Question

1. Solve the following transient heat equation by separation of variable method. ат ах Atx=L, T =To IC: At t = 0, T=T,

Answer 1

The solution is given in the images below.

The transiert heat equathon is 1 we will Solve this bl usi꺼 separation of van table method we as sume that re placin this value in the heat equatiom we ottain Tlt) CL) as eath term im the above ua umc hiom ot and the oahe equal, 9o they must oe Cims tont whh have taku SPNouu an mommal derivative insted v ion ane cither pantial derivative and we have t oin difieTenhal equation agiven om diman dTLt) C2 K Tx) (3) /8-K2-as it is Constant Selvina the cnuation (2) we have TLI) (4 In is the in tesral coms tant dyt2 So the cmanol Snlution is the pro duetot () ond Tv

Now, we have to use the bs un d any and ini hal Cond- tiom to ind out value of dicrem Constant usin2 the mitial Condition that at tFa· T-Ti is Ti independent f z,so we asmitiall usimg the boun dany Con ditiom that at at all time, so"we have L T- T то Sim KL diffevemtiatimg T with res we have, χ u simg the looun dary Can dition that a 27 t all t o CL

So, the emeral Solution Heat equatiom the Con stamt aru given toe low e Lo 0 To Coseek Ti Cot k