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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