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# 95notes - ACM 95c Notes Alex Alemi Spring Term 2007...

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ACM 95c Notes Alex Alemi Spring Term 2007

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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 . . . . . . . . . . . . . 51 4.3.1 Inhomogeneous Eq - Resonances . . . . . . . . . . . . . . 52 4.3.2 Inhomogeneous w/ some other BCs . . . . . . . . . . . . . 53 4.3.3 Inhomogeneous BCs - Finite Transforms . . . . . . . . . . 53 4.3.4 Transform Methods . . . . . . . . . . . . . . . . . . . . . 54 5 Week 5 56 5.1 Laplace Transforms in Infinite domains . . . . . . . . . . . . . . . 56 5.1.1 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . 56 5.1.2 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 58 5.2 Wednesday . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.2.1 Wave Equation in R . . . . . . . . . . . . . . . . . . . . . 59 5.2.2 Wave Equation in a Semi- infinite domain . . . . . . . . . 59 5.2.3 Radiation BC . . . . . . . . . . . . . . . . . . . . . . . . 60 5.2.4 Green’s Function for wave equation . . . . . . . . . . . . 62 5.2.5 Using the Green’s function . . . . . . . . . . . . . . . . . 63 5.3 Friday . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.3.1 Heat Equation in 2D and 3D . . . . . . . . . . . . . . . . 64 5.3.2 Separation of Variables for 2D, 3D . . . . . . . . . . . . . 66 6 Week 6 67 6.1 Separation of Variables for 3 or more Independent Variables . . . 67 6.1.1 Solving the 2-D wave equation . . . . . . . . . . . . . . . 68 6.1.2 Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . 69 6.1.3 Helmholtz’s Equation . . . . . . . . . . . . . . . . . . . . 69 6.1.4 Waves in a Rectangle . . . . . . . . . . . . . . . . . . . . 69 6.1.5 Properties of the Helmholtz’s equation . . . . . . . . . . . 72 6.2 More Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . 73 6.2.1 Helmholtz Returns . . . . . . . . . . . . . . . . . . . . . . 73 6.2.2 Rayleigh Quotient . . . . . . . . . . . . . . . . . . . . . . 74 6.2.3 Application to a vibrating circular membrane . . . . . . . 75 6.3 Bessel Functions Return! . . . . . . . . . . . . . . . . . . . . . . . 77 6.3.1 Finish up Vibrating circular membrane . . . . . . . . . . 77 2

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7 Week 7 81 7.1 Spheres and the Laplace Equation . . . . . . . . . . . . . . . . . 81 7.1.1 Wave Equation in Spherical Coordinates . . . . . . . . . . 81 7.2 Laplace Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.2.1 In Cartesian coordinates . . . . . . . . . . . . . . . . . . . 87 7.2.2 in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . 89 7.3 Laplace + Poisson . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.3.1 Properties of the Laplace Equation solution in 2D . . . . 91 7.3.2 Qualitative properties of the Laplace equation . . . . . . . 92 8 Week 8 94 8.1 More Laplace solutions . . . . . . . . . . . . . . . . . . . . . . . . 94 8.1.1 Laplace’s equation in spherical coordinates . . . . . . . . 94 8.1.2 Semi-infinite and infinite domains . . . . . . . . . . . . . 96 8.2 Wednesday . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.3 Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.3.1 Poisson’s Equation . . . . . . . . . . . . . . . . . . . . . . 98 8.3.2 Computing the Green’s function . . . . . . . . . . . . . . 101 9 Week 9 103 9.1 Monday . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9.2 Wednesday . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9.3 Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 10 Week 10 107 10.1 Numerical solution of PDE
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