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Solve using the Fourier Transform Method.
2.24) Solve Laplace's equation in a strip using Fourier transforms:...
(a) Find the Fourier transform of the following function (b) Using Fourier transforms, solve the wave...
(a) Find the Fourier transform of the following function (b) Using Fourier transforms, solve the wave equation , -∞<x<∞ t>0 and bounded as ∞ f(r)e We were unable to transcribe this imageu(r, 0)e 4(r.0) =0 , t ur. We were unable to transcribe this image f(r)e u(r, 0)e 4(r.0) =0 , t ur.
3. Consider the equation a(x, 0) = f(x) u(x,t) lim 0 Using a Fourier transform, solve this equati...
Fourier transform:
3. Consider the equation a(x, 0) = f(x) u(x,t) lim 0 Using a Fourier transform, solve this equation. Evaluate your solution in the case when f(x)-δ(x).
3. Consider the equation a(x, 0) = f(x) u(x,t) lim 0 Using a Fourier transform, solve this equation. Evaluate your solution in the case when f(x)-δ(x).
Problem 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms au(x,t) The domain of the problem is-oo < x < oo and is 0 < t < oo . At...
Problem 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms au(x,t) The domain of the problem is-oo < x < oo and is 0 < t < oo . At time t = 0, the initial condition is given by u (x,0)-0 a) Take the Fourier transform on x and show that the above PDE can be transformed into the following ODE where G() is the Fourier transform of g(x) and U(w,...
2) (Fourier Transforms Using Properties) - Given that the Fourier Transform of x(t) e Find the Fourier Transform of the following signals (using properties of the Fourier Transform). Sketch each sign...
2) (Fourier Transforms Using Properties) - Given that the Fourier Transform of x(t) e Find the Fourier Transform of the following signals (using properties of the Fourier Transform). Sketch each signal, and sketch its Fourier Transform magnitude and phase spectra, in addition to finding and expression for X(f): (a) x(t) = e-21,-I ! (b) x(t)-t e 21 1 (c) x(t)-sinc(rt ) * sinc(2π1) (convolution) [NOTE: X(f) is noLI i (1 + ㎡fy for part (c)]
2) (Fourier Transforms Using Properties)...
Q4. Laplace Transforms a) (20 points) Solve the differential equation using Laplace transform methods y" +...
Q4. Laplace Transforms a) (20 points) Solve the differential equation using Laplace transform methods y" + 2y + y = t; with initial conditions y(0) = y(O) = 0 |(s+2) e-*) b) (10 points) Determine L-1 s? +S +1
Using Fourier transform, prove that a solution of the Laplace equation in the half plane: Urn+...
Using Fourier transform, prove that a solution of the Laplace equation in the half plane: Urn+ Uyy=0,- << ,y>0, with the boundary conditions u(1,0) = f(t), - <I< u(x,y) +0,31 +0,+0, is given by r(2, y) == Love you > 0. Hint: 1. Take Fourier transform on the variable r, 2. Observe U(k, y) +0 as y → 00, 3. Use pt {e-Mliv = Vice in
3) (Fourier Transforms Using Properties) - Given that the Fourier Transform of a signal x(t) is X(f) - rect(f/ 2), find the Fourier Transform of the following signals using properties of the Fourier...
3) (Fourier Transforms Using Properties) - Given that the Fourier Transform of a signal x(t) is X(f) - rect(f/ 2), find the Fourier Transform of the following signals using properties of the Fourier Transform: (a) d(t) -x(t - 2) (d) h(t) = t x( t ) (e) p(t) = x( 2 t ) (f) g(t)-x( t ) cos(2π) (g) s(t) = x2(t ) (h)p()-x(1)* x(t) (convolution)
3) (Fourier Transforms Using Properties) - Given that the Fourier Transform of a signal...
Answers are: 9. (a) Use the Tables of Fourier transforms, along with the operational theorems, to...
