Answer Happy

2. This question carries [30 MARKS] in total. This question concerns the heat equation in two spatial dimensions for u(t, r, 0) with a source function F(t, r, 0) in polar coordinates: 𐐀и 18 - För (10). - F, 18u 700 (1) on a unit radius disk subject to boundary and initial conditions u(t, 1,0) -0, u(0,r, 0)-0. (2) The source function F' is continuous, bounded and obeys the same boundary conditions as u. Recall that the Bessel equation of order n is given by z¹²y + xy + (x² − n²)y-0 (3) and it has two independent solutions, J₁(x) and Y₂(2). Y₁ diverges as →0, while J,, is finite in this limit. Throughout this question you may find the following identities useful, 272 Ju-1(2) + Ju+1(2) Ju(2), (4) Jn-1(2) - Jn+1(2) 2.J'(z). (5) (a) [8 MARKS] Show that the ODE satisfied by J₂ (Fam) can be brought into Sturm-Liouville form, where Zm, m=1,2,3,... denote the zeros of J₂(z), ordered as Inl <In2 <... < Inm <... Identify the functions p(z), q(z) and w(z) appearing the Sturm-Liouville operator, as well as the Sturm-Liouville eigenvalue >. Verify that this Sturm-Liouville operator is self-adjoint in the interval [0, 1]. (b) [2 MARKS] Show that 8" (0) = €(0), is a Sturm-Liouville problem, where or is a constant and is periodic with period 2. (c) [10 MARKS] Using the method of separation of variables, write down the most general solution to (1), (2) which is bounded as r → 0. (d) [10 MARKS] Using the orthogonality relation associated to the Sturm-Liouville problem in part (b), and given the following orthogonality relation associated to the Sturm-Liouville problem in part (a): [Jn (Fagst) Ja (Eng) 2ilx = (Jn (Eup))² &per compute the bounded solution to (1), (2) for the source function F(t,r,0)(1-²)r cos 0. In addition to (4) and (5) you may find the following Bessel function identities useful, (2+¹Jn+1(x)) J₂(x) = (6) J₁(x) = (7) (I ¹¹ J₂ 1(x))' -n+1