2-40
Consider the following wave equation:
utt = c2 uxx, 0<x<a, 0<y<b
Subject to the following boundary conditions:
u(0,y, t) = 0, u(a, y, t) = 0, 0<y<b, t>0
u(x,0, t) = 0, u(x, b, t) = 0, 0<x<a, t>0
Find an expression for the solution if the initial conditions ar...

2-1 a, b
Consider the heat equation for a rectangular region, 0 < x < a, 0 < y < b, t > 0
ut = k(uxx + uyy) , 0 < x < a, 0 < y < b, t > 0
subject to the initial conditions: u(x,y) = f(x,y)
a) ux (0, y, t) = 0, ux (a, y, t) = 0, 0 < y < b, t > 0
uy (x, 0, t) = 0, uy...

The displacement u(x, t) from the vertical at distance x from its left endpoint, at time t, of a string of length L, fastened at both endpoints, satisfies the PDE
utt + aut = c2uxx, where a is a positive constant, with initial conditions
u(x, 0) = f(x), ut(x, 0) = g(x).
1. Solve the equation b...

please show all work in detail
Solve:
ut = k1 uxx + k2 uyy
on a rectangle (0<x<L, 0<y<H) Subject to
u(0 , y, t) = 0 uy = (x, 0, t) = 0
u(x, y, 0) = f(x,y)
u(L, y, t) = 0 uy = (x, H, t) = 0
Left and Right sides are kept at zero temp top and bottom ar...

7-4
Consider the displacement of ,u(r,,t) , a "pie-shaped" membrane of radius a and angle /3 that satisfies:
utt = c22u
Assume that >0. Determine the natural frequencies of oscillation if the boundary conditions are:
Problem a.
a) u(r, 0, t) = 0, u(r, /3, t) = 0,...

Consider the following problem; it can be interpreted as modeling the temperature distribution along a rod of length 1 with temperature decreasing along every point of the rod at a rate of bx (x the distance from the left endpoint, b a constant) while a heat source increases at each point the temper...

Concerning heat flow
I am confused about turning a non homogeneous equation (heat generation) into a homogeneous equation; could this process be explained in detail with an example....i unfortunately need this by noon on Thursday (EDT)
Thank You.

ut = 3uxx + 2, 0 < x < 4, t > 0,
u(0,t) = 0, u(4,t) = 0, t = 0
u(x,0) = 5sin2πx,0 < x < 4.
(a) Find the steady state solution uE(x)
(b) Find an expression for the solution.
(c) Verify, from the expression of the solution, that limt→∞ u(x, t) = uE (x)
for all x, 0 < x < 4...

see attached
Consider the following problem
ut = 4uxx 0<x<Pi, t>0
u(0,t)=a(t), u(Pi,t)=b(t) t>0
u(x,0)=f(x) 0<x<Pi
(a) Show that the solution (which exists and is unique for reasonably nice functions f,a,b) u(x,t) is of the form
U(x,t) = v(x,t)+(1-x/Pi) a(t)+x/Pi b(t)
wh...

Y,
I hope you and your family are well.
I am currently taking a course in PDE's and would like a few things explained if possible.
• Consider a bar insulated on both sides with the ends held at some constant temperature (other than 0) my analysis gives, that as times goes to infinity, t...

Find the region in the xy plane in which the equation [(x - y)^2 - 1] u_xx + 2u_xy + [(x - y)^2 - 1] u_yy = 0 is hyperbolic. The complete problem is in the attached file.

Using the chain rule of partial differentiation, to show that the differential equation
Au_xx + Bu_xy + Cu_yy + Du_x + Eu_y + Fu = G
transforms into the differential equation ...

Please show all steps.
1. Let f(x) be a 2pi- periodic function such that f(x) = x^2 −x for x ∈ [−pi,pi].
Find the Fourier series for f(x).
2. Let f(x) be a 2pi- periodic function such that f(x) = x^2 for x ∈ [−1,1]. Using
the complex form, find the Fourier series of the functi...

A general form of Parseval's Theorem says that if two functions are expanded in a Fourier Series
f(x) =1/2 ao + Sigma [(an cos(nx)) + bn sin(nx)]
g(x) 1/2 ao' + Sigma [(an' cos(nx)) + bn' (sin(nx)]
Then the average value, < f(x)g(x)>, is:
1/4 ao = sigma[an an' + bn bn'] prove this and ...

Consider a square wave. Write down a Fourier series for this function, and plot the original function and the series approximation for n = 1, 2, 3, 4, 5, 10, 20, 50. There should be 9 curves, including the original.

For the function:
f(x) = x; -L < x < L
f(x+2L) = f(x); - ? < x < ?
Plot the original function for -3L < x < 3L, and then also plot the Fourier series for values of n up to n = 1, 2, 3, 4, 5, 10, 20, 50. There should be a total of 9 curves including the original.

Using the method of separation of variables, solve the partial differential equation
u subscript(tt)+2(pi)u subscript(t)-u subscript(xx)=-3sin(3(pi)x)
for 0 less than or equal to x less than or equal to 1
with boundary conditions u(0,t)=u(1,t)=0
and initial conditions u(x,0)=u subscript (t)(x,0...

Hi, l want the Fourier series of this function by hand and using Matlab or any mathmatical program or programming language.
f (x) = 0, -π < x < 0
cos x 0 ≤ x < π
Find the Fourier series using Matlab?

Hi there, I have a question regarding Fourier Series which can be located here http://nullspace8.blogspot.com/2011/10/13.html. can someone please take a look? Full working step by step solution in pdf or word please. If you think the bid is insufficient and you can do it, please respond with a count...

Please see the attached file.
a) Sketch the periodic function y=ex, ‐2< x <2 and y(x)=y(x+4) for
values of x from ‐6 to 6.
State the period and whether the function is odd even or neither
b) give the Fourier series for the odd periodic extension of: y=ex, 0< x <2
c) Confirm Dirichlets the...

The row and column indices in the nxn Fourier matrix A run from 0 to n-1, and the i,j entry is E^ij, where E^ij = e^(2*PI*i/n). This matrix solves the following interpolation problem: Given complex numbers b_0, ... b_(n-1), find a complex polynomial f(t) = c_0 + c_1 + ... + c_(n-1) t^(n-1) such that...

Recall that S_N (f)(x)= sum (n=-N to N) c_n e^{inx}=
1/2pi integral (from -pi to pi) f(x-t)sin ((N+1/2)t)/sin(t/2) dt
Prove that if f in R[-pi, pi] and integral (from -1 to 1) |f(t)/t)|dt < infinity (convergent) then
lim( as N goes to infinity) S_N(f)(0)=0
Hint: Use the Riemann-Lebesgue ...

Hello,
I need help to inverse fourier transform below equation (with the prove) from frequency domain to its time domain form:
2 * pi * j^m * Jm(wd)
where
j = sqrt(-1)
m = Order of bessel function
Jm = Bessel function of m-th order.
w = Angular frequency
d = A constant