Integral equation

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:$${\displaystyle (\mathcal{V}y)(t)=g(t)}$$can be described by the following uniqueness and existence theorem. Recall that the Volterra integral operator ${\displaystyle \mathcal{V}:C(I)\to C(I)}$, can be defined as follows:$${\displaystyle (\mathcal{V}\phi )(t):={\int }_{t_0}^{t}K(t,s)\phi (s)ds}$$where ${\displaystyle t\in I=[t_0,T]}$ and K(t,s) is called the kernel and must be continuous on the interval ${\displaystyle D:=\{(t,s):0\le s\le t\le T\le \mathrm{\infty }\}}$.

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

The solution to a linear Volterra integral equation of the second kind, given by the equation:$${\displaystyle y(t)=g(t)+(\mathcal{V}y)(t)}$$can be described by the following uniqueness and existence theorem.

Theorem—Let ${\displaystyle K\in C(D)}$ and let ${\displaystyle R}$ denote the resolvent Kernel associated with ${\displaystyle K}$. Then, for any ${\displaystyle g\in C(I)}$, the second-kind Volterra integral equation has a unique solution ${\displaystyle y\in C(I)}$ and this solution is given by: ${\displaystyle y(t)=g(t)+{\int }_0^{t}R(t,s)g(s)ds.}$

Volterra integral equations in R2

A Volterra Integral equation of the second kind can be expressed as follows:$${\displaystyle u(t,x)=g(t,x)+{\int }_0^x{\int }_0^{y}K(x,\xi ,y,\eta )u(\xi ,\eta )d\eta d\xi }$$where ${\displaystyle (x,y)\in \mathrm{\Omega }:=[0,X]\times [0,Y]}$, ${\displaystyle g\in C(\mathrm{\Omega })}$, ${\displaystyle K\in C(D_2)}$ and ${\displaystyle D_2:=\{(x,\xi ,y,\eta ):0\le \xi \le x\le X,0\le \eta \le y\le Y\}}$. This integral equation has a unique solution ${\displaystyle u\in C(\mathrm{\Omega })}$ given by:$${\displaystyle u(t,x)=g(t,x)+{\int }_0^x{\int }_0^{y}R(x,\xi ,y,\eta )g(\xi ,\eta )d\eta d\xi }$$where ${\displaystyle R}$ is the resolvent kernel of K.

Uniqueness and existence theorems of Fredholm–Volterra equations

As defined above, a VFIE has the form:$${\displaystyle u(t,x)=g(t,x)+(\mathcal{T}u)(t,x)}$$with ${\displaystyle x\in \mathrm{\Omega }}$ and ${\displaystyle \mathrm{\Omega }}$ being a closed bounded region in ${\displaystyle {\mathbb{R}}^d}$ with piecewise smooth boundary. The Fredholm–Volterrra Integral Operator ${\displaystyle \mathcal{T}:C(I\times \mathrm{\Omega })\to C(I\times \mathrm{\Omega })}$ is defined as:$${\displaystyle (\mathcal{T}u)(t,x):={\int }_0^t{\int }_{\mathrm{\Omega }}K(t,s,x,\xi )G(u(s,\xi ))d\xi ds.}$$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. With this in mind, we can now introduce the following theorem:

Theorem—If the linear VFIE given by: ${\displaystyle u(t,x)=g(t,x)+{\int }_0^t{\int }_{\mathrm{\Omega }}K(t,s,x,\xi )G(u(s,\xi ))d\xi ds}$ with ${\displaystyle (t,x)\in I\times \mathrm{\Omega }}$ satisfies the following conditions:

• ${\displaystyle g\in C(I\times \mathrm{\Omega })}$, and
• ${\displaystyle K\in C(D\times {\mathrm{\Omega }}^2)}$ where ${\displaystyle D:=\{(t,s):0\le s\le t\le T\}}$ and ${\displaystyle {\mathrm{\Omega }}^2=\mathrm{\Omega }\times \mathrm{\Omega }}$

