Integral Equations:

1
Integral Equations
2
(No Transcript)
3
(No Transcript)
4
There is yet another picture partway between the
above two pictures., which is called Dirac (or
Interaction) Picture.
5
(No Transcript)
6
(No Transcript)
7
(No Transcript)
8
The left hand side can be written as
Hence
9
(No Transcript)
10
Eq.(12) represents what is called Born
Approximation. It contains the lowest order term
(first one) and the first order term which is
proportional to e.
We can develop a series solution for u by writing
11
Eq.(15) can now be written as
12
Hence to lowest non-trivial order (first-order)
This is the Born approximation.
13
This procedure can obviously be repeated to
generate s formal series solution for u.
This is the example of what is called
Neumann-series solution. The key question, of
course, is whether this series converges.
14
Classification of Integral Equations
15
Specific Examples
16
Terminology
17
Methods of Solution of Integral Equations
There are no all-encompassing methods for solving
integral equations. We develop a collection of
techniques which apply in different situations.
Solution by Integral Transforms Consider the
equation
18
We can check the previous solution
19
We can understand the process of inverting Eq.(1)
as merely a change of basis in Hilbert space. To
see this consider an analogous discrete case as
an example
20
We can check the inversion in Eq.(5) as follows
This verifies that eq.(4) gives the correct
prescription for going from one basis to another
21
Other equations can also be solved by other
integral transforms
Im t
.
.
Re t
.
.
22
(No Transcript)