Homogeneity

 

Homogeneity[edit]

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

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

Regularity[edit]

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]

$${\displaystyle �(�)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{�}^{-���}�(�)��}$$

$${\displaystyle �[�(�)]={\int }_0^{\mathrm{\infty }}{�}^{-��}�(�)��}$$

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

$${\displaystyle {�}^2={\int }_0^{�}\frac{1}{\sqrt{�-�}}�(�)��.}$$

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]

$${\displaystyle �(�)={\int }_{�}^{�}\frac{�(�)}{\sqrt{�-�}}��}$$

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]

Integro-differential equations[edit]

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]

$${\displaystyle {�}^{\prime }(�)=�(�,�(�))+({�}_{�}�)(�)}$$

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

$${\displaystyle ({\mathcal{�}}_{�,�}�)(�):={\int }_{�(�)}^{�}(�-�{)}^{-�}\cdot {�}_2(�,�,�(�),{�}^{\prime }(�))��}$$

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

$${\displaystyle {�}^{\prime }(�)=�(�,�(�),�(�(�)))+({\mathcal{�}}_{�,�}�)(�).}$$

Volterra integral equations[edit]

Uniqueness and existence theorems in 1D[edit]

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

$${\displaystyle (\mathcal{�}�)(�)=�(�)}$$

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

$${\displaystyle (\mathcal{�}�)(�):={\int }_{{�}_0}^{�}�(�,�)�(�)��}$$

where ${\displaystyle �\in �=[{�}_0,�]}$ and K(t,s) is called the kernel and must be continuous on the interval ${\displaystyle �:=\{(�,�):0\le �\le �\le �\le \mathrm{\infty }\}}$.^[3]

Theorem — Assume that ${\displaystyle �}$ satisfies ${\displaystyle �\in �(�),\mathrm{\partial }�/\mathrm{\partial }�\in �(�)}$ and  for some ${\displaystyle �\in �.}$ Then for any ${\displaystyle �\in {�}^1(�)}$ with ${\displaystyle �(0)=0}$ the integral equation above has a unique solution in ${\displaystyle �\in �(�)}$.

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

$${\displaystyle �(�)=�(�)+(\mathcal{�}�)(�)}$$

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

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

Volterra integral equations in ${\displaystyle {\mathbb{�}}^2}$[edit]

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

$${\displaystyle �(�,�)=�(�,�)+{\int }_0^{�}{\int }_0^{�}�(�,�,�,�)�(�,�)����}$$

where ${\displaystyle (�,�)\in \mathrm{\Omega }:=[0,�]\times [0,�]}$, ${\displaystyle �\in �(\mathrm{\Omega })}$, ${\displaystyle �\in �({�}_2)}$ and ${\displaystyle {�}_2:=\{(�,�,�,�):0\le �\le �\le �,0\le �\le �\le �\}}$.^[3] This integral equation has a unique solution ${\displaystyle �\in �(\mathrm{\Omega })}$ given by:^[3]

$${\displaystyle �(�,�)=�(�,�)+{\int }_0^{�}{\int }_0^{�}�(�,�,�,�)�(�,�)����}$$

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

Uniqueness and existence theorems of Fredhom-Volterra equations[edit]

As defined above, a VFIE has the form:

$${\displaystyle �(�,�)=�(�,�)+(\mathcal{�}�)(�,�)}$$

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

$${\displaystyle (\mathcal{�}�)(�,�):={\int }_0^{�}{\int }_{\mathrm{\Omega }}�(�,�,�,�)�(�(�,�))����.}$$

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

Theorem — If the linear VFIE given by: ${\displaystyle �(�,�)=�(�,�)+{\int }_0^{�}{\int }_{\mathrm{\Omega }}�(�,�,�,�)�(�(�,�))����}$ with ${\displaystyle (�,�)\in �\times \mathrm{\Omega }}$ satisfies the following conditions:

• ${\displaystyle �\in �(�\times \mathrm{\Omega })}$, and
• ${\displaystyle �\in �(�\times {\mathrm{\Omega }}^2)}$ where ${\displaystyle �:=\{(�,�):0\le �\le �\le �\}}$ and ${\displaystyle {\mathrm{\Omega }}^2=\mathrm{\Omega }\times \mathrm{\Omega }}$

Then the VFIE has a unique solution ${\displaystyle �\in �(�\times \mathrm{\Omega })}$ given by ${\displaystyle �(�,�)=�(�,�)+{\int }_0^{�}{\int }_{\mathrm{\Omega }}�(�,�,�,�)�(�(�,�))����}$ where ${\displaystyle �\in �(�\times {\mathrm{\Omega }}^2)}$ is called the Resolvent Kernel and is given by the limit of the Neumann series for the Kernel ${\displaystyle �}$ and solves the resolvent equations: ${\displaystyle �(�,�,�,�)=�(�,�,�,�)+{\int }_0^{�}{\int }_{\mathrm{\Omega }}�(�,�,�,�)�(�,�,�,�)����=�(�,�,�,�)+{\int }_0^{�}{\int }_{\mathrm{\Omega }}�(�,�,�,�)�(�,�,�,�)����}$

Special Volterra equations[edit]

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

$${\displaystyle �(�)=�(�)+({�}_{�}�)(�)}$$

where ${\displaystyle �\in �=[{�}_0,�]}$, the function g(t) is continuous on the interval ${\displaystyle �}$, and the Volterra integral operator ${\displaystyle ({�}_{�}�)}$ is given by:

$${\displaystyle ({�}_{�}�)(�):={\int }_{{�}_0}^{�}(�-�{)}^{-�}\cdot �(�,�,�(�))��}$$

with .^[3]

Converting IVP to integral equations[edit]

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:

$${\displaystyle {�}^{\prime }(�)=2��(�),�\ge 0}$$

and the initial condition:

$${\displaystyle �(0)=1}$$

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

$${\displaystyle {\int }_0^{�}{�}^{\prime }(�)��={\int }_0^{�}2��(�)��}$$

and by the fundamental theorem of calculus, we obtain:

$${\displaystyle �(�)-�(0)={\int }_0^{�}2��(�)��}$$

Rearranging the equation above, we get the integral equation:

$${\displaystyle �(�)=1+{\int }_0^{�}2��(�)��}$$

which is a Volterra integral equation of the form:

$${\displaystyle �(�)=�(�)+{\int }_{�(�)}^{�(�)}�(�,�)\cdot �(�)��}$$

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