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The wave equation, subject to the given conditions. problem 6
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- Please solve following wave equation on [-1,1] utt(x,t) = uxx(x,t)u(−1,t) = 0u(1,t) = 0u(x,0) = f(x)ut(x,0) = g(x) Only have to discuss the case lambda > 0The technique that we used to solve the time-dependent Schrodinger equation in class is known as separation of variables. Use the same technique to investigate solutions of the wave equation: ∂2y(x,t)∂x2=1v2∂2y(x,t)∂t2Consider the wave equation utt = uxx, (x, t) ∈ R2. find 2 distinct solutions please
- 2(7) Normalize the following wavefunctions: (a) Ψ(x) = cos(2x) , - π/2 ≤ x ≤ π/2 (b) Ψ(x) = e( iπ x+2) , 0 ≤ x ≤ 1Auxx + Buxy + Cuyy + Dux + Euy = 0 is used for: Group of answer choices A. One-dimensional heat equation B. Non-homogenous PDE C. Homogenous PDE D. One-dimensional wave equationfrom wave equation ∇2u(r,t) + 1/c2 (∂2u(r,t)/∂t2) = 0, get the Helmholtz equation. Using that u(r,t) = u(r)ei2πνt
- State the solution formula for the Cauchy problem for the homogeneous wave equation on R3. Prove that solutions corresponding to a localized initial signal (i.e. initial position and derivative are supported in a small ball) have a wave fore front and a wave back front and that these fronts are close to each other. Conclude that music is possible in R3. Say in words what happens if we consider R2 instead.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.For the wave equation in R3(i.e., (x1 , x2 , x3 ) ∈ R3), let u = u(r, t) be theradial symmetric solution of the systemutt = ∆u, 0 ≤ r < ∞, t > 0u(r, 0) = 1, r ≥ 0ut(r, 0) =2, 0 ≤ r ≤ 1,0, r > 1 Find u(2, 1) and u(2, 2).
- Solve using wave equation: u(0, t) = u(π, t) = u(x, 0) = 0, du/dt | t = 0 = sin(x)Show that the function Z = sin(wct)sin(wx) satisfies the wave equationsolve 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