Question

4.4. Consider the standard equilibrium heat equation with a source u (D 1, cp on x E [0, L]. Given the following parameter values and boundary condi tions, determine (1) the equilibrium solution and draw a graph, and (2) compute the flux and indicate on the solution graph the direction and magnitude of the flux. Alternatively specify if the solution does not exist, or detail how it is not fully determined 0 uxxR(x) _ (a) R 1, u(0)= 1, u(L 10 (b) R A, u(0) - B, u(L) C, where A,B,C are constants. (c) R x(1x) u'(0) (d) R 2, u(0)= 1, u(L 1) 0. (e) R sin(x), u(0) = 0, u(L 2T) 0 0, u(L ) 0 _

Please ONLY work parts a, d, e

Answer 1

Anscs' Giiven that the standaxd eauiliboium heat eauation is E Co,L]-O 20=Ucx +R(2) eauiliboium (a)Naw at Giiven thot RE uco1, u(L:):o Uxx t4-o Ux 1 Integation gies Ue = ーズ+C、 Agin intgstion u(o) AOC (o) +C2 (T-7n

Cand C in cauatien put tte values u x) > ThePluis qiven by he 9ズ の:ーズ+。 lux d) Given hat R2 uco) - 1, ull=リ=0 The Cauation o he eauilibaium is Uzz+R:0 UcI7-R2 Tategoation gives ニ-2エ+C、 Again integatin gives 9+x'pt x-:cx)n

integation Constant C and C whese axe he general soluion is ulr,4)2-x+0x+02 Conditions Put the iniial and boundasy ulco uci) z0o 0ニー2x Co) and ○ニー1 +Co which is the eaieed genesal Solution, The he. ニー22 elux -x Given that R- sinc), ulo):0, uCl-2) - o .

The cauation of the cquilibaium is Sinx Integation gives ニ COSス+C, he Again intetion gives uzo U -c Puee om geneal Sduticnis given by The SinT + C13+C u boundary Conditions put tte initial and Since)+C, 2T uit):Siox ohich eauised sdution Plu is The COS .