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Integral equation

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In mathematical analysis, integral equations are equations in which an unknown function appears under an integral sign.<ref name=":0" /> In mathematical notation, integral equations may thus be expressed as being of the form: <math display="block">f(x_1,x_2,x_3,\ldots,x_n ; u(x_1,x_2,x_3,\ldots,x_n) ; I^1 (u), I^2(u), I^3(u), \ldots, I^m(u)) = 0</math> where <math>I^i(u)</math> is an integral operator acting on u. Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals. A direct comparison can be seen with the mathematical form of the general integral equation above with the general form of a differential equation which may be expressed as follows:<math display="block">f(x_1,x_2,x_3,\ldots,x_n ; u(x_1,x_2,x_3,\ldots,x_n) ; D^1 (u), D^2(u), D^3(u), \ldots, D^m(u)) = 0</math>where <math>D^i(u)</math> may be viewed as a differential operator of order i.<ref name=":0" /> Due to this close connection between differential and integral equations, one can often convert between the two. For example, one method of solving a boundary value problem is by converting the differential equation with its boundary conditions into an integral equation and solving the integral equation.<ref name=":0" /> In addition, because one can convert between the two, differential equations in physics such as Maxwell's equations often have an analog integral and differential form.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> See also, for example, Green's function and Fredholm theory.

Classification and overviewEdit

Various classification methods for integral equations exist. A few standard classifications include distinctions between linear and nonlinear; homogeneous and inhomogeneous; Fredholm and Volterra; first order, second order, and third order; and singular and regular integral equations.<ref name=":0" /> These distinctions usually rest on some fundamental property such as the consideration of the linearity of the equation or the homogeneity of the equation.<ref name=":0" /> These comments are made concrete through the following definitions and examples:

LinearityEdit

Template:Em: An integral equation is linear if the unknown function u(x) and its integrals appear linearly in the equation.<ref name=":0" /> Hence, an example of a linear equation would be:<ref name=":0" /><math display="block">u(x) = f(x) + \lambda\int_{\alpha(x)}^{\beta(x)}K(x,t) \cdot u(t) \, dt</math>As a note on naming convention: i) u(x) is called the unknown function, ii) f(x) is called a known function, iii) K(x,t) is a function of two variables and often called the Kernel function, and iv) λ is an unknown factor or parameter, which plays the same role as the eigenvalue in linear algebra.<ref name=":0" />

Template:Em: An integral equation is nonlinear if the unknown function ''u(x) or any of its integrals appear nonlinear in the equation.<ref name=":0" /> Hence, examples of nonlinear equations would be the equation above if we replaced u(t) with <math>u^2(x), \, \, \cos(u(x)), \, \text{or } \,e^{u(x)}</math>, such as:<math display="block">u(x) = f(x) + \int_{\alpha(x)}^{\beta(x)}K(x,t) \cdot u^2(t) \, dt</math>Certain kinds of nonlinear integral equations have specific names.<ref name=":2" /> A selection of such equations are:<ref name=":2" />

  • Nonlinear Volterra integral equations of the second kind which have the general form: <math> u(x) = f(x) + \lambda \int_a^x K(x,t) \, F(x, t, u(t)) \, dt, </math> where Template:Mvar is a known function.<ref name=":2" />
  • Nonlinear Fredholm integral equations of the second kind which have the general form: <math>f(x)=F\left(x, \int_a^b K(x,y,f(x),f(y)) \, dy\right)</math>.<ref name=":2" />
  • A special type of nonlinear Fredholm integral equations of the second kind is given by the form: <math>f(x)=g(x)+ \int_a^b K(x,y,f(x),f(y)) \, dy</math>, which has the two special subclasses:<ref name=":2" />
    • Urysohn equation: <math>f(x)=g(x)+ \int_a^{b} k(x,y,f(y)) \, dy</math>.<ref name=":2" />
    • Hammerstein equation: <math>f(x)=g(x)+ \int_a^b k(x,y) \, G(y,f(y)) \, dy</math>.<ref name=":2" />

More information on the Hammerstein equation and different versions of the Hammerstein equation can be found in the Hammerstein section below.

