## Method of Successive Approximation

### Solution Method 2 (Neumann Series Solution, Solution by Iterated Kernels)

#### For Fredholm IE

For the equation ,

Put g(s) = f(s) or any appropriate function of s.

Find iteratively.

where

and solution will be of the form

#### For Volterra IE

where

and solution will be of the form

#### Example

Solve the integral equation for .

Continuing, . Hence,

.

Hence, the solution,

## Some Properties of Resolvent Kernels

- is convergent absolutely and uniformly on $\latex |\lambda|<\frac1B$, where

A property of iterated kernel:

## Fredholm’s Theorems

### Solution Method 3 (Approximation)

For Fredholm IE with any integrable kernel, to find an approximate solution for

- Choose n, and let
- Let
- Find
- can be obtained by solving set of equations .
- can be approximated by putting , where

### Solution Method 4 (Using Fredholm’s First Theorem)

For Fredholm IE , where f(s) and g(s) are integrable, where

and where

#### Example

For the linear IE , find

and .

Hence,

### Lesson Quiz

Follow this link for lesson quiz.