The function 4 : R x (0, co) - R given by (I, t) = Ant is called the fundamental solution of the one-dimensional heat equation. Notice that the fundamental solution does not have the form of a separated variables solution. (a) Verify that o satisfies the = us for (x, f) E R x (0, Do). (b) Use L'Hopital's rule to determine lim $(x, (). (c) Make an animation of the spatial graph of the fundamental solution o with ani- mation parameter t. (d) Calculate the spatial L' norm I(!) = (x,t) 2 of the fundamental solution. Hint(s): Note that I (t) = 2J(t) where J(t) = (1,t) dr. Calculate J(t)2. Use y as a spatial variable of integration in one of the factors J(t). Write what you get as an iterated integral and then as an integral of a function of two variables over the first quadrant. Use polar coordinates. (e) How could you modify 4 so that it satisfies the = kuzz for non-unitary conductiv- ity? Hint(s): Remember Problem I from Assignment 3 = Exam 1 about scaling in space and time. Also read section 10.4 of Haberman and see what Haberman defines as the fundamental solution. (f) Given up : R - R with u E CO(R), the function u( x, t) = 1(x - 5,t) 40() VEER is called the spatial convolution of the fundamental solution with up. Show that this spatial convolution satisfies the initial value problem on R x (0, Do) "(I, 0) = 10(1), TER for the heat equation on the whole real line. (g) (Bonus) How can you modify the one-dimensional fundamental solution of the heat equation to obtain the fundamental solution of the heat equation 4 : R. x (0, co) - R on (all of) R"