## Recollect:

### General Linear Integral Equation

where

- h(s) and f(s) are known functions
- g(s) is the unknown function
- K(s,t) is called the kernel
- If f(s) = 0 and h(s) = 1, is called the eigenvalue.

### Classification

#### Based on limits

**Fredholm**: Limits are constant**Volterra**: Limits are functions of s**Singular**: Either one, or both the limits are infinity or kernel, K(s,t) tends to infinity at one or more points in the give region of integration.

#### Based on the value of known functions

**1st Kind**: h(s) = 0**2nd Kind**: h(s) = 1**Homogeneous 2nd Kind**: h(s) = 1, f(s) = 0**Carleman’s or 3rd Kind**(General form above)

#### Of Kernels

**Separable/Degenerate**:**Symmetric or Hermetian**: , where * denotes the conjugate**Convolution Type**:

### Eigenvalues and Eigenfunctions

For a Homogenous Fredholm IE of 2nd Kind, is called the eigenvalue (characteristic number) and corresponding solution is called the eigenfunction.

## IE with Separable Kernel

#### Solution Method 1

If IE is

- Fredholm
- 2nd Kind
- Separable Kernel
- ‘s and ‘s are linearly independent

We can reduce it to a system of algebraic equations and solve for

- Solution
- Resolvent Kernel
- Eigenvalues and Eigenfunctions
- Invert IE

#### Example

Solve the integral equation for .

Put . We get . Substituting it back in original equation,

Simplifying further, we get . Hence,

#### Number of Solution

Consider IE of Second Kind

- Find transposed homogenous equation,
- Solve it for eigenfunctions
- Find . If it is zero, the IE has solutions for particular f(s) and if it is nonzero, it has no solutions.

#### Approximate Solution

Kernel can be expanded using known series, like etc. Choosing first one, two or three terms we can approximate the solution using Method 1.

### Lesson Quiz

Continue to Lesson 5 Quiz (link). Problems there can be solved using techniques described here.