Answer Happy

This question concerns the heat equation in two spatial dimensions for ult, r, 0) with a source function F(t, r, 4) in polar coordinates: ди at 1 a rar ди ar д 1 au = F. p2 a82 (1) on a unit radius disk subject to boundary and initial conditions ut,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 a’y" + xy' + (x2 – n)y = 0 (3) and it has two independent solutions, J, (x) and Y.(x). Y, diverges as x + 0, while J is finite in this limit. Throughout this question you may find the following identities useful,
2n Jn-1(0) + In+1(2) Jn-1(2) - Jn+1(2) - Jn (3), 21(3). 7 - (4) (5) (a)[8 MARKS] Show that the ODE satisfied by Janmt) can be brought into Sturm- Liouville form, where x nm, m = 1, 2, 3, ... denote the zeros of Jn (2), ordered as 1n1 <0n2...<xnm.... Identify the functions p(2), (2) and w(x) appearing the Sturm-Liouville operator, as well as the Sturm-Liouville eigenvalue 1. Verify that this Sturm-Liouville operator is self-adjoint in the interval (0, 1]. (b) [2 MARKS] Show that O"@) = (e), is a Sturm-Liouville problem, where a 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): 1 I wonder nat)()}rder = (F (8 Mb)* Om (P'nqir 57 ? compute the bounded solution to (1), (2) for the source function F(t, r, 6) = (1 – p2)r cos 0. In addition to (4) and (5) you may find the following Bessel function identities useful, (204+1 Jn+1(x))' Jn(x) xn+1 (.x-n+1Jn-1(x)) Jn(2) x-1+1 (6) (7)