Heat equation and Wave equation problems

[dominic.tsy]

1.
Solve the Heat equation u_t = ku_xx for 0 < x < ∏, t > 0 with the initial condition

u(x, 0) = 1 + 2sinx

and the boundary conditions u(0, t) = u(∏, t) = 1

(Notice that the boundary condition is not homogeneous)

3.
Find the solution of the Wave equation u_tt = 4 u_xx with

u(0, t) = u (π/2, t) = 0, t > 0

u(x, 0) = sin (2x) - 2sin(6x), u_x (x, 0) = -3 sin4x, 0 ≤ x ≤ ∏/2

Please help! =[...

 

[tiny-tim]

welcome to pf!

hi dominic.tsy! welcome to pf!

(try using the X₂ button just above the Reply box )

show us what you've tried, and where you're stuck, and then we'll know how to help!

 

[HallsofIvy]

1.
Solve the Heat equation u_t = ku_xx for 0 < x < ∏, t > 0 with the initial condition

u(x, 0) = 1 + 2sinx

and the boundary conditions u(0, t) = u(∏, t) = 1

(Notice that the boundary condition is not homogeneous)

Define v(x, t)= u(x,t)- 1. What differential equation, boundary condition, and initial condition does that satisfy?

3.
Find the solution of the Wave equation u_tt = 4 u_xx with

u(0, t) = u (π/2, t) = 0, t > 0

u(x, 0) = sin (2x) - 2sin(6x), u_x (x, 0) = -3 sin4x, 0 ≤ x ≤ ∏/2

Please help! =[...

Try a Fourier series solution of the form $$\sum A_n(t) sin(nx/4)+ B)n(t)cos(nx/4)$$