{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
ACM 95c Notes Alex Alemi Spring Term 2007
Image of page 1

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

View Full Document Right Arrow Icon
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
Image of page 2
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
Image of page 3

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

View Full Document Right Arrow Icon
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
Image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern