MATH 300: Solutions for Problem Set 1

Transcription

1 Question.a: The de is MATH 300: Solutions fo Poblem Set 2u 4u 4u u 0: So a 2, b 2 and c 4. Hence b 2 ac 4 < 0. The de is ellitic. Question.b: The de is u 2u u u 0: So a, b and c. Hence b 2 ac 2. The de is ellitic when jj <, aabolic when jj and hebolic when jj >. Question : The de is u t u ; u (; 0) : The oosed solution is u (; t) e t. Cleal, u (; 0) e 0. And since b diect calculation u t (e t ) t e t and u (e t ) e t the de is cleal satis ed. Question 2.2. : It follows fom the de that u ) u 2 2 A () ) u (; ) 2 2 A 4 e () B () ; whee e A () and B () ae abita functions of thei aguments. Question : It follows fom the ode that u 0 atu sin (t) ) d dt ) e at2 2 u (t) u (0) ) u (t) e at2 2 u (0) e at2 2 u e at2 2 sin (t) Z t 0 Z t 0 e a2 2 d e a2 2 d : Question 2.3. : The oblem coesonds to the one dimensional heat equation. The initial condition is a function of osition. The temeatue is seci ed at the ight hand bounda (and is a function of time). At the left hand bounda the mateial is efectl insulated. Question : The de is u t 2u ; u (; 0) ; u (0; t) u (2; t) 0:

2 The oosed solution is u (; t) e 2t. Cleal, u (; 0) e 0 and u (0; t) e 2t sin (0) 0 and u (2; t) e 2t sin (2) 0. And since b diect calculation u t e 2t t 2e 2t and u e 2t e 2t the de is cleal satis ed. Question : The ode is u t (u 20) whee u (0) 00; whee u (t) is the temeatue at time t and is a ositive constant. (t) u (t) 20, then the ode is Let t whee (0) 80; which has the solution (t) 80e t ) u (t) 20 80e t : Question : Pat : Witing the Lalacian in clindical coodinates. We have the tansfomation with the invese tansfomations cos () ; ; z z; 2 2 ; tan () ; z z: Fom the chain ule u u u 2 2 u 2 2 u ; so that 2 2 u u 2 u 2 2 ( 2 2 ) u ( 2 2 ) 2 u u 2 2 u 2 (u ) u 2 2 u ( 2 2 ) 32 u 2 u 2 2 (u ) 2 2 u 2 2 u 2 ( 2 2 ) 32 u

3 Similal, so that 2 2 Hence, we see that ( 2 2 ) 2 u 2 ( 2 2 ) 2 u : 2 u u u u u 2 u (u ) 2 ( 2 2 ) u ( 2 2 ) 2 u u 2 u u ; 2 2 u u u 2 2 (u ) 2 2 u 2 u u 2 ( 2 2 ) 32 u 2 ( 2 2 ) 2 u 2 ( 2 2 ) 2 u : 2 ( 2 2 ) 32 u u u u u u u 2 u zz (u ) u 2 u zz: Pat 2 : Witing the Lalacian in sheical coodinates. Hee, we will go in the evese diection. We will show that the tansfomation when intoduced into cos () ; ; z cos () ; 2 2 u u 2u 2 ( u ) 2 sin 2 () u cot () u u 2 2 sin 2 () u ; gives the Lalacian in Catesian coodinates. Fom the chain ule u u u z u z cos () u u cos () u z : u cos () (u ) (u ) cos () (u z ) : 3

4 And since (u ) cos () u u cos () u z ; (u ) cos () u u cos () u z ; (u z ) cos () u z u z cos () u zz ; we have u cos () [ cos () u u cos () u z ] [ cos () u u cos () u z ] cos () [ cos () u z u z cos () u zz ] sin 2 () cos 2 () u sin 2 () sin 2 () u cos 2 () u zz 2 sin 2 () cos () u 2 cos () cos () u z 2 cos () u z : Likewise, u u u z u z cos () cos () u cos () u u z : u [ cos () cos () u cos () u u z ] cos () u u cos () u z cos () cos () (u ) cos () (u ) (u z ) cos () u u cos () u z cos () cos () [ cos () cos () u cos () u u z ] cos () [ cos () cos () u cos () u u z ] [ cos () cos () u z cos () u z u zz ] cos () u u cos () u z 2 cos 2 () cos 2 () u 2 cos 2 () sin 2 () u 2 sin 2 () u zz 2 2 cos 2 () cos () u 2 2 cos () cos () u z 2 2 cos () u z : Likewise, u u u z u z u cos () u : u [ u cos () u ] cos () u u 4

5 (u ) cos () (u ) cos () u u [ u cos () u ] cos () [ u cos () u ] cos () u u 2 sin 2 () sin 2 () u 2 sin 2 () cos 2 () u 2 2 sin 2 () cos () u : u 2u cot () u u 2 2 sin 2 () u sin 2 () cos 2 () u sin 2 () sin 2 () u cos 2 () u zz 2 sin 2 () cos () u 2 cos () cos () u z 2 cos () u z 2 [ cos () u u cos () u z ] cos () [cos () cos () u cos () u u z ] [ cos () u u cos () u z ] cos 2 () cos 2 () u cos 2 () sin 2 () u sin 2 () u zz 2 cos 2 () cos () u 2 cos () cos () u z 2 cos () u z [cos () u u ] sin 2 () u cos 2 () u 2 cos () u sin 2 () cos 2 () cos 2 () cos 2 () sin 2 () u sin 2 () sin 2 () cos 2 () sin 2 () cos 2 () u cos 2 () sin 2 () u zz 2 sin 2 () cos () 2 cos 2 () cos () 2 cos () u [2 cos () cos () 2 cos () cos ()] u z [2 cos () 2 cos () ] u z 2 cos () cos2 () cos () cos () cos () 2 cos2 () [2 cos () cos () cos ()] u u u zz : u u 5

6 Question : To ove that the clindical coodinate sstem in othogonal is equivalent to showing that nomal vectos associated with the coodinates lanes ae all othogonal. That is, we must show that and and z ae all othogonal whee (@ z ) : B diect calculation 2 2 (; ; 0) ; z (0; 0; ) ; 2 2 Cleal, tan () 2 ( ; ; 0) : 2 z z 0: Question : Fo the geneal tansfomation f (; ) ) d f d f d; g (; ) ) d g d g d: in this oblem, whee f and g, it follows that d d d and d d d; fom which it follows that (ds) 2 (d) 2 (d) 2 (dz) 2 (d d) 2 (d d) 2 (dz) h (d) 2 (d) 2i (dz) 2 : 6