Q3) Solve the Following one -dimensional wave equation utt - Uxx+u = 0 for The boundary conditions are u(0, t) = 0, u(n, t) = 0 t>0 The initial conditions are (i)u(x, 0) = f (x), (ii)u;(x, 0) = 0 ,0
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- Q.4 Solve the wave equation: U =U +t+1- 1) =5, U(7,t) = cost U(x,0) = , U,(x,0) =Q5) Solve the wave equation for vibration of organ pipe subject to the boundary condition: a- u(0,t)=0, t>=0 du(1,1) b- ax - 0,1 20 ди(х.0) -U, cons tant с- d- u(x.0) =0.0<=x<=La2u satisfies the wave equation əx² -n a²u Verify that U(x, t) = e¬Vkt cos\ax %3D k at2
- = Use variables separation method to solve the wave equation uxxutt. This function is defined on spatial domain 0 0. Subject to boundary conditions: ux(0, t) = u,(a, t) = 0 and initial conditions: u(x, 0) = 0 and u₁(x,0) = f(x)a²u B/ Solve the wave equation: 01² 3x² Under the condition: u=0 when x = 0 and x = 1 ди = 0 when t = 0 and u(x,0) = x², at a²! a = 1 0 < x < 1.Solve the wave equation u,- c u = xt, 0 0, cisa constant subject to boundary conditions u(0,1) = t u,(L,t)=r? and initial conditions u(x,0)= sin x u, (x, 0)= 0
- Q2) Solve the wave equation u=u 00 " XX 11 subject to u(0,t) = u(2,t) =0, t>0 and u(x,0)=0, u )=0, u,(x,0): = sin (3лx), 0Q1) Solve the wave equation of the circular membrane of radius 4 assuming the axisymmetry subjected to the following conditions u(r,0) = u(4,t) =0, t>0 and u,(r,0)=1. Q2) Solve the wave equation u It " = u 00 XX subject to u(0,t) = u(2,t) =0, t>0 and u(x,0)=0, u(x,0)= sin (3лx), 00, " 18 " 00 with u¸ (0,t) = . X |u(x, 0) = { 1, 0Consider the wave equation Utt Uzz; 0 < x,t < ñ, u (0, t) u (T, t) = 0,0 < t < «. (a) Show that u (x, t) = sin x sin t is a solution of the above problem. (b) Find the maximum of u (x, t) on [0, 7]² . (c) Show that the above wave equation do not necessarily satisfy the maximum principle.Solve the wave equation Utt = V²u u (x, y, 0) = 0 k constant u4 (x, y, t) = kQ1:- Find the domain of the following vector functions:- (a) f (t) = (cos t)i – Ln(t)j + vt – 2k (b) f (t) = Ln|t – 1|i + e'j + vtk Q2:- Find the domain and the range of the following equations:- 1 -1 (1)W (2)W = sin x y (3)W x²+y2 ху 1 (4)W (5)W = /x² + y2 + z² (6) W = x – y x²+y2+z² (7)W = Ln(x² + y²) (8) W = xy (9) W = 4x² + 9y²(b) An elastic string which is fixed at both ends is governed by the wave equation a?u a?u 0 0, atz Where, u(x, t) is the displacement of the string. The initial conditions are given by 0SEE MORE QUESTIONSRecommended textbooks for youAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage