# Separable Kernels and Approximation Methods (IEL5)

## Recollect:

### General Linear Integral Equation

$h(s)g(s)=f(s)+\lambda\int_\square^\square K(s,t)g(t)dt$

where

1. h(s) and f(s) are known functions
2. g(s) is the unknown function
3. K(s,t) is called the kernel
4. If f(s) = 0 and h(s) = 1, $\lambda$ is called the eigenvalue.

### Classification

#### Based on limits

1. Fredholm: Limits are constant
2. Volterra: Limits are functions of s
3. 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

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

#### Of Kernels

1. Separable/Degenerate: $K(s,t) = \sum a_i(s)b_i(t)$
2. Symmetric or Hermetian: $K(s,t) = K^*(t,s)$, where * denotes the conjugate
3. Convolution Type: $K(s,t) = K(s-t)$

### Eigenvalues and Eigenfunctions

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

## IE with Separable Kernel

#### Solution Method 1

If IE is

1. Fredholm
2. 2nd Kind
3. Separable Kernel
4. $a_i$‘s and $b_i$‘s are linearly independent

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

1. Solution
2. Resolvent Kernel $\Gamma(s,t;\lambda)$
3. Eigenvalues and Eigenfunctions
4. Invert IE

#### Example

Solve the integral equation $\phi(x) = f(x)+\int_0^1 K(x,y)\phi(y)dy$ for $K(x,y)=xy^2$.

Put $c= \int_0^1 y^2\phi(y)dy$. We get $\phi(x)=f(x)+xc$. Substituting it back in original equation,

$f(x)+xc = f(x)+\int_0^1 xy^2(f(y)+yc)dy$ Simplifying further, we get $c= \frac43 \int_0^1 y^2 f(y)dy$. Hence, $\phi(x)=f(x)+ \frac43 x\int_0^1 y^2 f(y)dy$

#### Number of Solution

Consider IE of Second Kind $g(s) = f(s) +\lambda\int K(s,t)g(t)dt$

1. Find transposed homogenous equation, $\psi(s) = \lambda\int K^*(t,s)g(t)dt$
2. Solve it for eigenfunctions $\psi_{0i}$
3. Find $\int f(s)\psi_{0i}(s)ds$. 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 $e^x=\sum_{n=0}^\infty {x^n\over n!}$ 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.

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