In mathematics, integral equations are equations in which an unknown function appears under an integral sign.^[1] In mathematical notation, integral equations may thus be expressed as being of the form:

$${\displaystyle f(x_{1},x_{2},x_{3},...,x_{n};u(x_{1},x_{2},x_{3},...,x_{n});I^{1}(u),I^{2}(u),I^{3}(u),...,I^{m}(u))=0}$$

${\displaystyle I^{i}(u)}$

$${\displaystyle f(x_{1},x_{2},x_{3},...,x_{n};u(x_{1},x_{2},x_{3},...,x_{n});D^{1}(u),D^{2}(u),D^{3}(u),...,D^{m}(u))=0}$$

${\displaystyle D^{i}(u)}$

Classification and overview

Various classification methods for integral equations exist. A few standard classifications include distinctions between linear and nonlinear; homogenous and inhomogeneous; Fredholm and Volterra; first order, second order, and third order; and singular and regular integral equations.^[1] These distinctions usually rest on some fundamental property such as the consideration of the linearity of the equation or the homogeneity of the equation.^[1] These comments are made concrete through the following definitions and examples:

Linearity

Linear: An integral equation is linear if the unknown function u(x) and its integrals appear linear in the equation.^[1] Hence, an example of a linear equation would be:^[1]

$${\displaystyle u(x)=f(x)+\lambda \int _{\alpha (x)}^{\beta (x)}K(x,t)\cdot u(t)dt}$$

Nonlinear: An integral equation is nonlinear if the unknown function u(x) or any of its integrals appear nonlinear in the equation.^[1] Hence, examples of nonlinear equations would be the equation above if we replaced u(t) with ${\displaystyle u^{2}(x),\,\,cos(u(x)),\,{\text{or ))\,e^{u(x)))$, such as:

$${\displaystyle u(x)=f(x)+\int _{\alpha (x)}^{\beta (x)}K(x,t)\cdot u^{2}(t)dt}$$

• Nonlinear Volterra integral equations of the second kind which have the general form: ${\displaystyle u(x)=f(x)+\lambda \int _{a}^{x}K(x,t)\,F(x,t,u(t))\,dt,}$ where F is a known function.^[3]
• Nonlinear Fredholm integral equations of the second kind which have the general form: ${\displaystyle f(x)=F(x,\int _{a}^{b}K(x,y,f(x),f(y))\,dy)}$.^[3]
• A special type of nonlinear Fredholm integral equations of the second kind is given by the form: ${\displaystyle f(x)=g(x)+\int _{a}^{b}K(x,y,f(x),f(y))\,dy}$, which has the two special subclasses:^[3]
  • Urysohn equation: ${\displaystyle f(x)=g(x)+\int _{a}^{b}k(x,y,f(y))\,dy}$.^[3]
  • Hammerstein equation: ${\displaystyle f(x)=g(x)+\int _{a}^{b}k(x,y)\,G(y,f(y))\,dy}$.^[3]

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 equation

First kind: An integral equation is called an integral equation of the first kind if the unknown function appears only under the integral sign.^[3] An example would be: ${\displaystyle f(x)=\int _{a}^{b}K(x,t)\,u(t)\,dt}$.^[3]

Second kind: An integral equation is called an integral equation of the second kind if the unknown function also appears outside the integral.^[3]

Third kind: An integral equation is called an integral equation of the third kind if it is a linear Integral equation of the following form:^[3]

$${\displaystyle g(t)u(t)+\lambda \int _{a}^{b}K(t,x)u(x)\,dx=f(t)}$$

Limits of Integration

Fredholm: An integral equation is called a Fredholm integral equation if both of the limits of integration in all integrals are fixed and constant.^[1] An example would be that the integral is taken over a fixed subset of ${\displaystyle \mathbb {R} ^{n))$.^[3] Hence, the following two examples are Fredholm equations:^[1]

