User:Rosenkranz421/Integral equation

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

$$f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ; I^1 (u), I^2(u), I^3(u), ..., I^m(u)) = 0$$

where $$I^i(u)$$ 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:

$$f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ; D^1 (u), D^2(u), D^3(u), ..., D^m(u)) = 0$$

where $$D^i(u)$$ may be viewed as a differential operator of order i. 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 (wazwaz). 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. See, for example, Green's function, Fredholm theory, and Maxwell's equations.

Overview of Classifications of Integral Equations
Various classification methods for integral equations exist. A few standard classifications include distinctions between linear and nonlinear; homogenous and inhomogenous; Fredholm and Volterra; first order, second order, and third order; and singular and regular integral equations. These distinctions usually rest on some fundamental property such as the consideration of the linearity of the equation or the homogeneity of the equation. 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. (wazwaz) Hence an example of a linear equation would be:

….

As a note on naming convention: i) u(x), or φ(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.

iv) The parameter λ is an unknown factor, which plays the same role as the eigenvalue in linear algebra. Nonlinear : An integral equation is nonlinear if the unknown function u or any of its integrals appear nonlinear in the equation. (wazwaz) Hence, examples of nonlinear equations would be the equation below if we replaced u(t) with $$u^2(x), \, \, cos(u(x)), \, \text{or } \,e^{u(x)}$$:

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

Certain kinds of nonlinear integral equations have specific names. A selection of such equation are: (Source 6)


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

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. (Source 6) An example would be: $$ f(x) = \int_a^b K(x,t)\,u(t)\,dt $$

Second kind : An integral equation is called an integral equation of the second kind if the unknown function appears also outside the integral. (Source 6)

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:

$$ g(t)u(t) + \lambda \int_a^b K(t,x)u(x) \, dx = f(t) $$

where g(t) vanishes at least once in the interval [a,b]^(cite 1) or where g(t) vanishes at a finite number of points in (a,b).^(cite 2)

( https://epubs.siam.org/doi/pdf/10.1137/0504053 and the source https://www.sciencedirect.com/science/article/pii/S2346809217300533 )( https://www.sciencedirect.com/science/article/pii/0022247X84900969#:~:text=The%20third%2Dkind%20linear%20integral,Riesz%20and%20Sz .)

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. An example would be that the integral is taken over a fixed subset of R^n. (source 6) Hence, the following two examples are Fredholm equations: Fredholm equation of the first type: $$ f(x) = \int_a^b K(x,t)\,u(t)\,dt $$. Fredholm equation of the second type: $$ u(x) = f(x)+ \lambda \int_a^b K(x,t) \, u(t) \, dt. $$ Note that we can express integral equations such as those above also using integral operator notation (source 2). For example, we can define the Fredholm integral operator as:

$$(\mathcal{F}y)(t) := \int_{t_0}^T K(t,s) \, y(s) \, ds.$$

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

$$y(t)=g(t)+\lambda(\mathcal{F}y)(t). (Source 2)$$

Volterra : An integral equation is called a Volterra integral equation if at least one of the limits of integration is a variable. Hence, the integral is taken over a domain varying with the variable of integration. (Source 6) Examples of Volterra equations would be: Volterrra Integral equation of the first kind: $ f(x) = \int_a^x K(x,t) \, u(t) \, dt $.

Volterrra Integral equation of the second kind: $ u(x) = f(x) + \lambda \int_a^x K(x,t)\,u(t)\,dt. $ (Source 2 for the rest until source 6). As with Fredholm equation, we can again adopt operator notation. Thus, we can define the linear Volterra integral operator $$\mathcal{V} : C(I) \to C(I)$$, as follows:

$$(\mathcal{V} \phi)(t) := \int_{t_0}^t K(t,s) \, \phi(s) \, ds$$

where $$t \in I = [t_0, T]$$ and K(t,s) is called the kernel and must be continuous on the interval $$D := \{(t,s) : 0 \leq s \leq t \leq T \leq \infty\}$$. Hence, the Volterra integral equation of the first kind may be written as:

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

with $$g(0)=0$$.

The linear first kind has a unique solution if:

Theorem: Assume that K satisfies K \in C(D), \partial K / \partial t \in C(D) and |K(t,t)| \geq k_0 > 0 for some t \in I. Then for any g\in C^1(I) with g(0)=0 the integral equation above has a unique solution in y \in C(I).

