Question

4. Reminder: Vều can be expressed in many different coordinate systems. In polar coordinates, = 10u1 02u rör 72 882 If a system is axisymmetric we get the added simplification that we 0. ar2 ди Use separation of variables to tum the following PDE into ODES. Be sure to specify the appropriate initial conditions and boundary conditions. You do NOT have to solve the ODES a) An axisymmetric system described by = kvều where u(0,t) = 0, u,(5,t) = 0 and u(r,0) = r2 b) For your answer to part a), circle the spatial ODE and label it "spatial". Circle the temporal ODE and label it "temporal". Which ODE should be solved first? In one sentence explain why.

Answer 1

4 sol - from the daba we given havef Reminder: D²u can be expressed in many different coordinate systems. In polar Coordinates, du do? if a system is o do? ymmetric get the added Simplication -0. Deu + | du dr ㅎ 읍 + I due axis- we lication that du de An a by du JE axisymmetric system system derribed by Ekvu where ucot) = 0, ur (sit) and u (8,0) = 82 :- du = kolu Jt and acro) =82 ucoit) - our (sit) =0 - ou du t 1 du r dr t tu r² soz dr² du do for so axisymmetric system, v²u du u = + I tu 91 8r I du du st = K 882 = By separation of vartable let ucrit) R(1) TIE) Ut=RT', 499 ERPT, un = R'I = K (R" T +R'T Divideng both side by 'RT' we get r RT' 11 R' T' T EK ( 91 R

since, both sides ane independent of each other so in the equality, both of them most be ondo91 to hold 7 equal to some Constant say's T' R" K d T R r R T-d T=0 Temporal 11 KR" 의지 K RI + el =d R 9 kar" + KR' = dB KrR" +KR' Ar R=0 Spatial u (ort) RLO) T 16) = 0 Rlo) =0 (TLL) TO) un ( 56) ľ (5) T(6)=0 → R'(5) = 0 (: 1 (t) +O) 2 u(no) = R(1) Tlo) = 92 Tlo) = 92 (R() to So the has been converted to geven parteal DE the following ODES: KOR" +KR'-VOR = 0 - 0 T-HT = 0 2 with, RIO) = 0 IR'(5-) = 0 TlO) = 9?

should → spaticel OPE be solved frost because b thes We can through find the value of t (tvel-ve) use fupture to solve the temporal which we will ODE, ... If you Impress with Answer. please give me like