Then the VFIE has a unique solution ${\displaystyle u\in C(I\times \mathrm{\Omega })}$ given by ${\displaystyle u(t,x)=g(t,x)+{\int }_0^t{\int }_{\mathrm{\Omega }}R(t,s,x,\xi )G(u(s,\xi ))d\xi ds}$ where ${\displaystyle R\in C(D\times {\mathrm{\Omega }}^2)}$ is called the Resolvent Kernel and is given by the limit of the Neumann series for the Kernel ${\displaystyle K}$ and solves the resolvent equations: ${\displaystyle R(t,s,x,\xi )=K(t,s,x,\xi )+{\int }_0^t{\int }_{\mathrm{\Omega }}K(t,v,x,z)R(v,s,z,\xi )dzdv=K(t,s,x,\xi )+{\int }_0^t{\int }_{\mathrm{\Omega }}R(t,v,x,z)K(v,s,z,\xi )dzdv}$

Special Volterra equations

A special type of Volterra equation which is used in various applications is defined as follows:$${\displaystyle y(t)=g(t)+(V_{\alpha }y)(t)}$$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 ${\displaystyle (0\le \alpha <1)}$.

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.

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

$${\displaystyle u^{\prime }(t)=2tu(t),x\ge 0}$$and the initial condition:

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

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

$${\displaystyle {\int }_0^x{u}^{\prime }(t)dt={\int }_0^{x}2tu(t)dt}$$

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.

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

${\displaystyle \sum _{j=1}^n{w}_{j}K(s_i,t_j)u(t_j)=f(s_i),i=0,1,\dots ,n.}$

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

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

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

$${\displaystyle y(t)=\lambda x(t)+{\int }_0^{\mathrm{\infty }}k(t-s)x(s)ds,0\le t<\mathrm{\infty }.}$$ 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:$${\displaystyle g(t)={\int }_0^{t}K(t,s)G(s,y(s))ds.}$$Under certain regularity conditions, the equation is equivalent to the implicit Volterra integral equation of the second-kind:$${\displaystyle G(t,y(t))=g_1(t)-{\int }_0^t{K}_1(t,s)G(s,y(s))ds}$$where:$${\displaystyle g_1(t):=\frac{g^{\prime }(t)}{K(t,t)}\text{and}{K}_1(t,s):=-\frac{1}{K(t,t)}\frac{\mathrm{\partial }K(t,s)}{\mathrm{\partial }t}.}$$The equation may however also be expressed in operator form which motivates the definition of the following operator called the nonlinear Volterra-Hammerstein operator:$${\displaystyle (\mathcal{H}y)(t):={\int }_0^{t}K(t,s)G(s,y(s))ds}$$Here ${\displaystyle G:I\times \mathbb{R}\to \mathbb{R}}$ is a smooth function while the kernel K may be continuous, i.e. bounded, or weakly singular. 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:$${\displaystyle y(t)=g(t)+(\mathcal{H}y)(t)}$$In certain applications, the nonlinearity of the function G may be treated as being only semi-linear in the form of:$${\displaystyle G(s,y)=y+H(s,y)}$$In this case, we the following semi-linear Volterra integral equation:$${\displaystyle y(t)=g(t)+(\mathcal{H}y)(t)=g(t)+{\int }_0^{t}K(t,s)[y(s)+H(s,y(s))]ds}$$In this form, we can state an existence and uniqueness theorem for the semi-linear Hammerstein integral equation.

Theorem—Suppose that the semi-linear Hammerstein equation has a unique solution ${\displaystyle y\in C(I)}$ and ${\displaystyle H:I\times \mathbb{R}\to \mathbb{R}}$ be a Lipschitz continuous function. Then the solution of this equation may be written in the form: ${\displaystyle y(t)=y_l(t)+{\int }_0^{t}R(t,s)H(s,y(s))ds}$ where ${\displaystyle y_l(t)}$ denotes the unique solution of the linear part of the equation above and is given by: ${\displaystyle y_l(t)=g(t)+{\int }_0^{t}R(t,s)g(s)ds}$ with ${\displaystyle R(t,s)}$ denoting the resolvent kernel.

We can also write the Hammerstein equation using a different operator called the Niemytzki operator, or substitution operator, ${\displaystyle \mathcal{N}}$ defined as follows:$${\displaystyle (\mathcal{N}\phi )(t):=G(t,\phi (t))}$$More about this can be found on page 75 of this book.

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