Location of the unknown equationEdit

Template:Em: An integral equation is called an integral equation of the first kind if the unknown function appears only under the integral sign.<ref name=":2" /> An example would be: <math> f(x) = \int_a^b K(x,t)\,u(t)\,dt </math>.<ref name=":2" />

Template:Em: An integral equation is called an integral equation of the second kind if the unknown function also appears outside the integral.<ref name=":2" />

Template:Em: An integral equation is called an integral equation of the third kind if it is a linear Integral equation of the following form:<ref name=":2" /><math display="block"> g(t)u(t) + \lambda \int_a^b K(t,x)u(x) \, dx = f(t) </math>where g(t) vanishes at least once in the interval [a,b]<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> or where g(t) vanishes at a finite number of points in (a,b).<ref>Template:Cite journal</ref>

Limits of IntegrationEdit

Fredholm: An integral equation is called a Fredholm integral equation if both of the limits of integration in all integrals are fixed and constant.<ref name=":0" /> An example would be that the integral is taken over a fixed subset of <math>\mathbb{R}^n</math>.<ref name=":2" /> Hence, the following two examples are Fredholm equations:<ref name=":0" />

  • Fredholm equation of the first type: <math> f(x) = \int_a^b K(x,t)\,u(t)\,dt </math>.
  • Fredholm equation of the second type: <math> u(x) = f(x)+ \lambda \int_a^b K(x,t) \, u(t) \, dt. </math>

Note that we can express integral equations such as those above also using integral operator notation.<ref name=":1" /> For example, we can define the Fredholm integral operator as:<math display="block">(\mathcal{F}y)(t) := \int_{t_0}^T K(t,s) \, y(s) \, ds.</math>Hence, the above Fredholm equation of the second kind may be written compactly as:<ref name=":1" /><math display="block">y(t)=g(t)+\lambda(\mathcal{F}y)(t).</math>

Template:Em: An integral equation is called a Volterra integral equation if at least one of the limits of integration is a variable.<ref name=":0" /> Hence, the integral is taken over a domain varying with the variable of integration.<ref name=":2" /> Examples of Volterra equations would be:<ref name=":0" />

  • Volterra integral equation of the first kind: <math> f(x) = \int_a^x K(x,t) \, u(t) \, dt </math>
  • Volterra integral equation of the second kind: <math> u(x) = f(x) + \lambda \int_a^x K(x,t)\,u(t)\,dt. </math>

As with Fredholm equations, we can again adopt operator notation. Thus, we can define the linear Volterra integral operator <math>\mathcal{V} : C(I) \to C(I)</math>, as follows:<ref name=":2" /><math display="block">(\mathcal{V} \varphi)(t) := \int_{t_0}^t K(t,s) \, \varphi(s) \, ds</math>where <math>t \in I = [t_0 , T]</math> and K(t,s) is called the kernel and must be continuous on the interval <math>D := \{(t,s) : 0 \leq s \leq t \leq T \leq \infty\}</math>.<ref name=":2" /> Hence, the Volterra integral equation of the first kind may be written as:<ref name=":2" /><math display="block">(\mathcal{V}y)(t)=g(t)</math>with <math>g(0)=0</math>. In addition, a linear Volterra integral equation of the second kind for an unknown function <math> y(t) </math> and a given continuous function <math> g(t) </math> on the interval <math> I </math> where <math> t \in I </math>:<math display="block">y(t)=g(t)+(\mathcal{V} y)(t).</math>Template:Em: In higher dimensions, integral equations such as Fredholm–Volterra integral equations (VFIE) exist.<ref name=":2" /> A VFIE has the form:<math display="block">u(t,x) = g(t,x)+(\mathcal{T}u)(t,x)</math>with <math>x \in \Omega</math> and <math>\Omega</math> being a closed bounded region in <math>\mathbb{R}^d</math> with piecewise smooth boundary.<ref name=":2" /> The Fredholm-Volterra Integral Operator <math>\mathcal{T} : C(I \times \Omega) \to C(I \times \Omega)</math> is defined as:<ref name=":2" />

<math display="block">(\mathcal{T}u)(t,x) := \int_0^t \int_\Omega K(t,s,x,\xi) \, G(u(s, \xi)) \, d\xi \, ds.</math>Note that while throughout this article, the bounds of the integral are usually written as intervals, this need not be the case.<ref name=":1" /> In general, integral equations don't always need to be defined over an interval <math>[a,b] = I</math>, but could also be defined over a curve or surface.<ref name=":1" />

