Differential Equations

1
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

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Does this make sense?
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(b) n apples
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Example n10
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4. Central limit theorem
Gaussian Normal
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5. Conclusion and things to think about
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Differential Equations
1. Introduction
Finite-length bar separation of variables
Fourier series
Infinite bar Fourier transform
8
Other equations
FT can help in solution of all above.
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• 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

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2. FT of derivatives
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3. Solving the 1D heat equation
a. Reduce to ODE
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b. Solve the ODE IFT
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4. Convolution solution
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Some examples
1.
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2.
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5. Conclusion