Question

PDE

Solve using the method of characteristics

Plot the intial conditions and then solve the parial differential equation

utt = c² uxx, -∞ < x < ∞, t > 0

u(x,0) = { 0 if x < -1 , 1-x² if -1≤ x ≤1, 0 if x > 0

ut(x,0) = 0

Answer #1

(PDE
Use the method of separation of variables and Fourier series to
solve where m is a real constant
And boundary value prob. Of Klein Gordon eqtn.
Given :
Utt - C^2 Uxx + m^2 U = 0 ,for 0 less than x less pi , t greater
than 0
U (0,t) = u (pi,t) =0 for t greater than 0
U (x,0) = f (x) , Ut (x,0)= g (x) for 0 less than x less than
pj

(PDE)
WRITE down the solutions to the ff initial boundary problem for
wave equation in the form of Fourier series :
1. Utt = Uxx ; u( t,0) = u(t,phi) = 0 ; u(0,x)=1 , Ut( (0,x) =
0
2. Utt = 4Uxx ; u( t,0) = u(t,1) = 0 ; u(0,x)=x , Ut( (0,x) =
-x

8. Find the solution of the following PDE:
utt − 9uxx = 0
u(0, t) = u(3π, t) = 0
u(x, 0) = sin(x/3)
ut (x, 0) = 4 sin(x/3) − 6 sin(x)
9. Find the solution of the following PDE:
utt − uxx = 0
u(0, t) = u(1, t) = 0
u(x, 0) = 0
ut(x, 0) = x(1 − x)
10. Find the solution of the following PDE:
(1/2t+1)ut − uxx = 0
u(0,t) = u(π,t) =...

Solve the wave equation Utt - C^2 Uxx = 0 with initial condtions
:
1) u(x,0) = log (1+x^2), Ut(x,0) = 4+x
2) U(x,0) = x^3 , Ut(x,0) =sinx
(PDE)

Consider the one dimensional heat equation with homogeneous
Dirichlet conditions and initial condition:
PDE : ut = k uxx, BC : u(0, t) = u(L, t) = 0, IC : u(x, 0) =
f(x)
a) Suppose k = 0.2, L = 1, and f(x) = 180x(1−x) 2 . Using the
first 10 terms in the series, plot the solution surface and enough
time snapshots to display the dynamics of the solution.
b) What happens to the solution as t →...

Solve the non homogenous wave equation , Utt - c^2Uxx =1 ,
u(x,0) = sin (x) , Ut(x,0) = 1+x
(PDE)

Partial differential equations
Solve using the method of characteristics
ut +1/2 ux + 3/2 vx = 0 , u(x,0) =cos(2x)
vt + 3/2 ux + 1/2 vx = 0 , v(x,0) = sin(2x)

uxx = ut - u (0<x<1, t>0),
boundary conditions: u(1,t)=cost, u(0,t)= 0
initial conditions: u(x,0)= x
i) solve this problem by using the method of separation of
variables. (Please, share the solution step by step)
ii) graphically present two terms(binomial) solutions for
u(x,1).

Solve the following wave equation using Fourier Series
a2uxx = utt, 0 < x < pi,
t > 0, u(0,t) = 0 = u(pi,t), u(x,0) = sin2x - sin3x,
ut(x,0) = 0

Solve the following wave equation using Fourier Series
a2uxx = utt, 0 < x < pi,
t > 0, u(0,t) = 0 = u(pi,t), u(x,0) = sinxcosx,
ut(x,0) = x(pi - x)

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