HomogeneityEdit

Template:Em: An integral equation is called homogeneous if the known function <math>f</math> is identically zero.<ref name=":0" />

Template:Em: An integral equation is called inhomogeneous if the known function <math>f</math> is nonzero.<ref name=":0" />

RegularityEdit

Template:Em: An integral equation is called regular if the integrals used are all proper integrals.<ref name=":1" />

Template:Em or Template:Em: An integral equation is called singular or weakly singular if the integral is an improper integral.<ref name=":1" /> This could be either because at least one of the limits of integration is infinite or the kernel becomes unbounded, meaning infinite, on at least one point in the interval or domain over which is being integrated.<ref name=":0" />

Examples include:<ref name=":0" /><math display="block">F(\lambda) = \int_{-\infty}^\infty e^{-i\lambda x} u(x) \, dx</math><math display="block">L[u(x)] = \int_0^\infty e^{-\lambda x} u(x) \, dx</math>These two integral equations are the Fourier transform and the Laplace transform of u(x), respectively, with both being Fredholm equations of the first kind with kernel <math>K(x,t)=e^{-i\lambda x}</math> and <math>K(x,t)=e^{-\lambda x}</math>, respectively.<ref name=":0" /> Another example of a singular integral equation in which the kernel becomes unbounded is:<ref name=":0" /> <math display="block">x^2= \int_0^x \frac{1}{\sqrt{x-t}} \, u(t) \, dt.</math>This equation is a special form of the more general weakly singular Volterra integral equation of the first kind, called Abel's integral equation:<ref name=":1" /> <math display="block">g(x)=\int_a^{x} \frac{f(y)}{\sqrt{x-y}} \, dy</math>Template:Em: An integral equation is called strongly singular if the integral is defined by a special regularisation, for example, by the Cauchy principal value.<ref name=":1" />

Integro-differential equationsEdit

An Integro-differential equation, as the name suggests, combines differential and integral operators into one equation.<ref name=":0" /> There are many version including the Volterra integro-differential equation and delay type equations as defined below.<ref name=":2" /> For example, using the Volterra operator as defined above, the Volterra integro-differential equation may be written as:<ref name=":2" /><math display="block">y'(t)=f(t, y(t))+(V_\alpha y)(t)</math>For delay problems, we can define the delay integral operator <math>(\mathcal{W}_{\theta , \alpha} y)</math> as:<ref name=":2" /><math display="block">(\mathcal{W}_{\theta , \alpha} y)(t) := \int_{\theta(t)}^t (t-s)^{-\alpha} \cdot k_2(t,s,y(s), y'(s)) \, ds </math>where the delay integro-differential equation may be expressed as:<ref name=":2" /> <math display="block">y'(t)=f(t, y(t), y(\theta (t)))+(\mathcal{W}_{\theta , \alpha} y)(t).</math>

Volterra integral equationsEdit

Uniqueness and existence theorems in 1DEdit

The solution to a linear Volterra integral equation of the first kind, given by the equation:<math display="block">(\mathcal{V}y)(t)=g(t)</math>can be described by the following uniqueness and existence theorem.<ref name=":2" /> Recall that the Volterra integral operator <math>\mathcal{V} : C(I) \to C(I)</math>, can be defined as follows:<ref name=":2" /><math display="block">(\mathcal{V} \varphi)(t) := \int_{t_0}^t K(t,s) \, \varphi(s) \, ds</math>where <math>t \in I = [t_0 , T]</math> and K(t,s) is called the kernel and must be continuous on the interval <math>D := \{(t,s) : 0 \leq s \leq t \leq T \leq \infty\}</math>.<ref name=":2" /> Template:Math theorem The solution to a linear Volterra integral equation of the second kind, given by the equation:<ref name=":2" /><math display="block">y(t)=g(t)+(\mathcal{V} y)(t)</math>can be described by the following uniqueness and existence theorem.<ref name=":2" /> Template:Math theorem

