Approximate the solution to the heat equation
du/dt-d2u/dx2=0, 0

Question

Approximate the solution to the heat equation
du/dt-d2u/dx2=0, 0<t<1, 0<=t
with boundary conditions
u(0,t)=u(1,t)=0,0<t
and initial condition
u(x,0)=sin(pi)x,0<=x<=1

Compare results at time t=0.5 at location x=0.2,0.4,0.6,0.8 to the exact solution u(x,t)=exp(-pi2t)sin(pi)x
a) use FTCS (forward time centered space) scheme with h=0.1, k=0.0005
b) use the fully implicit scheme with h=0.1, k=0.01

anonymous · Answer

Approximate the solution to the heat equation
$$\frac{ \delta u }{ \delta t }-\frac{ \delta^{2}u }{ \delta x^{2} }=0$$
$$0<t <1, 0 \le t$$
with boundary condition
$$u(0,t)=u(1,t)=0,0<t$$
and initial condition
$$u(x,0)=\sin(\pi x)$$
Compare results at time t=0.5 at location x=0.2,0.4,0.6,0.8 to the exact solution
$$u(x,t)=\exp (-\pi^{2} t)\sin(\pi x)$$