95notes - ACM 95c Notes Alex Alemi Spring Term 2007 Contents 1 Week 1 5 1.1 Introduction 5 1.1.1 Solving an Example 7 1.2 Second Order Linear PDEs

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ACM 95c Notes Alex Alemi Spring Term 2007 Contents 1 Week 1 5 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 Solving an Example . . . . . . . . . . . . . . . . . . . . . 7 1.2 Second Order Linear PDEs . . . . . . . . . . . . . . . . . . . . . 9 1.2.1 Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Day Three . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.1 Equilibrium solutions . . . . . . . . . . . . . . . . . . . . 14 1.3.2 Separable solutions . . . . . . . . . . . . . . . . . . . . . . 16 2 Week 2 18 2.1 General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.1 Series solutions to Separable functions . . . . . . . . . . . 20 2.2 A closer look at complete series solutions . . . . . . . . . . . . . 22 2.2.1 The whole shebang . . . . . . . . . . . . . . . . . . . . . 24 2.3 Properties of Fourier Series . . . . . . . . . . . . . . . . . . . . . 26 2.3.1 Convergence of Fourier Series . . . . . . . . . . . . . . . . 27 2.3.2 Differentiation of Fourier Series . . . . . . . . . . . . . . . 27 2.3.3 Integration of Fourier Series . . . . . . . . . . . . . . . . . 28 3 Week 3 30 3.1 Singular S-L Problems . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1.1 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . 30 3.1.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.3 Generalizing the Example . . . . . . . . . . . . . . . . . . 33 3.1.4 A Reformulation . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 Laplace Transform Solution for the finite boundary problem . . 36 3.2.1 Some Issues, Speedy Heat, and Approximations . . . . . 36 3.2.2 Laplace Transform Solution for the Finite Boundary Prob- lem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3 Deltas, Maximum Principle, and “Good Behavior” . . . . . . . . 39 3.3.1 The Role of the Delta Function . . . . . . . . . . . . . . . 40 3.3.2 Solving general problems, suggesting a new approach . . . 41 3.3.3 Tying it all together, Green’s functions, transforms, sep- arations of variables, S-L problems . . . . . . . . . . . . . 42 1 3.3.4 Superposition in t . . . . . . . . . . . . . . . . . . . . . . 42 3.3.5 An example . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4 Week 4 45 4.1 Maxima, Minima, and the Wave Equation . . . . . . . . . . . . . 45 4.1.1 Maxima and Minima for the Heat Equation . . . . . . . . 45 4.1.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.1.3 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2 Solving the Wave Equation . . . . . . . . . . . . . . . . . . . . . 46 4.2.1 Solution with no boundaries . . . . . . . . . . . . . . . . . 47 4.2.2 Finite Signal Speed . . . . . . . . . . . . . . . . . . . . . . 49 4.2.3 Two examples . . . . . . . . . . . . . . . . . . . . . . . . 49 4.2.4 Separation of variables: normal modes, resonance . . . . . 50 4.3 Down and Dirty with the Wave Equation . . . . . . . . . . . . .....
View Full Document

This note was uploaded on 02/22/2010 for the course ACM 95A taught by Professor Nilesa.pierce during the Fall '06 term at Caltech.

Page1 / 111

95notes - ACM 95c Notes Alex Alemi Spring Term 2007 Contents 1 Week 1 5 1.1 Introduction 5 1.1.1 Solving an Example 7 1.2 Second Order Linear PDEs

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online