### Application of Fourier Transfors to Diffusion

Using Fourier Transforms, solve the one-dimensional equation for a point source located at x=xo, i.e., at time zero, c(x,0) = (delta)(x-xo).

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Using Fourier Transforms, solve the one-dimensional equation for a point source located at x=xo, i.e., at time zero, c(x,0) = (delta)(x-xo).

f(x)= {0 -2<x<0 f(x+4) = f(x) {1 0<x<2 I have to find the Fourier series of the problem and sketch the graph of the function at 3 periods. I'm not sure if you need my text and what sections were covering or not so I'll just give it. "Elementary differential equations and boundary value prob

1. Find the Fourier series expansions of f(x) = Co + C1x^2 with respect to the following two orthonormal bases on the interval [0,L] (L>0) a) {(1/L)^.5, (2/L)^.5*cos*((k*pi*x)/L)|k=1,2,...} b) {(2/L)^.5*sin*((k*pi*x)/L)|k=1,2,....}

4. In this problem, you will devise a computer experiment to investigate Gibb's phenomenon, which is the presence of spurious oscillations in the graph of a truncated Fourier series near the places where the full Fourier series is discontinous. Choose any function you like that demonstrates Gibb's phenomenon. Your goal is to

What is the solution to Y''(x) + 2y'(x) + 5y(x) = f(x) Where f(x) is a given forcing function, and y and f both decay to 0 as x ---> + or - INF Note: should read "as x approaches plus or minus infinity"

Please view the attached file for the full description of the two questions being analyzed. Essentially, this posting is asking the following: Solve the Schrodinger equation with different potentials using the Fourier transform.

Suppose f(t) and g(t) are 2π periodic functions with Fourier series representations {see attachment}. Find the Fourier series of {see attachment}.

Calculate the inverse Fourier transform of 1/(w^2+2iw-2) in two ways: using the definition and using partial fractions.

Use the Fourier transform to solve the one-dimensional wave equation. See attached file for full problem description.

3. The nefarious Evil Corp is dumping radioactive pollutant into a river moving with speed c. Define x to be the downstream coordinate, with the pollutant being injected at x=0... (see attachment for rest of question)

Please see the attached file for full problem description. Q = e^ (i n pi alpha) + e^ (-i n pi alpha) and Q = 0. How can Q be two different things?

If the Fourier transform of the signal v(t) is v(w) = AT sinwt / wt then the energy contained in v(t) is a) (A^2)/2 b) A^2 c) (A^2)T d) (A^2)T/2

Find the Fourier series as well as the first three partial sums of the Fourier series. F(x)=x-x^2 if -1<x<1

Please see the attached file for the fully formatted problem. Let Y: R --> R be the periodic function whose restriction to [0,1] is X (0,1/2) - X(1/2,1) Y is an odd function. S 1--> 0 Y(x) cos 2pi kx dx = S 1/2-->-1/2 Y(x) cos 2pi kx dx Vk Conclude the the complex Fourier Series...can be expressed in the form...

Problem attached. "Eigenvalues and Eigenvectors of the Fourier Transform" Recall that the Fourier transform F is a linear one-to-one transformation from L2 (?cc, cc) onto itself. Let .. be an element of L2(?cc,cc). Let..= , the Fourier transform of.., be defined by ..... It is clear that ..... are square-integrable fu

"Equivalent Widths" Suppose we define for a square-integrable function f(t) and its Fourier transform ..... the equivalent width as .... and the equivalent Fourier width as .... a) Show that ..... is independent of the function f, and determine the value of this const. b) Determine the equivalent width and the equiva

Please see the attached file for the fully formatted problems. The convolution of two functions f and g is defined to be the new function f * g. whenever the integral converges. (a)Is it true that f*g = g*f? Why? (b) If F(k) = (i//) f°° exp (?ikx) f(x) dx and G(k) = (i//) f°° exp (?ikx) g(x) dx are the Fourier tra

Please see the attached file for the fully formatted problem. 2. Let p be a fixed and given square integrable function, i.e. 0 < S g(x)g(x) dx = ||g^2|| < inf The function g must vanish as |x| ---> inf. Consequently, one can think of g as a function whose non-zero values are concentrated in a small set around the origin x

Please see the attached file for the fully formatted function. Find the Fourier series, sketch the graph of the function for 3 periods. Is this a discontinuous graph? Is it an even or odd function, I know there are Fourier series rules for them.

(a) Explain the relationship between the spectral components indicated above and the corresponding graph showing the motion of the surface of the motor plotted as a function of time. What do they represent? (b) Given the amplitudes indicated in the above diagram, copy and complete the table below for the amplitude of the vib