18.306 Lecture 22 - Thu 2021 05 13 - Virtual
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Continue with Green's Functions:
Finish #S08
xtra: Show how to obtain Green's functions by reverting
      integration/summation for mode expansions.
#S09
#S09b Already done.  Green's function then \sum lambda_n^{-1} u_n(x)u^*_n(y)
      where \{u_n\} orthonormal basis of eigenvectors.
Plus
either #S10
or     #S04a-b [Normal modes and why they work + completeness].

Future:
#S10, #S10b, #S15.
Fundamental Solutions for Laplace.
#S13, #S14 [Random walks]
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 % -------------------------------------------------------------------- #S08
 Fundamental solution for heat equation. Dimension n = 1.
 Obtain similarity form of the solution by:
    Dimensional arguments.
    Stretch invariance of the problem, and (assumed) uniqueness.

 For u_t = D u_xx form is  u = (1/sqrt{D*t}} f(x/sqrt{D*t}),
 where f must have integral 1.
 --- Substitute form derived, get ode and solve.
 --- Fix free constants by constraint on integral.

       u = = (1/sqrt{4*pi*D*t}} exp(-x^2/4*D*t),

 --- Show solution yields delta function as t \to 0 (t > 0).
 --- 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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 % -------------------------------------------------------------------- #S09b
 Special point:
 Note that Green's function for heat equation is symmetric   [1]
    G(x, y) = G(y, x).
 Same thing observed for the Green's function for problems
    like u'' + u = f(x), 0 < x < 1, with u(0) = u(1) = 0.    [2]
 Why?
 Relate to self-adjoint problems.
 For [2], computing inverse of self-adjoint operator.
 For [1], solution to \dot{y} = A y [y = exp(A*t) y0] when
     A = self-adjoint operator with discrete spectrum only.
     G = Sum phi_n(y)^* phi_n(x) e^{\lambda_n*t}

 Finally: expression of Green's function for self-adjoint
          operator using orthonormal set of eigenfunctions.
          Poorly convergent [not surprising, given the delta].
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 % -------------------------------------------------------------------- #S10
 Fundamental solutions for special bounded and semi-bounded domains,
 with various BC [Dirichlet, Neumann, Periodic].   Exploit symmetry:
 Reflection principle. Method of images. Do 1D examples [2D in #S15].

 Stress role of symmetries to get Green functions for problems with
 boundary conditions. The method of "images" is based on the idea of
 "mirror" symmetry. But others can be exploited [e.g.: for Robin].
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 % -------------------------------------------------------------------- #S10b
 Green's functions and weak forms.
 Show how to write equations in "weak" form, using test functions.
 Example: heat equation.
    --- important for finite elements.
    --- gives clear mathematical meaning to the problems that
        Green functions solve.
 Do example of heat equation in a half line, with initial conditions
 and Dirichlet BC.   Use test functions that satisfy the homogeneous
 BC,  and show that the formulation yields back the original problem
 for a smooth enough solution of the weak problem.
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 Green function in multi-D and methods of images.
 --- Do generic case. Images NEED NOT be deltas.
     Examples with Robin boundary conditions.

     See:            Green's functions #04  [Assigned in Problem Set].
                     [IVP, heat equation in the semi-infinite line].
         Symmetry here is:
         --- if u solves heat equation, then v = u-u_x solves it too.
         --- if v solves heat equation, then u defined by
             u-u_x = v and u bounded at infinity solves it too.
             u = e^{x}*\int_x^\infty e^{-s} v ds
          Why?
          0 = v_t-v_xx = (u_t-u_xx) - (u_t-u_xx)_x
          so that  u_t-u_xx = a(t)*e^x.  From boundedness, a = 0.

     See:            Supplementary material.
                     Method of images. Robin BC in an interval.

     Both in the problem series: Point sources and Green's functions.
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