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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.

### Transforming Partial Differential Equations

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.

### 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

### 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 using any two functions Pl

### Fourier Series Calculation

Please determine the Fourier series for the attached function.

### Write down a Fourier series for a square wave

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.

### 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)=0

### l want the solution of this problem

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?

### 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 counter bid. Thank you.

### 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 theorem on both types of disconti

### Euler's Identity and Boundary Problems

Consider the boundary value problem: y" - 2y' + (1 + λ)y = 0, y(0) = 0, y'(1) = 0 For which values of λ does the problem have a non-trivial solution?

### 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)

### 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 f(E^v) = b_v. (i) Explain

### 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 lemma.

### 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