Question

Solving PDE with separation of variables

3. Solve the heat flow equation on a circle. (10 point) Otu(t,0) = o u(t,0). such that the initial condition is u(0,0) = cos? (0)

Answer 1

variables. The given quation is heart flow equation -alto) = Duct, o) Initial condition i uco,o) = cos Solutions. We use the method of separation of The given equations can be unitlen as su = du (1) Assume that the solution equ (1) is of the form a (t,0) = Tit) $0) ) where T is a function of t only and & is a function of only. Substituting this in (1), we obtain - TQT" & ie T = 6 T 0 thich gives us to ordinary differential equations as & T -KEco jedp - KP = 0 C Solering Equations (4) and we get W when kis positive and = pt say, then I TE Get tept p = Cz epet de ² do REDMI NOTE 8 O AI QUAD CAMERA

se pt say, then in when Kis-ve and = -p2 T= cu cospt to scnpt, a=ge (ii) when Kis zero, we have - Ta Cytte p-ca. Thus the various possible solutions of the given heat equction (1) are u= (e, et tett) sept (6) U = (accost & Casinpt) c, e pt (7) U = (t + G) Cq (8) Where Ci, i=1,2,... 9 are constants of lintegration. Now out of the above three solutions we have to choose the solution which is consistent with the physical situation. tere, since the qu. Ą heat equation We have to consider heat Conduction as the physical situation. Hence it must be a transient solution, ie u is to kontroase decrease with the chercase of to me t as a represents temperatur Accordingly the solution given by Eq. (6 (7) is the correct solution OO REDMI NOTE 8 CO AI QUAD CAMERA

which is of the form I ua (e, cospt + c sinpt) e o t (9) where d, and are new instants. Given initial condition is u(0,0) = cos²o, Teatt-o, u caso From Eqn. (9) we have U= coto = [c, cospcolt sy einpoje ie e, a cos²o The solution is u(t,0) = (costo cospt + c ronfpt) et REDMI NOTE 8 AI QUAD CAMERA