Solve the IBVP wave equation. d^2/dt^2=16d^2/dx^2 0<x<pi u(x,0)=sinx du(x,0)/dt=0 u(0,t)=u(pi,t) =0 t>0
Wave equation: (d^2u/dt^2) = 9(d^2u/dx^2) with u(0,t) = u(π,t) = 0 u(x,0) = 3sin4x + 8sin5x, ∂u/dt(x,0) = x, 0 < x < π/2 , π − x, π/2 < x < π.
Consider the second order partial differential equation du/dt=
d^2u/dx^2 +2du/dx+u over the domain x in [0,l) and t>=0. It is
given that u(0,t)=u(l,t)=0. Use the method of separation of
variables to prove that the general solution with the given
boundary condition is u(x,t)= infinity series n=1
bnsin(npix/l)exp(-x-((npi/l)^2)t) where bn is
a constant for every n N
Hint u(x,t)=X(x)T(t)
tnsit te Seind ond partial difertinl cuatan +2n St the dowain e To,e) an Use metod o Separet ion Vaiades to rore...
for the following parabolic PDEs heat equation for one variable d2/dx² u(x,t) = d/dt u(x,t) . Where u(0,t)=0 , u(1,t)=0 , u(x,0)=sinπx . Complete using crank nicolson method . With h=0.2 , k=0.02
2. Use eigenfunction expansion to solve the following IBVP: u,(x, t) ="-(x,t) + (t-1)sin(m), 0
3. Solve the wave equation subject to the conditions u(0,t)=0, u(z,t) = 0 at 2 2 u(x, 0) = 4 =0 at 2 =1
3. Solve the wave equation subject to the conditions u(0,t)=0, u(z,t) = 0 at 2 2 u(x, 0) = 4 =0 at 2 =1
slove the system eqution: d^3y(t)/dt^3 - 2 d^2y(t)/dt^2 - 5 dy(t)/dt +6 y(t) = 2 d^2u(t)/dt^2 +du(t)/dt +u(t) A) compute the transfer function Y(s)/U(s)? B)Find inverse Laplace for y(t) and x(t)? C) find the final value of the system? D)find the initial value of the system? Please solve clearly with steps.
Solve the following integrodifferential equation 2 dx + 5x +3 / x dt + 4 = sin 4t, x(0) = 1. 0 x(t) is calculated as e-t + e -1.5t , + cos t) + sind |t)]u(t).
Solve the following homogeneous wave equation: un (2. t) = 4uxx(x, t), u(0.t) = u(t) = 0, u(x,0) = 0, (3,0) = 1.
2. Use separation of variables to solve the IBVP: utt (z, y, t) u(0, y, t) u (x, y,0) uzz(z, y, t) + un, (x, y, t) = 0, 0 < x < 1, 0 < y < 1, 0, u(1,y,t)=0, u(z,0,t)=0, u(z, l,t) = 0 sin(r) sin (2my), ue (r, y,02 sin(2mx) sin(ry) t > 0, = =
For 0 x π , 0S9, π , and 120 , solve the 2-D wave equation subject to the following conditions. u(0,y,t)-0, u(T.yt):0, u(x,0,) u(x,π, t) 0, 0 Boundary condition: C11 1 u(x),0)-sin(x)sin(2y) + sin(2x)sin(4y), 0 at It=0 Initial condition:
For 0 x π , 0S9, π , and 120 , solve the 2-D wave equation subject to the following conditions. u(0,y,t)-0, u(T.yt):0, u(x,0,) u(x,π, t) 0, 0 Boundary condition: C11 1 u(x),0)-sin(x)sin(2y) + sin(2x)sin(4y), 0 at It=0 Initial condition: