Given a wave equation: 7.5 ,00 Subject to boundary conditions: u(0,t) = 0, u(2,t) = 1 for 0 ≤ t ≤ 0.4 An initial conditions: u(x,0)=2x Ju(x,0) at = 1 for 0≤x≤ 2 By using the explicit finite-difference method, analyse the wave equation by taking: h = Ax=0.5, k = At = 0.2
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- Solve the wave equation for the following cases: ii) For free free ends beam with following condition: du Boundry conditions: For x = 0 and x = 1 => əx Inetial conditions: inetial desplacement: u(x, 0) = f(x): du(x, 0) inetial velocity at = g(x). ||Consider the wave equation on an infinite line, with u(x, 0) = f (x) defined by f (x) = { 0, for x < 0, x^2, for 0 ≤x ≤1, (2 −x), for 1 ≤x ≤2, 0, for x > 2. Set ∂u/∂t(x, 0) = g(x) = 0 and c = 1/2. Draw the solution at t = 0 and t = 5. Find t, at which u(15, t) = 1/2 using the D'Alambertsolve the one dimensional wave equation with the boundary conditions and inital conditions as given below: δ2u/δt2 = 1/pi2.δ2u/δx2 u(0,t)= 0, t>0. u(1,t)=0, t>0 u(x,0)= sinππxcosπx, 0<x<1 δu/δt(x,0)=0 0<x<1 using the method of seperation of variable
- Let c>0 be a constant. Solve the one-dimensional wave equation Uxx=(1/c^2)U 0<x<1, with the boundry conditions U(1,0)=U(0,t)=0, subject to the U(x,0)=3sin(3xpi)+4sin(4xpi) and Ut=(x,0)=0.a²u Solve the wave equation: at² - 02.02 0x2 Under the condition: u=0 when x = 0 and x = π Du =0 when t= 0 and u(x, 0) = x², Ət 0Consider the wave equation 0 0, with u(0,1) = 1(xt)= 0, u(x,0) = sin x and =0 at t=0. Then u isa) Consider the wave equation with ² U = UII, -∞= 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)Let f(x,t)=cos(12x+4t). Find the value of K so that f satisfies the wave equation ∂^2f/∂x2=K∂^2f/∂t^2Solve the inhomogeneous wave equation on the real lineUtt − c2Uxx = sin x, x ∈ RU(x, 0) = 0, Ut(x, 0) = 0.Explain what theory you are using and show your full computations.(a) Show that u(x,t) = 4cos (2x+2ct)+ e**** is a solution of the wave equation d'u d²u dx² (9 marks) =A string of length 'L', that has been tightly stretched between two points at X-axis for t > 0, and satisfied one-dimensional wave equation of the following form: 8²u = 4. St2 82u 8x2 Produce the wave function, u(x,t) using separation of variables method with given boundary and initial conditions below. u(0, )- и (п,1) -0, u(x,0)= sin 5x t > 0 ди (at t=0) = 0 %3DSEE MORE QUESTIONSRecommended textbooks for youAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage