18.306 Lecture 21 - Tue 2021 05 11 - Virtual % ============================================================================== Begin with Green's Functions: #MMM Motivation. #S07 Just the review of Green's functions for ode BVP. Example: u'' + u = f(x), 0 < x < 1, u(0) = u(1) = 0. Show how to obtain Green's functions by reverting integration/summation for mode expansions. Aim to cover [at least partially] #S08, #S09, #S09b, #S10, #S10b, #S15. plus Fundamental Solutions for Laplace. Maybe cover also #S04a-b [Normal modes and why they work + completeness]. Maybe cover also: #S13, #S14 [Random walks] % % ============================================================================== % % -------------------------------------------------------------------- #MMM "Motivation": M1) In Linear Algebra one can write the solution to A*x = y as x = G*y, where G = A^{-1}. Component-wise: x_n = sum_m G_nm y_m. G is defined by A*G = id. Component-wise sum_m A_nm G_mk = \delta_nk [one set of equations for each k]. Continuum limit: A now a linear operator on functions. Then write solution to A f = h as f(x) = \int G(x, y) h(y) dy, where: for every y, A*g = \delta(x-y), where g(x) = G(x, y). M2) Relationship with spectrum [eigenvalues, etc]. For a matrix A, the matrix-function R(z) = (z-A)^{-1}, is called the "resolvent". R is an analytic function in the complex plane with poles at z=lambda, lambda an eigenvalue. For a linear operator R(z) = (z-A)^{-1} is also called the resolvent. Let G(x, y, z) be the associated Green's function. Then the "spectrum" of A occurs where G has singularities. Poles correspond to eigenvalues; other types [branch points, etc.] give rise to other types of spectrum [continuum, residual]. M3) Cauchy's theorem and eigenvalue expansions. Cauchy's theorem says that, for an analytic function f(z), f(u) = (1/(2*pi*i)) \int_Gamma f(z)dz/(z-u) where Gamma is a counter-clock-wise path enclosing u. This extends to matrices, and we can write f(A) = (1/(2*pi*i)) \int_Gamma f(z)(z-A)^{-1} dz = (1/(2*pi*i)) \int_Gamma f(z) R(z) dz In particular, take f \equiv 1, so f(A) = identity, and apply it to a vector v. Then v = (1/(2*pi*i)) \int_Gamma R(z)*v dz If you then evaluate the integral using Residues, this yields the expansion of v in terms of the eigenvalues of A. This can be extended to operators. Then the final result is a formula showing how to expand a function in terms of the eigenfunctions of the operator. e.g.: Fourier Series and Transforms, Bessel Transform, Chebyshev expansions, Hermite polynomials, etc. For time dependent equations, the analog of the ode \dot{y} = A*y, the Laplace Transform provides a way to do an analog calculation that yields the solution in terms of normal modes. % % -------------------------------------------------------------------- #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. % % -------------------------------------------------------------------- #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). % % -------------------------------------------------------------------- #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]. % % -------------------------------------------------------------------- #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]. % % -------------------------------------------------------------------- #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. % -------------------------------------------------------------------- #S15 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. % % ============================================================================== EOF