystyle k}$

Nonlinear integral equation

A nonlinear integral equation has the form

${\displaystyle \int _{\Omega }K(y,x,u(y))\mathrm {d} y=f(x)}$

with a suitable definition of the core function and a suitable integration area . The function you are looking for is now included in the core function in a non-linear manner. ${\displaystyle K}$ ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle u(y)}$

Classification of linear integral equations

Linear integral equations can be found in

• Integral equations first kind if , ${\displaystyle \lambda (x)\equiv 0}$
• Integral equations of the 2nd kind if , and ${\displaystyle \lambda (x)\equiv \lambda \in \mathbb {C} \setminus \{0\}}$
• Integral equations of the 3rd kind , for all others , ${\displaystyle \lambda }$

organize.

In the case of integral equations of the 1st type, the unknown function you are looking for occurs only in the integral, in the case of such 2nd type also outside. ${\displaystyle u(x)}$

This classification appears arbitrary, but is necessary due to the different analytical properties of the respective types of integral equations. For example, integral equations of the 2nd kind (under weak prerequisites for the kernel) are uniquely solvable for almost all values ​​of , and the solution continuously depends on . In general, this does not apply to integral equations of the first kind (under the same conditions for the kernel). Integral equations of the first kind are such as B. the Laplace transform almost always incorrectly posed problems . The Fourier transform is one of the few exceptions. Type 3 integral equations are usually incorrectly posed problems. ${\displaystyle \lambda }$ ${\displaystyle f}$

If the known function occurring in an integral equation , then the equation is homogeneous , otherwise inhomogeneous . For homogeneous linear equations, the scaled function is also a solution. ${\displaystyle f\equiv 0}$ ${\displaystyle u(x)}$ ${\displaystyle \alpha \cdot u(x)}$

In addition, one can classify integral equations according to their integration limits. If all limits are constant, one speaks of Fredholm integral equations, if one of the limits is variable, the equation is called a Volterra integral equation .

Another classification is based on properties of the core. There are weakly singular and strongly singular integral equations.

Examples

• (linear) Fredholm integral equation type 1, inhomogeneous case:

  ${\displaystyle \int _{a}^{b}k(s,t)u(t)dt=f(s)}$

• (linear) Fredholm integral equation 2nd type, inhomogeneous case:

  ${\displaystyle \lambda \cdot \int _{a}^{b}k(s,t)u(t)dt+f(s)=u(s)}$

The parameter plays a role similar to that of an eigenvalue in linear algebra. ${\displaystyle \lambda }$

• (linear) Fredholm integral equation of the 2nd kind, homogeneous case:

  ${\displaystyle \int _{a}^{b}k(s,t)u(t)dt=u(s)}$

• (linear) Volterra integral equation type 1, inhomogeneous case:

  ${\displaystyle \int _{a}^{s}k(s,t)u(t)dt=f(s)}$

• (linear) Volterra integral equation 2nd type, inhomogeneous case:

${\displaystyle \lambda \cdot \int _{a}^{s}k(s,t)u(t)dt+f(s)=u(s)}$

• Nonlinear Volterra integral equation of the 2nd kind, inhomogeneous case:

${\displaystyle \lambda \cdot \int _{a}^{s}k(s,t)F(s,t,u(t))dt+f(s)=u(s)}$

with a given non-linear function ${\displaystyle F(s,t,u(t))}$

Operator theoretical approach

→ Main article : integral operator

With

${\displaystyle (Ku)(x)=\int _{\Omega }k(x,y)u(y)\,\mathrm {d} y}$

a linear operator is defined for a sufficiently integrable kernel . The theory of compact operators is essential for the theory of (not strongly singular) integral equations . This theory is somewhat similar to that of linear equations in the finite dimensional. Compact operators essentially have pure eigenvalue spectra . More precisely this means: the spectrum consists (possibly apart from zero) only of eigenvalues ​​and these accumulate in at most one point, the zero. All eigenspaces (possibly apart from the zero) are finite-dimensional. ${\displaystyle k(x,y)}$ ${\displaystyle K}$

Historically, too, the theory of integral equations was developed at the beginning of the 20th century as a continuous limit value transition, for example from eigenvalue equations of linear algebra, with eigenvectors now corresponding to eigenfunctions and the matrix a core function.

