Question

3) Considere o problema do Calor Ut x para 0 x<T et > 0 u (0, t)u (T, t) = 0 para t 0 u (x, 0) f (x) para 0 x<T Encontre a solução explicitamente por separação de variáveis, nos seguintes casos: a) f (x) sin b) f (x) sin3 x c) f (x) (x)

3) consider the heat equation:

Find explicit solution by separation of variables for the following cases:
a f(x) = sin x

b f(x)= (sin^3)x

c f(x)= x(pi - x)

** para means 'for' and 'e' means ' and'

Answer 1

SOIuction 410t)= 47t) o 4010)3f(U)- Let ulut) x(u) Tit) xy then vet u- appy Bc of euetion A O

from Tit TH T(t)= ce genenrece solutrien 91t T En sin nx e appy mitfas Condition Hence Ed Thus sinxe (9 419 [ingx3inn-usin7

f(u) 3 sina-in Hence ift t 3 4 ushen fu) xin- 2- En sin nx En= cos nx - 1x cashx +h n2 n 3casnx +xcesn ausinx + n2 ncas nn + ncusnn y1-cas nn) -sh 1 -n2t 1ence nsn