Lecture 8 Differential Equations

- OUTLINE
- Link between normal distribution and convolution

(Lecture 7 contd.). - Fourier transforms of derivatives
- The heat equation
- Solving differential equations with FTs

- Refs
- The Fourier Transform and its Applications, RN

Bracewell, 2nd Ed Ch 16, 3rd Ed Ch. 18 - Mathematical Methods for Physics and Engineering

(2nd Ed), Riley, Hobson, and Bence, Section 19.4 - Integral transforms for engineers and applied

mathematicians, LC Andrews, Ch. 3

Does this make sense?

(b) n apples

Example n10

4. Central limit theorem

Gaussian Normal

5. Conclusion and things to think about

Differential Equations

1. Introduction

Finite-length bar separation of variables

Fourier series

Infinite bar Fourier transform

Other equations

FT can help in solution of all above.

- Refs
- The Fourier Transform and its Applications, RN

Bracewell, 2nd Ed Ch 16, - 3rd Ed Ch. 18
- Mathematical Methods for Physics and Engineering

(2nd Ed), KF Riley, - MP Hobson, SJ Bence, Ch. 19.4
- Integral transforms for engineers and applied

mathematicians, LC Andrews, Ch. 3

2. FT of derivatives

3. Solving the 1D heat equation

a. Reduce to ODE

b. Solve the ODE IFT

4. Convolution solution

Some examples

1.

2.

5. Conclusion