Fact 1: Volterra Integral of Second kind
Linear Volterrra Integral equation of the second kind for the unknown function y(t) and a given continuous function g(t) on the interval I where t \in I: $$y(t)=g(t)+(\mathcal{V} y)(t)$$

The solution to this equation is summarised by the theorem:

Let K \in C(D) and let R denote the resolvent Kernel associated with K. Ten for any g \in C(I) the second-kind VOlterra integral equation has a unique solution y \in C(I) and this solution is given by:

$$y(t)=g(t)+\int_0^t R(t,s) \, g(s) \, ds$$

\break\

some unique stuff:

$$y(t)=g(t)+(V_\alpha y)(t)$$

where $$t \in I = [t_0, T]$$, the function g(t) is continuous on the interval I, and the Volterra integral operator $$(V_\alpha t)$$ is given by:

$$(V_\alpha t)(t) := \int_{t_0}^t (t-s)^{-\alpha} \cdot k(t,s,y(s)) \, ds $$

with $$(0 \leq \alpha < 1)$$ and the kernel being k.

In higher dimensions:

Volterra integral equation in R^2
second-kind VIE with $$(x,y) \in \Omega := [0,X] \times [0,Y]$$, $$g \in C( \Omega)$$, $$K \in C(D_2)$$ where $$D_2 := \{(x, \xi,y,\eta): 0 \leq \xi \leq x \leq X, 0 \leq \eta \leq y \leq Y\}$$

$$u(t,x) = g(t,x)+\int_0^x \int_0^y K(x,\xi, y, \eta) \, u(\xi, \eta) \, d\eta \, d\xi$$This integral equation with the condiitions above has a unique solution $$u \in C( \Omega)$$ given by:

$$u(t,x) = g(t,x)+\int_0^x \int_0^{y} R(x,\xi, y, \eta) \, g(\xi, \eta) \, d\eta \, d\xi$$

Where R is the resolvent kernel of K.

There are also

Volterra-Fredholm Integral Equations
A VFIE has the form:

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

with $$x \in \Omega$$ and $$\Omega$$ being a closed bounded region in $$\mathbb{R}^d$$ with piecewise smooth boundary

The Fredholm-Volterrra Integral Operator $$\mathcal{T} : C(I \times \Omega) \to C(I \times \Omega)$$ is defined as:

$$(\mathcal{T}u)(t,x) := \int_0^t \int_\Omega K(t,s,x,\xi) \, G(u(s, \xi)) \, d\xi \, ds$$

In the case where the Kernel K may be written as $$K(t,s,x,\xi) = k(t-s)H(x, \xi)$$, k is called the positive memory kernel.

Theorem (solution):

If

a) $$g \in C(I \times \Omega)$$

b) $$K \in C(D \times \Omega^2)$$ where $$D:= \{(t,s): 0 \leq s \leq t \leq T \}$$ and $$\Omega^2 = \Omega \times \Omega$$

then the linear VFIE with $$(t,x) \in I \times \Omega$$: $$u(t,x) = g(t,x)+\int_0^t \int_\Omega K(t,s,x,\xi) \, G(u(s, \xi)) \, d\xi \, ds$$

has a unique solution $$u \in C(I \times \Omega)$$ given by: $$u(t,x) = g(t,x)+\int_0^t \int_\Omega R(t,s,x,\xi) \, G(u(s, \xi)) \, d\xi \, ds$$

where $$R \in C(D \times \Omega^2)$$ is the resolvent Kernel and is the limit of the Neumann series for the Kernel K and solves the resolvent equations:

$$R(t,s,x,\xi) = K(t,s,x,\xi)+\int_0^t \int_\Omega K(t,v,x,z) R(v,s,z,\xi) \, dz \, dv = K(t,s,x,\xi)+\int_0^t \int_\Omega R(t,v,x,z) K(v,s,z,\xi) \, dz \, dv  $$

Note that while throughout this article, the bounds of the integral are usually written as intervals, this need not be the case. In general, integral equations don't always need to be defined over an interval $$[a,b] = I$$, but could also be defined over a curve or surface. (Source 6).

Homogeneity
Homogenous : An integral equation is called homogeneous if the known function f is identically zero. For a Volterra integral equation of the second kind, an example would be: …. (wazwaz)

Inhomogeneous : An integral equation is called homogeneous if the known function f is nonzero. (wazwaz)

Regularity
Regular : An integral equation is called regular if the integrals used are all proper integrals. (Source 6)

Singular or weakly singular : An integral equation is called singular or weakly singular if the integral is an improper integral. (Source 6) 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. (wazwaz)

Examples:

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

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

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 $$K(x,t)=e^{-i\lambda x}$$ and $$K(x,t)=e^{-\lambda x}$$, respectively.

Another example of a singular integral equation in which the kernel becomes unbounded is: (Source: wazwaz.)

$$x^2= \int_0^x \frac{1}{\sqrt{x-t}} \, u(t) \, dt.$$

This equation is a special form of the more general weakly singular Volterra integral equation of the first kind, called Abel's integral equation: (source 6)

$$g(x)=\int_a^{x} \frac{f(y)}{\sqrt{x-y}} \, dy$$

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. (Source 6)

Integro-differential equations
Volterra Integro-Differnetial Equations: An integro-differential equation of the Volterra type may be written, using the Volterra operator defined above, as follows:

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

(same info on t and V as above) (Source 2).

