Integral equation

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]

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

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.[1] 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:

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

Certain kinds of nonlinear integral equations have specific names.[3] A selection of such equations are:[3]

• 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.

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]

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

where g(t) vanishes at least once in the interval [a,b][4][5] or where g(t) vanishes at a finite number of points in (a,b).[6]

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:

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

Hence, the above Fredholm equation of the second kind may be written compactly as:[7]

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

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]

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

where ${\displaystyle t\in I=[t_{0},T]}$ and K(t,s) is called the kernel and must be continuous on the interval ${\displaystyle D:=\{(t,s):0\leq s\leq t\leq T\leq \infty \}}$.[3] Hence, the Volterra integral equation of the first kind may be written as:[3]

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

with ${\displaystyle g(0)=0}$. In addition, a linear Volterra integral equation of the second kind for an unknown function ${\displaystyle y(t)}$ and a given continuous function ${\displaystyle g(t)}$ on the interval ${\displaystyle I}$ where ${\displaystyle t\in I}$:

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

Volterra-Fredholm: In higher dimensions, integral equations such as Fredholm-Volterra integral equations (VFIE) exist.[3] A VFIE has the form:

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

with ${\displaystyle x\in \Omega }$ and ${\displaystyle \Omega }$ being a closed bounded region in ${\displaystyle \mathbb {R} ^{d}}$ with piecewise smooth boundary.[3] The Fredholm-Volterrra Integral Operator ${\displaystyle {\mathcal {T}}:C(I\times \Omega )\to C(I\times \Omega )}$ is defined as:[3]

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

Note that while throughout this article, the bounds of the integral are usually written as intervals, this need not be the case.[7] In general, integral equations don't always need to be defined over an interval ${\displaystyle [a,b]=I}$, but could also be defined over a curve or surface.[7]

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]

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]

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

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

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 ${\displaystyle K(x,t)=e^{-i\lambda x}}$ and ${\displaystyle K(x,t)=e^{-\lambda x}}$, respectively.[1] Another example of a singular integral equation in which the kernel becomes unbounded is:[1]

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

This equation is a special form of the more general weakly singular Volterra integral equation of the first kind, called Abel's integral equation:[7]

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

Strongly singular: An integral equation is called strongly singular if the integral is defined by a special regularisation, for example, by the Cauchy principal value.[7]

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]

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

For delay problems, we can define the delay integral operator ${\displaystyle ({\mathcal {W}}_{\theta ,\alpha }y)}$ as:[3]

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

where the delay integro-differential equation may be expressed as:[3]

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

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

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

can be described by the following uniqueness and existence theorem.[3] Recall that the Volterra integral operator ${\displaystyle {\mathcal {V}}:C(I)\to C(I)}$, can be defined as follows:[3]

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

where ${\displaystyle t\in I=[t_{0},T]}$ and K(t,s) is called the kernel and must be continuous on the interval ${\displaystyle D:=\{(t,s):0\leq s\leq t\leq T\leq \infty \}}$.[3]

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

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

can be described by the following uniqueness and existence theorem.[3]

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

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

where ${\displaystyle (x,y)\in \Omega$ :=[0,X]\times [0,Y]} , ${\displaystyle g\in C(\Omega )}$, ${\displaystyle K\in C(D_{2})}$ and ${\displaystyle D_{2}:=\{(x,\xi ,y,\eta ):0\leq \xi \leq x\leq X,0\leq \eta \leq y\leq Y\}}$.[3] This integral equation has a unique solution ${\displaystyle u\in C(\Omega )}$ given by:[3]

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

where ${\displaystyle R}$ is the resolvent kernel of K.[3]

As defined above, a VFIE has the form:

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

with ${\displaystyle x\in \Omega }$ and ${\displaystyle \Omega }$ being a closed bounded region in ${\displaystyle \mathbb {R} ^{d}}$ with piecewise smooth boundary.[3] The Fredholm-Volterrra Integral Operator ${\displaystyle {\mathcal {T}}:C(I\times \Omega )\to C(I\times \Omega )}$ is defined as:[3]

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

In the case where the Kernel K may be written as ${\displaystyle K(t,s,x,\xi )=k(t-s)H(x,\xi )}$, K is called the positive memory kernel.[3] With this in mind, we can now introduce the following theorem:[3]

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

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

where ${\displaystyle t\in I=[t_{0},T]}$, the function g(t) is continuous on the interval ${\displaystyle I}$, and the Volterra integral operator ${\displaystyle (V_{\alpha }t)}$ is given by:

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

with ${\displaystyle (0\leq \alpha <1)}$.[3]