Consider the wave equationUtt - c²Uxx1 the half-line x = [0, ∞) with boundary conditionUx(0, t) = 0nd initial conditions0U(x,0) = f(x), Ut(x,0) = g(x).this problem we construct the solution to the above problem using even extensions. Foris, letF(x) = { {(2)) +²0 +f(x) x>0f(-x) x<0'1

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Answered: Consider the wave equation Utt- c²Uxx =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Consider the wave equation: utt = uxx, −∞ < x < ∞, t > 0 with ut(x, 0) = 0 and u(x, 0) = ( 0, x < 0 and x > π) u(x, 0) = cos2 x, 0 ≤ x ≤ π a)Find a solution to this problem using the D’Alembert’s formula. Then, plot the solution at t = 0, π /4 , π /2 , π b)Find a solution to this problem using the Fourier transform and show that your solution is the same as part (a).
(a) D'Alembert's formula for the solution of the wave equation on R is given by 1 rx+ct U(x,t) -(* + ct) + f(x - ct) + 8()ds. %3D Provide a brief discussion of the meaning of the two terms in the right-hand side of the above formula. (b) Show by direct computation that D'Alembert's formula is a solution to the problem Un - eux = 0, U(x,0) = f(x), U(x,0) = g(x). xE R, t> 0, (c) Find the solution to the problem Un - cU = 0, U(x,0) = 0, U;(x,0) x E R, t > 0, = COS X. (d) What is the main difference between the wave equation and the heat equation in terms of the speed of propagation of information?
4) Find the Frenet (TNB) frame for the following function at the point (√3,-1,3π): r(t) = (2 sin(t).-2 cos (t), 3t) (Hint: once you find T and N, you can plug in the value of t before finding B since it doesn't involve taking a derivative!)
Determine if the lines intersect. If they intersect, find the intersection point.If the lines do not intersect find the exact distance between them (do not round). Formulate the final statement for your answer.L1: [x, y, z] = [1, 2, 1] + s[1, –1, –1]L2: [x, y, z] = [0, 2, 5] + t[0, 1, –1]
Which of the following functions are solutions to the 1-D wave equation? a² f a² f Ət² əx² 4 f = cos(2x + t) f = sin(x - 2t) f = sin(2x - t) f = cos(x + 2t) f=tan(x - 2t) Submit Question
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The equations rey +uz – cos(v) – 2 = 0 and u cos(y) + 4a?v – 5yz2 – 1 = 0 can be solved for (u,v) as functions of (xy,z) near the point P(xy,zu,v)=(2,0,1,1,0). Find ()zy at (2,0,1). (rey + uz – cos(v) – 2 = 0 ve u cos(y) + 4z?v – 5yz2 – 1 = 0 denklemleri P(xy.z,u,v)=(2,0,1,1,0) noktası civarında (xy.z)'nin fonksiyonu olmak üzere (u,v) için çözümlüdür. (2,0,1) noktasında ()zy yi hesaplayınız.) Lütfen birini seçin: 00 01 O 0,5 O-0,5 O-1
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Consider the wave equation Utt- c²Uxx = 0 ᏆᏆ the half-line x = [0, ∞) with boundary condition U (0, t) = 0 d initial conditions U(x,0) = f(x), Ut(x,0) = g(x). this problem we construct the solution to the above problem using even extensions. For s, let F(x) = { √(-²2) +²0 + f(x) x>0 1