Delay Problems: In applications, so called delay type problems often correspond to integro-differential equations. An example of one a delay integro-differential equation may be expressed as follows:

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

where the delay integral operator $$(\mathcal{W}_{\theta, \alpha} y)$$ is defined as:

$$(\mathcal{W}_{\theta, \alpha} y)(t) := \int_{\theta(t)}^t (t-s)^{-\alpha} \cdot k_2(t,s,y(s), y'(s)) \, ds $$

(Source 2).

Converting BVP to integral equations
Why convert IVP to integral equations? Because integral equations can often be "more suitable for proving existence and uniqueness" theorems for solutions.(source 6).

Source: A First course in integral equations by abdul-Majid Wazwaz, page 1-2:

Take the boundary value problem given by:

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

and the intial condition:

$$u(0)=1$$

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

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

and by the fundamental theorem of calculus, we obtain:

$$u(x)-u(1) = \int_{0}^{x}2tu(t)dt$$

Rearrainging we get:

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

which is an integral equation of the form:

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

which is called a Volterra Integral Equation, where K(x,t) is called the kernel and equal to 2t, and f(x)=1.

Numerical solutions
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


 * $$ \sum_{j=1}^n w_j K\left (s_i,t_j \right ) u(t_j)=f(s_i), \qquad i=0, 1, \dots, n. $$

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


 * $$u(t_0),u(t_1),\dots,u(t_n).$$

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


 * $$ g(s) = s \int_0^\infty K(st) \, f(t) \, dt $$

in the form of a power series


 * $$ f(t)= \sum_{n=0}^\infty \frac{a_n}{M(n+1)} t^n $$

where


 * $$ 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.

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 $M = [M_{i,j}]$ 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


 * $$ \int K(x,y) \, \varphi(y) \, dy = \lambda \, \varphi(x),$$

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
$$ y(t) = \lambda x(t) + \int_0^\infty k(t-s) \, x(s) \, ds, \qquad 0 \leq t < \infty.$$

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.

Hammerstein equations
A Hammerstein equation is a nonlinear first-kind Volterra integral equation of the form: $$g(t) = \int_0^t K(t,s) \, G(s,y(s)) \, ds.$$

Under certain regularity conditions, the equation is equivalent to the implicit Volterra integral equation of the second-kind: $$G(t, y(t)) = g_1(t) - \int_0^t K_1(t,s) \, G(s,y(s)) \, ds$$ where:

$$g_1(t) := \frac{g'(t)}{K(t,t)}$$$$K_1(t,s) := -\frac{1}{K(t,t)} \frac{\partial K(t,s)}{\partial t}$$

Volterra-Hammerstein Operator
The Volterra-Hammerstein Operator may be written as:

$$(\mathcal{H}y)(t):= \int_0^t K(t,s) \, H(s, y(s)) \,ds$$

The following is a type of semi-linear Volterra Integral equation:

$$y(t)=g(t)+(\mathcal{V} y)(t)+(\mathcal{H}y)(t) = g(t) = \int_0^t K(t,s)[y(s)+H(s,y(s))] \, ds$$

Existence and Uniquness theorem: Suppose that the nonlinear integral equation above has a unique solution $$y \in C(I)$$ and $$H:I \times \mathbb{R} \to \mathbb{R}$$ be a Lipschitz continuous function. Then the solution of this eqution may be written in the form:

$$y(t)=y_l(t)+\int_0^t R(t,s) \, H(s, y(s)) \, ds$$

where $$y_l(t)$$ denotes the unique solution of the linear part of the equation above and is given by:

$$y_l(t) = g(t) + \int_0^t R(t,s) \, g(s) \, ds$$

and R(t,s) deniting the resolvent kernel.

The above stuff however is apparently some special case of the more general Volterra-hammerstein Intgeral Equation defined below:

Definition: The nonlinear Volterra-Hammerstein Operator may be written as:

$$(\mathcal{H}y)(t):= \int_0^t K(t,s) \, G(s, y(s)) \,ds$$

Here $$G:I \times \mathbb{R} \to \mathbb{R}$$ is smoth whilt the kernel K may be continous (bounded) or weakly singular. The coressponding second-kind Volterra integral equation called the Volterrra-Hammerstein IE of the second kind:

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

In the special case above: the nonlinearity G is given by $$G(s,y) = y+ H(s,y)$$

Niemytzki Operator
We can rewrite Hammerstein stuff as above if we define the following operator:

Definition: Niemytzki Operator, also called the substitution operator, $$\mathcal{N}$$ defined as follows:

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

page 75 for more

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.

Other applications include:


 * Actuarial science (ruin theory )
 * Computational electromagnetics
 * Boundary element method
 * Inverse problems
 * Marchenko equation (inverse scattering transform)
 * Options pricing under jump-diffusion
 * Radiative transfer
 * Viscoelasticity
 * Fluid mechanics