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

Overview Changes More black-board work-througs! Monday exercises extended (10-13)! No Maple TA (still Maple) on Wednesdays! Sections : 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!

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!

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!

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

Separation of variables in polar coordinates As before, we need each operator to boil down to essentially a constant factor  r 2 u +  Θ 2 u +  φ 2 u = 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!

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

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

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

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

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

Laplace’s equation, plane

Enough for today! Good luck with the Exercises!