Question

(PDE)

Solve the ff boundary value problems using Laplace Equationnon the square , omega= { o<x<phi, 0< y <phi}:

u(x,0) =0, u(x,phi) = 0 ; u(0,y)= siny , u(phi,y) =0

Answer #1

(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

7.4 Solve the Laplace equation u = 0 in the square 0 < x, y
< π, subject to the boundary condition u(x, 0) = u(x, π) = 1,
u(0, y) = u(π, y) = 0.

Using the Laplace transform, solve the system of initial value
problems:
w ′′(t) + y(t) + z(t) = −1
w(t) + y ′′(t) − z(t) = 0
−w ′ (t) − y ′ (t) + z ′′(t) = 0
w(0) = 0, w′ (0) = 1, y(0) = 0, y′ (0) = 0, z(0) = −1, z′ (0) =
1

Solve the below boundary value equation
1. Ut=2uxx o<x<pi 0<t
2. u(0,t) = ux(pi,t) 0<t
3. u(x,0) = 1-2x 0<x<pi

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

(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

Solve the PDE using the change of variables v = x, w = y/x
x2uxx + 2xyuxy +
y2uyy = 4x2

Solve the PDE using the change of variables v = x, w = y/x
x2uxx + 2xyuxy +
y2uyy = 4x2

Solve the system of differential equations using Laplace
transform:
y'' + x + y = 0
x' + y' = 0
with initial conditions
y'(0) = 0
y(0) = 0
x(0) = 1

Solve this Initial Value Problem using the Laplace
transform.
x''(t) - 9 x(t) = cos(2t),
x(0) = 1,
x'(0) = 3

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