Answers are:
9. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the inverse Fourier transform of iw 4 + 9 w2 9 w2 (b) The function f(t) satisfies the integral equation: OO -4u Н(u) du + 6sgn(t) е З, f(t) 0- ft - u) е" = 4 e -OO Find the Fourier transform of the function f(t) and hence find the solution f(t) 7 "(1-)н, (b) Transform the equation by using the convolution Theorem:...
Solve the differential equation y' + 3y = 0 by means of Fourier transforms
Solve the differential equation y' + 3y = 0 by means of Fourier transforms
Solve Laplace's equation on
Solve Laplace's equation on \(-\pi \leq x \leq \pi\) and \(0 \leq y \leq 1\),$$ \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0 $$subject to periodic boundary conditions in \(x\),$$ \begin{aligned} u(-\pi, y) &=u(\pi, y) \\ \frac{\partial u}{\partial x}(-\pi, y) &=\frac{\partial u}{\partial x}(\pi, y) \end{aligned} $$and the Dirichlet conditions in \(y\),$$ u(x, 0)=h(x), \quad u(x, 1)=0 $$
Fourier transform:
3. Consider the equation a(x, 0) = f(x) u(x,t) lim 0 Using a Fourier transform, solve this equation. Evaluate your solution in the case when f(x)-δ(x).
3. Consider the equation a(x, 0) = f(x) u(x,t) lim 0 Using a Fourier transform, solve this equation. Evaluate your solution in the case when f(x)-δ(x).
Problem 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms au(x,t) The domain of the problem is-oo < x < oo and is 0 < t < oo . At time t = 0, the initial condition is given by u (x,0)-0 a) Take the Fourier transform on x and show that the above PDE can be transformed into the following ODE where G() is the Fourier transform of g(x) and U(w,...
2) (Fourier Transforms Using Properties) - Given that the Fourier Transform of x(t) e Find the Fourier Transform of the following signals (using properties of the Fourier Transform). Sketch each signal, and sketch its Fourier Transform magnitude and phase spectra, in addition to finding and expression for X(f): (a) x(t) = e-21,-I ! (b) x(t)-t e 21 1 (c) x(t)-sinc(rt ) * sinc(2π1) (convolution) [NOTE: X(f) is noLI i (1 + ㎡fy for part (c)]
2) (Fourier Transforms Using Properties)...
Q4. Laplace Transforms a) (20 points) Solve the differential equation using Laplace transform methods y" + 2y + y = t; with initial conditions y(0) = y(O) = 0 |(s+2) e-*) b) (10 points) Determine L-1 s? +S +1
Using Fourier transform, prove that a solution of the Laplace equation in the half plane: Urn+ Uyy=0,- << ,y>0, with the boundary conditions u(1,0) = f(t), - <I< u(x,y) +0,31 +0,+0, is given by r(2, y) == Love you > 0. Hint: 1. Take Fourier transform on the variable r, 2. Observe U(k, y) +0 as y → 00, 3. Use pt {e-Mliv = Vice in
3) (Fourier Transforms Using Properties) - Given that the Fourier Transform of a signal x(t) is X(f) - rect(f/ 2), find the Fourier Transform of the following signals using properties of the Fourier Transform: (a) d(t) -x(t - 2) (d) h(t) = t x( t ) (e) p(t) = x( 2 t ) (f) g(t)-x( t ) cos(2π) (g) s(t) = x2(t ) (h)p()-x(1)* x(t) (convolution)
3) (Fourier Transforms Using Properties) - Given that the Fourier Transform of a signal...
Answers are:
9. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the inverse Fourier transform of iw 4 + 9 w2 9 w2 (b) The function f(t) satisfies the integral equation: OO -4u Н(u) du + 6sgn(t) е З, f(t) 0- ft - u) е" = 4 e -OO Find the Fourier transform of the function f(t) and hence find the solution f(t) 7 "(1-)н, (b) Transform the equation by using the convolution Theorem:...
Solve the differential equation y' + 3y = 0 by means of Fourier transforms
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