# Definitions and Types of Integral Equations

In this article, I will explain **what Integral Equations are**, how they are **structured** and what are certain **types of Integral Equations**.

**Table of Contents**

## What is an Integral Equation?

An integral equation is an equation in which an unknown function appears under one or more integration signs.

Any integral calculus statement like — $ y= \int_a^b \phi(x) dx$ can be considered as an integral equation.

If you noticed, I have used two integration limits (a and b) in the above integral equation – they are special in this discussion, and their significance will be discussed later in the article.

## Linear Integral Equations

A general type of integral equation, $ g(x) y(x) = f(x) + \lambda \int_a^\Box K(x, t) y(t) dt$ is called **linear integral equation** as only linear operations are performed in the equation.

The one, which is not linear, is obviously called a **"Non-linear integral equation"**.

We generally mean linear integral equation when we say integral equation. When a non-linear integral equation is to be called, it is called exclusively.

In the general type of the linear equation

$ g(x) y(x) = f(x) + \lambda \int_a^\Box K(x, t) y(t) dt$

we have used a 'box $ \Box$' to indicate the higher limit of the integration.

## Types of Integral Equations

Integral Equations can be of two types according to whether the box $ \Box$ (the upper limit) is a *constant, $b$* or a *variable, $x$*.

The first type of integral equations which involve constants as both the limits — are called **Fredholm Type Integral equations**.

On the other hand, when one of the limits is a variable ($x$, the independent variable of which $y$, $f$ and $K$ are functions), such integral equations are called **Volterra’s Integral Equations**.

Thus

- $ g(x) y(x) = f(x) + \lambda \int_a^b K(x, t) y(t) dt$ is a Fredholm Integral Equation and
- $ g(x) y(x) = f(x) + \lambda \int_a^x K(x, t) y(t) dt$ is a Volterra Integral Equation.

In an integral equation, $ y$ is to be determined with $ g$, $ f$ and $ K$ being known and $ \lambda$ being a non-zero complex parameter. The function $ K (x,t)$ is called the ‘kernel’ of the integral equation.

**Structure of an Integral Equation**

## Types of Fredholm Integral Equations

As the general form of Fredholm Integral Equation is $ g(x) y(x) = f(x) + \lambda \int_a^b K(x, t) y(t) dt$, there may be following other types of it according to the values of $ g$ and $ f$ :

**Fredholm Integral Equation of First Kind**—when — $ g(x) = 0$

$ f(x) + \lambda \int_a^b K(x, t) y(t) dt=0$**Fredholm Integral Equation of Second Kind**—when — $ g(x) =1$

$ y(x) = f(x) + \lambda \int_a^b K(x, t) y(t) dt$**Fredholm Integral Equation of Homogeneous Second Kind**—when $ f(x)=0$

and $ g(x)=1$

$ y(x) = \lambda \int_a^b K(x, t) y(t) dt$- The general equation of Fredholm equation is also called
**Fredholm Equation of Third/Final kind**, with $ f(x) \neq 0, 1 \neq g(x)\neq 0$.

## Types of Volterra Integral Equations

As the general form of Volterra Integral Equation is $ g(x) y(x) = f(x) + \lambda \int_a^x K(x, t) y(t) dt$, there may be following other types of it according to the values of $ g$ and $ f$ :

**Volterra Integral Equation of First Kind**—when — $ g(x) = 0$

$ f(x) + \lambda \int_a^x K(x, t) y(t) dt=0$**Volterra Integral Equation of Second Kind**—when — $ g(x) =1$

$ y(x) = f(x) + \lambda \int_a^x K(x, t) y(t) dt$**Volterra Integral Equation of Homogeneous Second Kind**—when $ f(x)=0$

and $ g(x)=1$

$ y(x) = \lambda \int_a^x K(x, t) y(t) dt$- The general equation of Volterra equation is also called the
**Volterra Equation of Third/Final kind**, with $ f(x) \neq 0, 1 \neq g(x)\neq 0$.

## Singular Integral equations

In the general *Fredholm/Volterra Integral equations*, there arise two singular situations:

- the limit $ a \to -\infty$ and $ \Box \to \infty$.
- the kernel $ K(x,t) = \pm \infty$ at some points in the integration limit $ [a, \Box]$.

Then, such integral equations are called Singular (Linear) Integral Equations. Based on these two singular situations, here are two examples of the Singular Integral equations.

**Type-1**

$ a \to -\infty$ and $ \Box \to \infty$

**General Form:** $ g(x) y(x) = f(x) + \lambda \int_{-\infty}^{\infty} K(x, t) y(t) dt$

* Example:* $ y(x) = 3x^2 + \lambda \int_{-\infty}^{\infty} e^{-|x-t|} y(t) dt$

**Type-2:** $ K(x,t) = \pm \infty$ at some points in the integration limit $ [a, \Box]$

* Example: *$ y(x) = f(x) + \int_0^x \dfrac{1}{(x-t)^n} y(t)$ is a singular integral equation as the integrand reaches to $ \infty$ at $ t=x$.

