Introduction to Integral Equation & its types

integral equations are equations in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations
Basically integral equation is classified into two main subtypes

An integral equation is an equation in which the unknown function u(x) to be determined
appears under the integral sign. A typical form of an integral equation in u(x) is of the
form, as in video . where K(x, t) is called the kernel of the integral equation, and α(x) and β(x) are the limits of integration. In (1), it is easily observed that the unknown function u(x) appears
under the integral sign as stated above, and out of the integral sign in most other cases as
will be discussed later. It is important to point out that the kernel K(x, t) and the function
f(x) in (1) are given in advance. Our goal is to determine u(x) that will satisfy (1), and
this may be achieved by using different techniques that will be discussed in the
forthcoming chapters. The primary concern of this text will be focused on introducing
these methods and techniques supported by illustrative and practical examples.
Integral equations arise naturally in physics, chemistry, biology and engineering
applications modelled by initial value problems for a finite interval [a, b]. More details
about the sources and origins of integral equations can be found in [12] and [14]. In the
following example we will discuss how an initial value problem will be converted to
the form of an integral equation.

The most frequently used linear integral equations fall under two main classes namely
Fredholm and Volterra integral equations. However, in this text we will distinguish four
more related types of linear integral equations in addition to the two main classes. In
what follows, we will give a list of the Fredholm and Volterra integral equations, and
the four related types:
1. Fredholm integral equations
2. Volterra integral equations
3. Integro-differential equations
4. Singular integral equations
5. Volterra-Fredholm integral equations
6. Volterra-Fredholm integro-differential equations
In the following we will outline the basic definitions and properties of each type.

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[KhanNasar-sy]

Thank you Sir for giving the lecture in smooth easy and easy understandable way

[azad]

Hello sir u are doing a good job, sir aapne advanced calculus k lectures diye hain, numbering kro playlist mae

[mshamiilahi]

Video deik kae samjh aa gayii mashallah acha method hai sir

[siya]

Sir calculus of variations k lecture v provide kra dijiye

[raven_iv]

Great work sir, please could you tell me what is the reference that you used??
I really need it

[saqibss]

Sir can you make vedio on
Fredholm's equation as a limit of a finite system of linear equations
Hadamard's theorem
Convergence proof
Fredholm's two fundamental relation