18.306 Lecture 22 - Thu 2021 05 13 - Virtual
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Continue with Green's Functions:
#S08 --- a few points left to do.
xtra: Show how to obtain Green's functions by reverting
      integration/summation for mode expansions.
Normal modes [last class proved self-adjoint theorem for matrices].
Show examples in infinite dimension.
#S09  Heat fundamental solution, n > 1.

TO DO next
#S22  Fundamental Solutions for Laplace [this part assigned pset]
      Sketch BIMs
      Method of images and convergence issues.

Probably will not have time to do.
#S13  Random Walks
#S14  Random Walks
#S15  Green function in multi-D and methods of images.
      Do just for Neumann and Dirichlet

SKIP
#S10b SKIP
#S11  Does not exist
#S12  Does not exist

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 % -------------------------------------------------------------------- #S08
 Fundamental solution for heat equation. Dimension n = 1.
     u = = (1/sqrt{4*pi*D*t}} exp(-x^2/4*D*t).
 --- Note solution becomes non-zero everywhere for any t > 0,
     hence "infinite propagation speed".
 --- On the other hand "the bulk of the heat" stays within a region
     of size O(\sqrt{D\,t}). Confirms early dimensional arguments.
 --- Use form of the solution just obtained to show what happens with
     initial discontinuities, discontinuities in derivative (integrate
     by parts), ... etc.
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 % -------------------------------------------------------------------- #S09
 Fundamental solution, n > 1 --- see [A] below.
 --- Again, as t \to 0, get delta in n-D.
 --- Solution initial value problem. Generally smooth for any t > 0.

 Derivation by similarity arguments, same as for #S08. -------- % [A]
    Final formula involves "area" of unit sphere S_{n-1}. ----- % [A]
 Derivation using tensorial nature of n-dim heat equation. ---- % [A]
    Get formula for "area" of unit sphere S_{n-1}, any n. ----- % [A]

 For [A] use the problem:
 --- Problem series: Point Sources and Green functions.
 --- problem:        Nonlinear diffusion from a point seed.
 --- Subsection:     Example: Green function for the heat equation
                     in R^d (The area of a sphere in d-dimensions).
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 % -------------------------------------------------------------------- #S22c
 Various lecture notes for 18306.
 Section:    Laplace and Poisson equations - harmonic functions. 
 Subsection: The fundamental solutions.
 Plus [A] & [B] below.
 
 [A] ................................................................. [PTS]
 Point out the limitations that the slow decay (or none in 2-D) of
 the fundamental solutions imposes on the method of images for the
 Laplace operator [as opposed to the exponential decay in the heat
 equation,  which allows good approximations to a Green's function
 with only a few terms --- for moderate times].
 Not even convergence is guaranteed!

 [1] To obtain convergence in dimensions 2 and 3, the "infinities"
     must be subtracted. Illustrate this with
     f(x) = sum_{n \neq 0} 1/|x-n|         (not convergent),
            which can be re-defined as
     f(x) = sum_{n \neq 0} (1/|x-n|-1/|n|) (convergent)
            upon subtracting the "infinity".

 [2] The approach in (1) can also be used to accelerate
     convergence, e.g.:
     f(x) = sum_{n \neq 0} (1/|x-n|-1/|n|-|x|/n^2)
            + |x| sum (1/n^2),
     where the second term can be done exactly.
      
    However: convergence is always slow
             (i.e.: no better than 1/n^p, for some p).

    A similar situation arises in other elliptic problems. For
    example, Stokes equation. This leads to serious difficulties
    --- e.g.: lack of good approximations for suspensions at
    concentrations which are not very small.
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