Duality of integral and differential equations

Integral operators often (but not exclusively) appear in the solution of differential equations , for example in Sturm-Liouville problems , or in partial differential equations in the form of Green's function .

Integro differential equation

An integro-differential equation is an equation in which the derivative of the function to be determined occurs as well as an integral in whose integrand the function sought occurs.

Such equations, like integral or differential equations, can be linear or non-linear. If only ordinary derivatives of the function sought occur, one speaks of an ordinary integro differential equation, if partial derivatives occur, then one speaks of a partial integro differential equation.

An example of this is from the kinetic theory of gases originating Boltzmann equation .

Wiener-Hopf equation and Wiener-Hopf method

The Wiener-Hopf equation is an integral equation which is defined on the positive real semi-axis and in which the kernel depends on the difference of the arguments: ${\displaystyle k(x)}$

${\displaystyle \lambda \cdot u(x)+\int _{0}^{\infty }k(x-s)u(s)=f(x)}$

for . There is a given function (in the case of the homogeneous equation ) and the function you are looking for. is a parameter as above. The core is translation invariant. ${\displaystyle 0\leq x<\infty }$ ${\displaystyle f(x)}$ ${\displaystyle f(x)=0}$ ${\displaystyle u(x)}$ ${\displaystyle \lambda }$

It is essential that one of the edges is in infinity and one in finite.

It is named after Eberhard Hopf and Norbert Wiener , who developed a solution method (Wiener-Hopf method) for it, and is used, for example, for the problem of radiation transport in astrophysics (Milne equation, it is of the Wiener-Hopf type Equation).

The Wiener-Hopf method (also factoring method) is a general method for solving certain integral equations and boundary value problems of certain partial differential equations (such as the wave equation or Laplace equation, for example in optics or electromagnetism), whereby boundaries typically occur that extend into infinity like at the half plane. The Fourier transformation (or also the Laplace transformation or Mellin transformation ) of the functions sought are considered and their complex analytical properties are used. The function and its transform are broken down into two parts , which are defined as analytical functions in the upper and lower complex half-planes (these should only have polynomial growth behavior), but have a section of the real axis in the domain in common. ${\displaystyle \Phi _{\pm }}$

literature

• David Hilbert : Fundamentals of a general theory of linear integral equations . BG Teubner, Leipzig, Berlin 1912.
• Adolf Kneser : The integral equations and their applications in mathematical physics . Friedr. Vieweg & Son, Braunschweig 1922.
• Richard Courant , David Hilbert: Methods of Mathematical Physics Volume 1 . Julius Springer, Berlin 1924. (Third chapter)
• Heinz W. Engel: Integral equations , Springer Verlag Vienna, 1997.

Individual evidence

1. ↑ Eberhard Schock : Integral Equations of the Third Kind. Studia Mathematica, Vol. 81, 1985, pp. 1-11
2. ↑ Integro differential equation . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .
3. ^ VI Dmitriev: Wiener-Hopf equation , Encyclopedia of Mathematics, Springer
4. ^ Wiener, Hopf, On a class of singular integral equations, Preuss meeting reports. Academy of Science Berlin, 1931, pp. 696-706
5. ^ Wiener-Hopf method , Encyclopedia of Mathematics, Springer
6. ^ Wiener-Hopf Method , Mathworld
7. ^ B. Noble: Methods based on the Wiener-Hopf technique for the solution of partial differential equations, Pergamon 1959
8. ^ Morse, Feshbach: Methods of theoretical physics, McGraw Hill 1953, Volume 1, pp. 978ff
9. ↑ Michio Masujima: Applied mathematical methods in theoretical physics, Wiley 2009
10. ^ LA Weinstein: The theory of diffraction and the factorization method, Golem Press, Boulder 1969
11. ↑ Vito Daniele, Rodolfo Zilch: The Wiener-Hopf method in electromagnetics, Scitech Publ. 2014