1. Integral Transform Method
Chapter 15
Integral Transform Method

2. Outline
15.1 Error Function 15.2 Applications of the Laplace Transform 15.3 Fourier Integral 15.4 Fourier Transforms 15.5 Fast Fourier Transform

3. Error Function Error function Complementary error function
The functions are related by the identity

4. Error Function (cont’d.)
Useful Laplace transforms

5. Applications of the Laplace Transform
Assume that the operational properties of Laplace transforms of one-variable functions discussed in Chapter 4 apply to functions of two variables
For example ℒ
It follows that
ℒ ℒ

6. Applications of the Laplace Transform (cont’d.)
Example: Transform of a PDE
Find the Laplace transform of the wave equation
From the transform properties of partial derivatives,
ℒ ℒ
Which becomes
ℒ ℒ or

7. Fourier Integral
The Fourier integral of a function f defined on is given by
where and

8. Fourier Integral (cont’d.)
The Fourier integral of an even function on is the cosine integral
where
The Fourier integral of an odd function on is the sine integral

9. Fourier Integral (cont’d.)
The Fourier integral also possesses an equivalent complex form, or exponential form

10. Fourier Transforms
An integral transform and its inverse is a transform pair
If ℒ , then the inverse Laplace transform is the contour integral ℒ
If f(x) is transformed into F() by an integral transform then the function can be recovered by the inverse transform

11. Fourier Transforms (cont’d.)
The Fourier integral is the source of three new integral transforms

12. Fourier Transforms (cont’d.)
Examining the transforms of derivatives will aid in applying the concepts to BVPs
Fourier transform
Fourier sine transform
Fourier cosine transform
The choice of which transform to use on a given BVP depends on the type of BC specified at zero

13. Fast Fourier Transform
Consider a function f that is defined and continuous on the interval
are equally spaced points on the interval
Corresponding function
values
represent a discrete
sampling of f

14. Fast Fourier Transform (cont’d.)
Sampling (cont’d.)
T is the sampling rate, or length of sampling interval
is the fundamental angular frequency
2p is the fundamental period
If f is continuous at T, the sample of f at T is the product of the function f and the Dirac delta function, and the discrete version of f, or discrete signal is

15. Fast Fourier Transform (cont’d.)
Sampling (cont’d.)
Applying the Fourier transform and the sifting property of the Dirac delta function, we arrive at the discrete Fourier transform (DFT)
Consider the function values f(x) at N equally spaced points,
The finite (discrete) Fourier series gives

16. Fast Fourier Transform (cont’d.)
Sampling (cont’d.)
The finite (discrete) Fourier series and letting
gives

17. Fast Fourier Transform (cont’d.)
Sampling (cont’d.)
This system in matrix notation is

18. Fast Fourier Transform (cont’d.)
Sampling (cont’d.)
Let the N  N matrix be
If is the matrix of complex conjugates of and I denotes the identity matrix, we have
It follows that

19. Fast Fourier Transform (cont’d.)
If a signal is band-limited (the range of frequencies of the signal like in a band ), the signal can be reconstructed by sampling two times for every cycle of the highest frequency present

20. Fast Fourier Transform (cont’d.)
The DFT can be written as
Or, for simplicity of notation,
Recall the matrix notation

21. Fast Fourier Transform (cont’d.)
The key to the Fast Fourier Transform (FFT) is properties of and matrix factorization
If we can write Fn in the following way
and is the identity matrix and P is the permutation matrix that rearranges c