Given the following 1-D heat equation, use Fourier transform to show the following result and then use the initial condition to prove u(x,t) for all t>0. x goes from - infinity to positive infinity
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Given the following 1-D heat equation, use Fourier transform to show the following result and then use the initial condi...
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,...
Problem 6 [30 points Use Fourier transform to solve the heat equation U = Ura -o0<x<...
Problem 6 [30 points Use Fourier transform to solve the heat equation U = Ura -o0<x< t> 0 subject to the initial condition -1, 1 u(x,0) = -1 < x < 0 0 < x <1 x € (-00, -1) U (1,00)
1. Use the Fourier Transform to solve the following problem with W1 21 (a) Find the Fourier Trans...
1. Use the Fourier Transform to solve the following problem with W1 21 (a) Find the Fourier Transform of u by applying F to the equation and initial condition; denote this function U(w, t). (b) Find u u(z, t) by taking the inverse transform of the U(w, t) you found in part (a).
1. Use the Fourier Transform to solve the following problem with W1 21 (a) Find the Fourier Transform of u by applying F to the equation and...
Question 3 Fourier transform] Find the Fourier transform of the following functions. (i) f(z) = H...
Question 3 Fourier transform] Find the Fourier transform of the following functions. (i) f(z) = H (t-k)e-4. (ii) f(x) = 5e-4H21 (im)(xe 0, otherwise. IV) f(x) = Fourier transform
Question 3 Fourier transform] Find the Fourier transform of the following functions. (i) f(z) = H (t-k)e-4. (ii) f(x) = 5e-4H21 (im)(xe 0, otherwise. IV) f(x) = Fourier transform
12 points) Note: The formulas for the Fourier transform on half intervals are often given in...
12 points) Note: The formulas for the Fourier transform on half intervals are often given in the form { ). f(x)cos L dx, with the Ź outside the integral. Computing these integrals will often involve u-substitutions, integration by parts, and other integration techniques that will produce all kinds of constants. With the way these problems are asked in WebWork, it would be hard to keep track of when constants should be factored out or not, so we will adopt the...
4. The Fourier transform of a rectangular pulse 1 비 r/2 0 otherwise is given by...
4. The Fourier transform of a rectangular pulse 1 비 r/2 0 otherwise is given by (a) Use pr(t) and properties of the Fourier transform to find the Fourier transform, D(w), of d(t) shown below, in terms of P(. First state the approach that you are using to find D(), then show all of the details. d(t)
Given LTI system with following input response (can use properties of the Fourier transform like, sinc(x)...
Given LTI system with following input response (can use properties of the Fourier transform like, sinc(x) = sin(πx)/πx ): h(t) = 8/π sinc(8t/π) where input x(t) of the LTI system is the following continuous-time signal x(t) = cos(t) cos(8t) a) find the Fourier transform of x(t) b) find the Fourier transform of h(t) c) Is this LTI system BIBO stable? Prove d) find the output y(t) of the LTI system
Please finish these questions. Thank you Given find the Fourier transform of the following: (a) e dt 2T(2 1) 4 cos (2t) (Using properties of Fourier Transform to find) a) Suppose a signal m(t) is giv...
Please finish these questions. Thank you
Given find the Fourier transform of the following: (a) e dt 2T(2 1) 4 cos (2t) (Using properties of Fourier Transform to find) a) Suppose a signal m(t) is given by m()-1+sin(2 fm) where fm-10 Hz. Sketch the signal m(t) in time domain b) Find the Fourier transform M(jo) of m(t) and sketch the magnitude of M(jo) c) If m(t) is amplitude modulated with a carrier signal by x(t)-m(t)cos(27r f,1) (where fe-1000 Hz), sketch...
9. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the...
9. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the inverse Fourier transform of iw 45iw(iw2 (b) The function f(t) satisfies the integral equation: f(t)2 f(t - u) sgn(u) du — 6е" H(). Find the Fourier transform of the function f (t) and hence find the solution f(t) The sign function sgn(t) = 1 if t 0, 0 if t 0 and -1 if t < 0 H(t) is the Heaviside unit step function...
4. Using the properties of the Fourier transform, evaluate the following integrals in terms of a...
4. Using the properties of the Fourier transform, evaluate the following integrals in terms of a and/or 6, where a is positive. You have to express real-valued evaluations. eat sinc(t)dt -nt - atcos (Bt)dt (10 pts) (a) (10 pts) (b) e 0 0 d [In (x)] dx Hint: teldt -1디미플
4. Using the properties of the Fourier transform, evaluate the following integrals in terms of a and/or 6, where a is positive. You have to express real-valued evaluations. eat sinc(t)dt...
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,...
Problem 6 [30 points Use Fourier transform to solve the heat equation U = Ura -o0<x< t> 0 subject to the initial condition -1, 1 u(x,0) = -1 < x < 0 0 < x <1 x € (-00, -1) U (1,00)
1. Use the Fourier Transform to solve the following problem with W1 21 (a) Find the Fourier Transform of u by applying F to the equation and initial condition; denote this function U(w, t). (b) Find u u(z, t) by taking the inverse transform of the U(w, t) you found in part (a).
1. Use the Fourier Transform to solve the following problem with W1 21 (a) Find the Fourier Transform of u by applying F to the equation and...
Question 3 Fourier transform] Find the Fourier transform of the following functions. (i) f(z) = H (t-k)e-4. (ii) f(x) = 5e-4H21 (im)(xe 0, otherwise. IV) f(x) = Fourier transform
Question 3 Fourier transform] Find the Fourier transform of the following functions. (i) f(z) = H (t-k)e-4. (ii) f(x) = 5e-4H21 (im)(xe 0, otherwise. IV) f(x) = Fourier transform
12 points) Note: The formulas for the Fourier transform on half intervals are often given in the form { ). f(x)cos L dx, with the Ź outside the integral. Computing these integrals will often involve u-substitutions, integration by parts, and other integration techniques that will produce all kinds of constants. With the way these problems are asked in WebWork, it would be hard to keep track of when constants should be factored out or not, so we will adopt the...
4. The Fourier transform of a rectangular pulse 1 비 r/2 0 otherwise is given by (a) Use pr(t) and properties of the Fourier transform to find the Fourier transform, D(w), of d(t) shown below, in terms of P(. First state the approach that you are using to find D(), then show all of the details. d(t)
Please finish these questions. Thank you
Given find the Fourier transform of the following: (a) e dt 2T(2 1) 4 cos (2t) (Using properties of Fourier Transform to find) a) Suppose a signal m(t) is given by m()-1+sin(2 fm) where fm-10 Hz. Sketch the signal m(t) in time domain b) Find the Fourier transform M(jo) of m(t) and sketch the magnitude of M(jo) c) If m(t) is amplitude modulated with a carrier signal by x(t)-m(t)cos(27r f,1) (where fe-1000 Hz), sketch...
9. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the inverse Fourier transform of iw 45iw(iw2 (b) The function f(t) satisfies the integral equation: f(t)2 f(t - u) sgn(u) du — 6е" H(). Find the Fourier transform of the function f (t) and hence find the solution f(t) The sign function sgn(t) = 1 if t 0, 0 if t 0 and -1 if t < 0 H(t) is the Heaviside unit step function...
4. Using the properties of the Fourier transform, evaluate the following integrals in terms of a and/or 6, where a is positive. You have to express real-valued evaluations. eat sinc(t)dt -nt - atcos (Bt)dt (10 pts) (a) (10 pts) (b) e 0 0 d [In (x)] dx Hint: teldt -1디미플
4. Using the properties of the Fourier transform, evaluate the following integrals in terms of a and/or 6, where a is positive. You have to express real-valued evaluations. eat sinc(t)dt...
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