General Linear Integral Equation
- 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.
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)
- 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
- 2nd Kind
- Separable Kernel
- ‘s and ‘s are linearly independent
We can reduce it to a system of algebraic equations and solve for
- Resolvent Kernel
- Eigenvalues and Eigenfunctions
- Invert IE
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.
Kernel can be expanded using known series, like etc. Choosing first one, two or three terms we can approximate the solution using Method 1.
Continue to Lesson 5 Quiz (link). Problems there can be solved using techniques described here.