Mat-F February 28, 2005 Separation of variables: Plane, Cylindrical & Spherical cases

1. Mat-FFebruary 28, 2005Separation of variables: Plane, Cylindrical & Spherical cases Åke Nordlund Niels Obers, Sigfus Johnsen Kristoffer Hauskov Andersen Peter Browne Rønne

2. Overview • Changes • More black-board work-througs! • Monday exercises extended (10-13)! • No Maple TA (still Maple) on Wednesdays! • Sections 19.3-19.5 • 19.3: Polar coordinates (today) • 19.4: Integral transform method • assumes Fourier and Laplace transforms  skip! • 19.5: Greens functions (Wednesday – cursory) • assumes vector analysis  postpone details!

3. Optimizing the time spent • Preparations • Read / browse before Monday lecture & exercise • Exercises: “OK – I know how” / “NOK – I need to see more” • Lectures • Detailed examples & work-throughs • Exercises • Actually doing it (mostly) yourself • Don’t panic! • New topics always appear confusing at first • This section (19.3) has a lot of material • this is a good thing – more help!

4. 19.3: Polar coordinates • Why? • Lots of physics (cylindrical or rotational symmetry) • Equations are mostly similar to before • waves, diffusion, Laplace & Poisson • How? • Derivatives in cylindrical & spherical coordinates • Lots of examples • on the black board! • But first the principles!

5. Physics PDEs in polar coordinates • The diffusion equation where 2u= u/t 2u r2u + Θ2u + φ2u Ansatz: u(r,Θ,φ,t) = F(r) G(Θ) H(φ) T(t)

6. Separation of variables in polar coordinates • As before, we need each operator to boil down to essentially a constant factor r2u + Θ2u + φ2u = 0 - au - bu- cu = 0 With polar coordinates as with Cartesian coordinates, all that remains of the PDE after separation is an algebraic equation and a set of ODEs – much easier to solve!

7. Cases • Laplace’s equation • Plane polar coordinates • Cylindrical coordinates • Spherical coordinates • The wave equation • Plane polar coordinates • Cylindrical coordinates • Spherical coordinates

8. Common features • The angular part is the same in all cases • Sinusoidal in plane and cylindrical coordinates • Spherical harmonics in spherical coordinates • The radial part differs • Depends on the equation • Depends on the number of coordinates • plane, cylindrical, spherical

9. Laplace’s equation • The angular part • Spherical harmonics in spherical coordinates • Sinusoidal in plane and cylindrical coordinates • special case of spherical harmonics • The radial part • Polynomial in plane polar case with origin included • Bessel functions in cylindrical coordinates • Legendre functions in spherical coordinates

10. The wave equation • The angular part • Spherical harmonics in spherical coordinates • Sinusoidal in plane and cylindrical coordinates • special case of spherical harmonics • The radial part • Bessel functions in plane polar coordinates • Bessel functions in cylindrical coordinates • Spherical Bessel functions in spherical coordinates

11. Time for the black board! • Laplace’s equation • plane • cylindrical • spherical • Wave equation • plane • cylindrical • spherical

12. Laplace’s equation, plane

13. Enough for today! Good luck with the Exercises!