• Fredholm equation of the first type: ${\displaystyle f(x)=\int _{a}^{b}K(x,t)\,u(t)\,dt}$.
• Fredholm equation of the second type: ${\displaystyle u(x)=f(x)+\lambda \int _{a}^{b}K(x,t)\,u(t)\,dt.}$

Note that we can express integral equations such as those above also using integral operator notation.^[7] For example, we can define the Fredholm integral operator as:

$${\displaystyle ({\mathcal {F))y)(t):=\int _{t_{0))^{T}K(t,s)\,y(s)\,ds.}$$

$${\displaystyle y(t)=g(t)+\lambda ({\mathcal {F))y)(t).}$$

Volterra: An integral equation is called a Volterra integral equation if at least one of the limits of integration is a variable.^[1] Hence, the integral is taken over a domain varying with the variable of integration.^[3] Examples of Volterra equations would be:^[1]

• Volterrra integral equation of the first kind: ${\displaystyle f(x)=\int _{a}^{x}K(x,t)\,u(t)\,dt}$
• Volterrra integral equation of the second kind: ${\displaystyle u(x)=f(x)+\lambda \int _{a}^{x}K(x,t)\,u(t)\,dt.}$

As with Fredholm equations, we can again adopt operator notation. Thus, we can define the linear Volterra integral operator ${\displaystyle {\mathcal {V)):C(I)\to C(I)}$, as follows:^[3]

$${\displaystyle ({\mathcal {V))\phi )(t):=\int _{t_{0))^{t}K(t,s)\,\phi (s)\,ds}$$

${\displaystyle t\in I=[t_{0},T]}$

${\displaystyle D:=\{(t,s):0\leq s\leq t\leq T\leq \infty \))$

$${\displaystyle ({\mathcal {V))y)(t)=g(t)}$$

${\displaystyle g(0)=0}$

${\displaystyle y(t)}$

${\displaystyle g(t)}$

${\displaystyle I}$

${\displaystyle t\in I}$

$${\displaystyle y(t)=g(t)+({\mathcal {V))y)(t).}$$

$${\displaystyle u(t,x)=g(t,x)+({\mathcal {T))u)(t,x)}$$

${\displaystyle x\in \Omega }$

${\displaystyle \Omega }$

${\displaystyle \mathbb {R} ^{d))$

${\displaystyle {\mathcal {T)):C(I\times \Omega )\to C(I\times \Omega )}$

$${\displaystyle ({\mathcal {T))u)(t,x):=\int _{0}^{t}\int _{\Omega }K(t,s,x,\xi )\,G(u(s,\xi ))\,d\xi \,ds.}$$

${\displaystyle [a,b]=I}$

Homogeneity

Homogenous: An integral equation is called homogeneous if the known function ${\displaystyle f}$ is identically zero.^[1]

Inhomogenous: An integral equation is called inhomogeneous if the known function ${\displaystyle f}$ is nonzero.^[1]

Regularity

Regular: An integral equation is called regular if the integrals used are all proper integrals.^[7]

Singular or weakly singular: An integral equation is called singular or weakly singular if the integral is an improper integral.^[7] 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.^[1]

Examples include:^[1]

$${\displaystyle F(\lambda )=\int _{-\infty }^{\infty }e^{-i\lambda x}u(x)\,dx}$$

$${\displaystyle L[u(x)]=\int _{0}^{\infty }e^{-\lambda x}u(x)\,dx}$$

${\displaystyle K(x,t)=e^{-i\lambda x))$

${\displaystyle K(x,t)=e^{-\lambda x))$

$${\displaystyle x^{2}=\int _{0}^{x}{\frac {1}{\sqrt {x-t))}\,u(t)\,dt.}$$

$${\displaystyle g(x)=\int _{a}^{x}{\frac {f(y)}{\sqrt {x-y))}\,dy}$$

Integro-differential equations

An Integro-differential equation, as the name suggests, combines differential and integral operators into one equation.^[1] There are many version including the Volterra integro-differential equation and delay type equations as defined below.^[3] For example, using the Volterra operator as defined above, the Volterra integro-differential equation may be written as:^[3]

