Answer to Question #253098 in Differential Equations for Anuj

Question

Solve the following initial value problem

Ut(x,t)=10Uxx(x,t) -10

U(-1,t)=U(1,t) Ux(-1,t)=Ux(1,t) t>0

Ux(x,0)=x+1 -1

Answer 1

Solution of heat equation:

$u_n(x,t)=B_nsin(\pi nx/L)e^{kt(\pi n/L)^2}$

$k=10$

$B_n=\frac{2}{L}\int^L_0f(x)sin(\frac{\pi nx}{L})dx$ , n=1,2,3,...

$f(x)=u(x,0)=10$

Then:

$-10u(-1,t)=u(1,t)$ :

$10\sum B_nsin(\pi n/L)e^{10t(\pi n/L)^2}=\sum B_nsin(\pi n/L)e^{10t(\pi n/L)^2}$

$\implies sin(\pi n/L)=0 \implies L=1$

$u_x=\sum B_n \frac{\pi n}{L} cos(\pi nx/L)e^{10t(\pi n/L)^2}=\sum B_n \pi n cos(\pi nx)e^{10t(\pi n)^2}$

$u_x(x,0)=\sum B_n \pi n cos(\pi nx)=x+1$

$B_n=20\int^1_0sin(\pi nx)dx=-\frac{20}{\pi n}cos(\pi nx)|^1_0=20/(\pi n)-20cos(\pi n)/(\pi n)$

$B_n=0$ for even n

$B_n=40/(\pi n)$ for odd n

$u_x(x,0)=40\sum cos(\pi nx)=x+1$ , n=1,3,5,...

$u(x,t)=\sum\frac{40}{\pi n}sin(\pi nx)e^{10t(\pi n)^2}$ , n=1,3,5,...