Question 4 Find the explicit finite difference solution of the wave equation utt - Uxx = 0, with the boundary conditions and the initial conditions 0 < x < 1, u(0, t) = u(1,t) = 0, t> 0 U₁,1 and u₁, 2. Use Ax=0.25, At = 0.1 t>0 u(x, 0) = сosπx, u₁(x, 0) = 0, 0≤x≤1
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- 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), 0a²u 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.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<=LWhich of the following is most suitable solution for wave equation = c2? %3D at2 ax2 .(a) y = (AeP* + Be-P*)(Ce pt + De ept) (b) y = (Acos px + Bsin px)(Ccos cpt + Dsin cpt) (c) y = (Ax + B)(Ct + D) (d) y = (Aepx + Be-px)(Cep*t + De-ep*r) O a O b O c O dSolve 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.4. Consider a wave equation on an infinite line, J²u J²u 9 Ət² əx² = 0. = Find the characteristics though the point (1,3). Draw the domains of depen- dence and influence of the point (1,3).Q.4 Solve the wave equation: U =U +t+1- 1) =5, U(7,t) = cost U(x,0) = , U,(x,0) == 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)Show whether the following functions are wave functions or not. 1. У(х, t) еxp(ikx) = A- exp (i(ot-Ф)) кЗх3-0313-3kоxt(kx-ot)-iф)) 2 У(х, t) 3. y(x, t) Aexp(i(k³x³-w³t³-3kwxt(kx-wt)-ip)) Аехр (i(-kx? + оt))Consider the family of functions uc(x,t) of two variables x,t, indexed by the parameter c,uc(x,y)=ln(x+ct)(cos(ct)cos(x)−sen(ct)sin(x)).Determine the value of the parameter c>0 so that the function uc(x,y) is a solution of the wave equation 13.∂^(2)uc/∂x^(2)−8.∂^(2)uc/∂t^(2)=0.for wave equation, seperation of vairables u(x,t)=X=(x)T(t)Find the solution to the wave equation on the half - line: utt = c^(2) uxx, x > 0 , t > 0. u(0,t) = 0, t > 0. u(x,0) = 0, ut(x,0) = e^(-2x), x > 0.SEE MORE QUESTIONS