$${\displaystyle y'(t)=f(t,y(t))+(V_{\alpha }y)(t)}$$

${\displaystyle ({\mathcal {W))_{\theta ,\alpha }y)}$

$${\displaystyle ({\mathcal {W))_{\theta ,\alpha }y)(t):=\int _{\theta (t)}^{t}(t-s)^{-\alpha }\cdot k_{2}(t,s,y(s),y'(s))\,ds}$$

$${\displaystyle y'(t)=f(t,y(t),y(\theta (t)))+({\mathcal {W))_{\theta ,\alpha }y)(t).}$$

Volterra integral equations

Uniqueness and existence theorems in 1D

The solution to a linear Volterra integral equation of the first kind, given by the equation:

$${\displaystyle ({\mathcal {V))y)(t)=g(t)}$$

${\displaystyle {\mathcal {V)):C(I)\to C(I)}$

$${\displaystyle ({\mathcal {V))\phi )(t):=\int _{t_{0))^{t}K(t,s)\,\phi (s)\,ds}$$

${\displaystyle t\in I=[t_{0},T]}$

${\displaystyle D:=\{(t,s):0\leq s\leq t\leq T\leq \infty \))$

The solution to a linear Volterra integral equation of the second kind, given by the equation:^[3]

$${\displaystyle y(t)=g(t)+({\mathcal {V))y)(t)}$$

Volterra integral equations in ℝ²

A Volterra Integral equation of the second kind can be expressed as follows:^[3]

$${\displaystyle u(t,x)=g(t,x)+\int _{0}^{x}\int _{0}^{y}K(x,\xi ,y,\eta )\,u(\xi ,\eta )\,d\eta \,d\xi }$$

${\displaystyle (x,y)\in \Omega :=[0,X]\times [0,Y]}$

${\displaystyle g\in C(\Omega )}$

${\displaystyle K\in C(D_{2})}$

${\displaystyle D_{2}:=\{(x,\xi ,y,\eta ):0\leq \xi \leq x\leq X,0\leq \eta \leq y\leq Y\))$

${\displaystyle u\in C(\Omega )}$

$${\displaystyle u(t,x)=g(t,x)+\int _{0}^{x}\int _{0}^{y}R(x,\xi ,y,\eta )\,g(\xi ,\eta )\,d\eta \,d\xi }$$

${\displaystyle R}$

Uniqueness and existence theorems of Fredhom-Volterra equations

As defined above, a VFIE has the form:

$${\displaystyle u(t,x)=g(t,x)+({\mathcal {T))u)(t,x)}$$

${\displaystyle x\in \Omega }$

${\displaystyle \Omega }$

${\displaystyle \mathbb {R} ^{d))$

${\displaystyle {\mathcal {T)):C(I\times \Omega )\to C(I\times \Omega )}$

$${\displaystyle ({\mathcal {T))u)(t,x):=\int _{0}^{t}\int _{\Omega }K(t,s,x,\xi )\,G(u(s,\xi ))\,d\xi \,ds.}$$

${\displaystyle K(t,s,x,\xi )=k(t-s)H(x,\xi )}$

Special Volterra equations

A special type of Volterra equation which is used in various applications is defined as follows:^[3]

$${\displaystyle y(t)=g(t)+(V_{\alpha }y)(t)}$$

${\displaystyle t\in I=[t_{0},T]}$

${\displaystyle I}$

${\displaystyle (V_{\alpha }t)}$

$${\displaystyle (V_{\alpha }t)(t):=\int _{t_{0))^{t}(t-s)^{-\alpha }\cdot k(t,s,y(s))\,ds}$$

${\displaystyle (0\leq \alpha <1)}$

Converting IVP to integral equations

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.^[7]