## Kernels

The nature of the solution of integral equations solely depends on the nature of the *Kernel* of the integral equation K(x,t).

Kernels are of the following special types:

### Symmetric Kernel

When the kernel $ K(x,t)$ is symmetric or complex symmetric or Hermitian, if $ K(x,t)= \bar{K}(t,x)$ .

Here bar $ \bar{K}(t,x)$ denotes the complex conjugate of $ K(t,x)$.

That’s if there is no imaginary part of the kernel, then $ K(x, t) = K(t, x)$ implies that $ K$ is a symmetric kernel.

**For example **$ K(x,t)= \sin (x+t)$ is symmetric kernel.

### Separable or Degenerate Kernel

A kernel $ K(x,t)$ is called separable if it can be expressed as the sum of a finite number of terms, each of which is the product of ‘a function’ of *x *only and ‘a function’ of *t* only, i.e., $ K(x,t)= \displaystyle{\sum_{n=1}^{\infty}} \phi_i (x) \psi_i (t)$

### Difference Kernel

When $ K(x,t) = K(x-t)$, the kernel is called *difference kernel*.

### Resolvent or Reciprocal Kernel

The solution of the integral equation $ y(x) = f(x) + \lambda \int_a^\Box K(x, t) y(t) dt$ is of the form $ y(x) = f(x) + \lambda \int_a^\Box \mathfrak{R}(x, t;\lambda) f(t) dt$.

The kernel $ \mathfrak{R}(x, t;\lambda)$ of the solution is called *resolvent *or *reciprocal kernel.*

## Integral Equations of Convolution Type

The integral equation $ g(x) y(x) = f(x) + \lambda \int_a^\Box K(x, t) y(t) dt$ is called of *integral equation of convolution type *when the kernel $ K(x,t)$ is difference kernel, i.e., $ K(x,t) = K(x-t)$.

Let $ y_1(x)$ and $ y_2(x)$ be two continuous functions defined for $ x \in E \subseteq\mathbb{R}$ then the convolution of $ y_1$ and $ y_2$ is given by

$$ y_1 * y_2 = \int_E y_1 (x-t) y_2(t) dt$$

$$=\int_E y_2 (x-t) y_1(t) dt$$

**For standard convolution**, the limits are $ -\infty$ and $ \infty$.

## Eigenvalues and Eigenfunctions of Integral Equations

The homogeneous integral equation $ y(x) = \lambda \int_a^\Box K(x, t) y(t) dt$ has the obvious solution $ y(x)=0$ which is called the *zero solution *or *the trivial solution of the *integral equation.

Except this, the values of $ \lambda$ for which the integral equation has **non-zero **solution $ y(x) \neq 0$, are called the **eigenvalues of integral equation** or **eigenvalues of the kernel***.*

Every non-zero solution $ y(x)\neq 0$ is called an **eigenfunction** corresponding to the obtained *eigenvalue *$ \lambda$*.*

- Note that $ \lambda \neq 0$
- If $ y(x)$ an eigenfunction corresponding to eigenvalue $ \lambda$ then $ c \cdot y(x)$ is also an eigenfunction corresponding to $ \lambda$

## Leibnitz Rule of Differentiation under the integral sign

Let $ F(x,t)$ and $ \dfrac{\partial F}{\partial x}$ be continuous functions of both *x *and *t *and let the first derivatives of $ G(x)$ and $ H(x)$ are also continuous, then

$ \dfrac{d}{dx} \displaystyle {\int_{G(x)}^{H(x)}} F(x,t) dt$

$= \displaystyle {\int_{G(x)}^{H(x)}}\dfrac{\partial F}{\partial x} dt + F(x, H(x)) \dfrac{dH}{dx} – \\ F(x, G(x)) \dfrac{dG}{dx}$.

This formula is called Leibnitz’s Rule of differentiation under integration sign. In a special case, when G(x) and H(x) both are absolute (constants) –let $ G(x) =a$, $ H(x)=b \iff dG/dx =0=dH/dx$ ; then

$ \dfrac{d}{dx} \displaystyle {\int_a^b} F(x,t) dt = \displaystyle {\int_a^b}\dfrac{\partial F}{\partial x} dt$

## Changing Integral Equation with Multiple Integrals into Standard Simple Integral

(Multiple Integral Into Simple Integral — The magical formula)

Say, the integral of order n is given by $ \displaystyle{\int_{\Delta}^{\Box}} f(x) dx^n$

We can prove that $ \displaystyle{\int_{a}^{t}} f(x) dx^n = \displaystyle{\int_{a}^{t}} \dfrac{(t-x)^{n-1}}{(n-1)!} f(x) dx$

**Example:** *Solve $ \int_0^1 x^2 dx^2$*

**Solution**: $ \int_0^1 x^2 dx^2$

$ = \int_0^1 \dfrac{(1-x)^{2-1}}{(2-1)!} x^2 dx$

(since *t=1*)

$ =\int_0^1 (1-x) x^2 dx$

$ =\int_0^1 (1-x) x^2 dx$

$ =\int_0^1 (x^2-x^3) dx =1/12$