Volterra integral equations in Template:MathEdit

A Volterra Integral equation of the second kind can be expressed as follows:<ref name=":2" /><math display="block">u(t,x) = g(t,x)+\int_0^x \int_0^y K(x,\xi, y, \eta) \, u(\xi, \eta) \, d\eta \, d\xi</math>where <math>(x,y) \in \Omega := [0,X] \times [0,Y]</math>, <math>g \in C( \Omega)</math>, <math>K \in C(D_2)</math> and <math>D_2 := \{(x, \xi,y,\eta): 0 \leq \xi \leq x \leq X, 0 \leq \eta \leq y \leq Y\}</math>.<ref name=":2" /> This integral equation has a unique solution <math>u \in C( \Omega)</math> given by:<ref name=":2" /><math display="block">u(t,x) = g(t,x)+\int_0^x \int_0^{y} R(x,\xi, y, \eta) \, g(\xi, \eta) \, d\eta \, d\xi</math>where <math>R</math> is the resolvent kernel of K.<ref name=":2" />

Uniqueness and existence theorems of Fredholm–Volterra equationsEdit

As defined above, a VFIE has the form:<math display="block">u(t,x) = g(t,x)+(\mathcal{T}u)(t,x)</math>with <math>x \in \Omega</math> and <math>\Omega</math> being a closed bounded region in <math>\mathbb{R}^d</math> with piecewise smooth boundary.<ref name=":2" /> The Fredholm–Volterrra Integral Operator <math>\mathcal{T} : C(I \times \Omega) \to C(I \times \Omega)</math> is defined as:<ref name=":2" /><math display="block">(\mathcal{T}u)(t,x) := \int_0^t \int_\Omega K(t,s,x,\xi) \, G(u(s, \xi)) \, d\xi \, ds.</math>In the case where the Kernel K may be written as <math>K(t,s,x,\xi) = k(t-s)H(x, \xi)</math>, K is called the positive memory kernel.<ref name=":2" /> With this in mind, we can now introduce the following theorem:<ref name=":2" /> Template:Math theorem

Special Volterra equationsEdit

A special type of Volterra equation which is used in various applications is defined as follows:<ref name=":2" /><math display="block">y(t)=g(t)+(V_\alpha y)(t)</math>where <math>t \in I = [t_0 , T]</math>, the function g(t) is continuous on the interval <math>I</math>, and the Volterra integral operator <math>(V_\alpha t)</math> is given by:<math display="block">(V_\alpha t)(t) := \int_{t_0}^t (t-s)^{-\alpha} \cdot k(t,s,y(s)) \, ds </math>with <math>(0 \leq \alpha < 1)</math>.<ref name=":2" />

Converting IVP to integral equationsEdit

In the following section, we give an example of how to convert an initial value problem (IVP) into an integral equation. There are multiple motivations for doing so, among them being that integral equations can often be more readily solvable and are more suitable for proving existence and uniqueness theorems.<ref name=":1" />

The following example was provided by Wazwaz on pages 1 and 2 in his book.<ref name=":0" /> We examine the IVP given by the equation:

<math display="block">u'(t) = 2tu(t), \, \, \,\,\, \,\, x \geq 0 </math>and the initial condition:

<math display="block">u(0)=1</math>

If we integrate both sides of the equation, we get:

<math display="block">\int_{0}^{x}u'(t) \, dt = \int_0^x 2tu(t) \, dt</math>

and by the fundamental theorem of calculus, we obtain:

<math display="block">u(x)-u(0) = \int_0^x 2tu(t) \, dt</math>

Rearranging the equation above, we get the integral equation:

<math display="block">u(x)= 1+ \int_0^x 2tu(t) \, dt</math>

which is a Volterra integral equation of the form:

<math display="block">u(x) = f(x) + \int_{\alpha(x)}^{\beta(x)}K(x,t) \cdot u(t) \, dt</math>

where K(x,t) is called the kernel and equal to 2t, and f(x) = 1.<ref name=":0" />

Numerical solutionEdit

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

<math> \sum_{j=1}^n w_j K(s_i,t_j) u(t_j)=f(s_i), \qquad i=0, 1, \dots, n. </math>

Then we have a system with Template:Mvar equations and Template:Mvar variables. By solving it we get the value of the Template:Mvar variables

<math>u(t_0),u(t_1),\dots,u(t_n).</math>

Integral equations as a generalization of eigenvalue equationsEdit

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

<math> \sum _j M_{i,j} v_j = \lambda v_i</math>

where Template:Math is a matrix, Template:Math is one of its eigenvectors, and Template:Mvar is the associated eigenvalue.