The following example was provided by Wazwaz on pages 1 and 2 in his book.^[1] We examine the IVP given by the equation:

$${\displaystyle u'(t)=2tu(t),\,\,\,\,\,\,\,x\geq 0}$$

$${\displaystyle u(0)=1}$$

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

$${\displaystyle \int _{0}^{x}u'(t)dt=\int _{0}^{x}2tu(t)dt}$$

and by the fundamental theorem of calculus, we obtain:

$${\displaystyle u(x)-u(0)=\int _{0}^{x}2tu(t)dt}$$

Rearranging the equation above, we get the integral equation:

$${\displaystyle u(x)=1+\int _{0}^{x}2tu(t)dt}$$

which is a Volterra integral equation of the form:

$${\displaystyle u(x)=f(x)+\int _{\alpha (x)}^{\beta (x)}K(x,t)\cdot u(t)dt}$$

where K(x,t) is called the kernel and equal to 2t, and f(x)=1.^[1]

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

${\displaystyle g(s)=s\int _{0}^{\infty }K(st)\,f(t)\,dt}$

in the form of a power series

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

where

${\displaystyle g(s)=\sum _{n=0}^{\infty }a_{n}s^{-n},\qquad M(n+1)=\int _{0}^{\infty }K(t)\,t^{n}\,dt}$

are the Z-transform of the function g(s), and M(n + 1) is the Mellin transform of the Kernel.

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

${\displaystyle \sum _{j=1}^{n}w_{j}K\left(s_{i},t_{j}\right)u(t_{j})=f(s_{i}),\qquad i=0,1,\dots ,n.}$

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

${\displaystyle u(t_{0}),u(t_{1}),\dots ,u(t_{n}).}$

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

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

where M = [M_i,j] is a matrix, v is one of its eigenvectors, and λ is the associated eigenvalue.

Taking the continuum limit, i.e., replacing the discrete indices i and j with continuous variables x and y, yields

${\displaystyle \int K(x,y)\,\varphi (y)\,dy=\lambda \,\varphi (x),}$

where the sum over j has been replaced by an integral over y and the matrix M and the vector v have been replaced by the kernel K(x, y) and the eigenfunction φ(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.

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 equations

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

Hammerstein equations

A Hammerstein equation is a nonlinear first-kind Volterra integral equation of the form:^[3]

$${\displaystyle g(t)=\int _{0}^{t}K(t,s)\,G(s,y(s))\,ds.}$$

$${\displaystyle G(t,y(t))=g_{1}(t)-\int _{0}^{t}K_{1}(t,s)\,G(s,y(s))\,ds}$$

$${\displaystyle g_{1}(t):={\frac {g'(t)}{K(t,t)))\,\,\,\,\,\,\,{\text{and))\,\,\,\,\,\,\,K_{1}(t,s):=-{\frac {1}{K(t,t))){\frac {\partial K(t,s)}{\partial t)).}$$

$${\displaystyle ({\mathcal {H))y)(t):=\int _{0}^{t}K(t,s)\,G(s,y(s))\,ds}$$

${\displaystyle G:I\times \mathbb {R} \to \mathbb {R} }$

$${\displaystyle y(t)=g(t)+({\mathcal {H))y)(t)}$$

$${\displaystyle G(s,y)=y+H(s,y)}$$

$${\displaystyle y(t)=g(t)+({\mathcal {H))y)(t)=g(t)+\int _{0}^{t}K(t,s)[y(s)+H(s,y(s))]\,ds}$$

We can also write the Hammerstein equation using a different operator called the Niemytzki operator, or substitution operator, ${\displaystyle {\mathcal {N))}$ defined as follows:^[3]

$${\displaystyle ({\mathcal {N))\phi )(t):=G(t,\phi (t))}$$

Applications

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.

• Actuarial science (ruin theory^[8])
• Computational electromagnetics
  • Boundary element method
• Inverse problems
  • Marchenko equation (inverse scattering transform)
• Options pricing under jump-diffusion^[9]
• Radiative transfer
• Renewal theory^[10]
• Viscoelasticity
• Fluid mechanics^[11][12]