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Fourier Analysis


Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series. It showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat propagation. The subject of Fourier analysis encompasses a vast spectrum of mathematics.

The decomposition process itself is called a Fourier transform. The transform often gives a more specific name which depends upon the domain and other properties of the function being transformed. The original concept of Fourier analysis has been extended over time to apply to more abstract and general solutions and the general field is often known as harmonic analysis.

Fourier analysis has many scientific applications including areas of physics, partial differential equations, number theory, combinatorics, signal processing, imaging, probability theory and many more. This wide applicability stems from many useful properties of the transforms linear operators, invertibility, exponential functions, convolution theorem and the discrete version. It was also useful as a compact representation of a signal. 

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 are: a) u(x, y, 0) = xy(a-

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 (x, b, t) = 0, 0 < x < a, t

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 by separation of variables. The

2D Heat equation with mixed boundary conditions

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 are insulated

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, ur(a, , t) = 0 proble

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 Solve using separ

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 temperature by a rate proportional t

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) where v solves a heat equation o

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, the temperature goes to 0 whic

The relation between Beta and Gamma functions

               Prove that               B(p, q) = [Gamma (p) Gamma (q)]/Gamma (p + q)                     The detail problem is in the attached file.

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 function f(x). 3. See attachmen

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 using any two functions Pl

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)=0

Fourier Series

Determine the Fourier series for the function: f(t) = (0, for -2 â?¤ t <0 (t, for 0 â?¤ t <2

Evaluate functions.

Determine if the functions below are odd, even or neither. (a) f(x) = x^2 + 2 (b) f(x) = (x^2 + 2)tan(x^2) (c) f(x) = (x^2 + 2)sin (x)tan(x^2)

PDE solutions using Fourier Transforms

See attached PDF for the compiled LaTex. I would prefer any solution include the Latex Source, however, if you are unable to use LaTex, you may use another format. Exercise 1 Solve, using Fourier Transforms frac{partial^{2}u}{partial x^{2}}+frac{partial^{2}u}{partial y^{2}}=0 for 0<y<H, -infty<x<infty subject to the i

Find the 7 Fourier coefficients of the function...

There is an attached file with further information regarding the problem. Find by inspection the first seven Fourier coefficients {a0, a1, b1, a2, b2, a3, b3} of the function: f(x) = 14-cos(Pi*x/10) + 3sin(Pi*x/10) + 0.5cos(Pi*x/5) + 5sin(3*Pi*x/10)

Fourier transform

Please see the attached file and include an explanation of problem. Thank you. 1. Compute the Fourier transform for x(t) = texp(-t)u(t) 2. The linearity property of the Fourier transform is defined as: 3. Determine the exponential Fourier series for: 4. Using complex notation, combine the expressions to form a singl

How to Find Normal Modes for a Wave Equation

Could you please show me how to do the problem attached? You don't have to do the first part (proving solutions to the wave equation by a separation of variables) as I know how to do that. Please start where it asks what is a normal mode, etc. See the attached file.

Fourier transforms

1. Find the Fourier sine series of f(x)=1, 0<x<L/2. Use this to prove that 1-1/3+1/5-1/7+...=Pi/4 2. Solve df/dt=d^2 f/dx^2 - f subject to the initial condition f(0,x)=1 if |x|<L/2 or f(0,x)=0 if |x|>L/2 Please see the attachment to view the questions with correct mathematical notation (and also phrased slightly differe

The Fourier coefficients of a derivative

Let f be a 2 pi periodic, differentiable function with Fourier coefficients a_n and b_n. Let (a_n)*, (b_n)* be the Fourier coefficients of f'. a) Show that (a_0)*=0 b) Use integration by parts to find a formula for the Fourier coefficients of f' in terms of the Fourier coefficients of f. (The attachment contains the

Waveforms : Fourier and Laplace Transforms

Have the following question regarding extracting information from a waveform a. Write down an expression, in the time domain, for the signal in the diagram above. b.Derive the Laplace transform for this signal. c.Use Laplace transform analysis to derive the Fourier transform in its simplest form. All details in the a

Heat Problem on the Circle

So I am able to get to u(x,t) = sum_{n=-infty}^{infty} Cexp(iBx - ktB^2) + Dexp(-iBx - ktB^2) because the general series is sum_{n=-infty}^{infty} X(x)T(t) You have 1 coefficient in your general series, yet I have 2. How do I get it into 1?

Heat Equation on Circle

Solve the heat problem on the circle u_t = ku_{xx} u(x,0) = phi(x) where phi(x) is the 2l periodic extension of phi using the separation of variables. I am able to go as far as u = XT -X''/X = lambda where lambda = beta^2 usually the solution for X'' + beta^2 * X = 0 is Ccos(beta * L) + D sin