Taking the continuum limit, i.e., replacing the discrete indices Template:Mvar and Template:Mvar with continuous variables Template:Mvar and Template:Mvar, yields

<math> \int K(x,y) \, \varphi(y) \, dy = \lambda \, \varphi(x),</math>

where the sum over Template:Mvar has been replaced by an integral over Template:Mvar and the matrix Template:Math and the vector Template:Math have been replaced by the kernel Template:Math and the eigenfunction Template:Math. (The limits on the integral are fixed, analogously to the limits on the sum over Template:Mvar.) This gives a linear homogeneous Fredholm equation of the second type.

In general, Template:Math can be a distribution, rather than a function in the strict sense. If the distribution Template:Mvar has support only at the point Template:Math, then the integral equation reduces to a differential eigenfunction equation.

In general, Volterra and Fredholm integral equations can arise from a single differential equation, depending on which sort of conditions are applied at the boundary of the domain of its solution.

Wiener–Hopf integral equationsEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} <math display="block"> y(t) = \lambda x(t) + \int_0^\infty k(t-s) \, x(s) \, ds, \qquad 0 \leq t < \infty.</math> 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.

Hammerstein equationsEdit

A Hammerstein equation is a nonlinear first-kind Volterra integral equation of the form:<ref name=":2" /><math display="block">g(t) = \int_0^t K(t,s) \, G(s,y(s)) \, ds.</math>Under certain regularity conditions, the equation is equivalent to the implicit Volterra integral equation of the second-kind:<ref name=":2" /><math display="block">G(t, y(t)) = g_1(t) - \int_0^t K_1(t,s) \, G(s,y(s)) \, ds</math>where:<math display="block">g_1(t) := \frac{g'(t)}{K(t,t)} \,\,\,\,\,\,\, \text{and} \,\,\,\,\,\,\, K_1(t,s) := -\frac{1}{K(t,t)} \frac{\partial K(t,s)}{\partial t}.</math>The equation may however also be expressed in operator form which motivates the definition of the following operator called the nonlinear Volterra-Hammerstein operator:<ref name=":2" /><math display="block">(\mathcal{H}y)(t):= \int_0^t K(t,s) \, G(s, y(s)) \,ds</math>Here <math>G:I \times \mathbb{R} \to \mathbb{R}</math> is a smooth function while the kernel K may be continuous, i.e. bounded, or weakly singular.<ref name=":2" /> The corresponding second-kind Volterra integral equation called the Volterra-Hammerstein Integral Equation of the second kind, or simply Hammerstein equation for short, can be expressed as:<ref name=":2" /><math display="block">y(t)=g(t)+(\mathcal{H}y)(t) </math>In certain applications, the nonlinearity of the function G may be treated as being only semi-linear in the form of:<ref name=":2" /><math display="block">G(s,y) = y+ H(s,y)</math>In this case, we the following semi-linear Volterra integral equation:<ref name=":2" /><math display="block">y(t)=g(t)+(\mathcal{H}y)(t) = g(t) + \int_0^t K(t,s)[y(s)+H(s,y(s))] \, ds</math>In this form, we can state an existence and uniqueness theorem for the semi-linear Hammerstein integral equation.<ref name=":2" /> Template:Math theorem We can also write the Hammerstein equation using a different operator called the Niemytzki operator, or substitution operator, <math>\mathcal{N}</math> defined as follows:<ref name=":2" /><math display="block">(\mathcal{N} \varphi )(t) := G(t, \varphi(t))</math>More about this can be found on page 75 of this book.<ref name=":2" />

ApplicationsEdit

Integral equations are important in many applications. Problems in which integral equations are encountered include radiative transfer, and the oscillation of a string, membrane, or axle. Oscillation problems may also be solved as differential equations.

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See alsoEdit

BibliographyEdit

  • Agarwal, Ravi P., and Donal O'Regan. Integral and Integrodifferential Equations: Theory, Method and Applications. Gordon and Breach Science Publishers, 2000.<ref>Template:Cite book</ref>
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  • Jerri, Abdul J. Introduction to Integral Equations with Applications. Sampling Publishing, 2007.<ref>Template:Cite book</ref>
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ReferencesEdit

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Further readingEdit

External linksEdit

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