#### Wave equation on a rectangular domain

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...

#### Heat equation on a rectangular domain

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...

#### Damped 1D wave equation on a clamped string

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...

#### 2D wave equation on a wedge

7-4 Consider the displacement of ,u(r,,t) , a "pie-shaped" membrane of radius a and angle /3 that satisfies: utt = c22u 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,...

#### 1D wave equation

Solve the following string equation problem  utt = 1/4* uxx, 0<x<1,t>0, u(0, t) = 0, u(1,t)=0, t>0 1/2 * x, 0< x<1/2 u(x,0) = 1−x, 1/2<x<1. ut (x,0) = 0 ...

#### 1 dimensional non homogeneous heat equation

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...

#### The non-homogeneous heat equation

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.

#### Non homogeneous 1D heat equation

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...

#### 1D heat equation

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...

#### One dimensional heat equation

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...

#### Fourier Series and Fourier Transform

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...

#### Damped Driven Wave Equation

By the method of separation of variables, solve the equation: u(subscript tt) + 4u(subscript t) −u(subscript xx) = 5sin(2πx) for 0 ≤ x ≤ 1, with boundary conditions u(0, t) = u(1, t) = 0 and initial conditions u(x, 0) = u(subscript t)(x, 0) = 0.

#### Working with Parseval's Theorem

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 ...

#### Plot the Fourier series

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.

#### Fourier methods in one dimension

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...

#### Fourier Series

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...

#### Dirichlet's theorem on both types of discontinuity

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...

#### Determine the eigenvalues of Fourier matrix.

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...

#### Fourier series question

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 ...

#### Inverse fourier transform of an equation

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