A linear integral equation of the form

\[g(x)y(x) = f(x) + \lambda \int_a^b {K(x,t)y(t)dt}\]

Where a,b are both constant \[g(x)y(x)\] and \[K(x,t)\] are known functions while

is unknown function and is a non-zero real or complex parameter, is called fredholm Integral equation of third kind. The function \[K(x,t)\]is known as the kernel of the integral equation.

The following special cases of the above equation are of our main interest.

**Fredholm integral equation of the first kind**

A linear integral equation of the form (g(x)=0 in above equation)

\[f(x) + \lambda \int_a^b {K(x,t)y(t)dt} = 0\]

**Fredholm integral equation of the second kind**

A linear integral equation of the form (=1 in above equation)

\[y(x) = f(x) + \lambda \int_a^b {K(x,t)y(t)dt} \]

**Homogeneous fredholm integral equation of the second kind**

A linear integral equation of the form (f(x)=0 in above equation)

\[y(x) = \lambda \int_a^b {K(x,t)y(t)dt} \]

**VOLTERRA INTEGRAL EQUATION **

A linear integral equation of the form

\[g(x)y(x) = f(x) + \lambda \int_a^x {K(x,t)y(t)dt} \]

Where a,b are both constant \[g(x)y(x)\] and \[K(x,t)\] are known functions while

\[y(x)\]is unknown function and \[\lambda \] is a non-zero real or complex parameter, is called

Volterra integral equation of the third kind. The function \[K(x,t)\]is known as the kernel of the integral equation.

The following special cases of the above equation are of our main interest.

**Volterra integral equation of the first kind**

A linear integral equation of the form (\[g(x)\]=0 in above equation)

\[f(x) + \lambda \int_a^x {K(x,t)y(t)dt} = 0\]

**Volterra integral equation of the second kind**

A linear integral equation of the form (\[g(x)=1\] in above equation)

\[y(x) = f(x) + \lambda \int_a^x {K(x,t)y(t)dt} \]

**Homogeneous Volterra integral equation of the second kind**

A linear integral equation of the form (f(x)=0 in above equation)

\[y(x) = \lambda \int_a^x {K(x,t)y(t)dt} \]