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- Consider the wave equation utt = uxx, (x, t) ∈ R2. find 2 distinct solutions pleasesolve 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 variableTransform the vector A = 3i – 2j – 4k at P(2,3,3) to cylindrical coordinates
- from wave equation ∇2u(r,t) + 1/c2 (∂2u(r,t)/∂t2) = 0, get the Helmholtz equation. Using that u(r,t) = u(r)ei2πνtPlease 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 > 0Find 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.
- A string is stretched between two fixed pegs a distance πapart. The mathematical model representing the physicalphenomenon can be described by 1D wave equation as ∂2u∂t2=∂2u∂x2; 0 < x < π, t > 0 u(0,t) = 0, u(π,t) = 0,u(x, 0) = sin x , ut (x, 0) = 0. Find the displacement function u(x,t) subject to the giveninitial and boundary conditions.A particle at (1, 0, 0) starts moving in space in such a way that its position vector at any time t ≥ 0 is R~ (t) = (cost + tsin t)ˆi + (sin t − t cost)ˆj + t 2ˆk, t ≥ 0.1. (Section 17.7) Use Stokes’ Theorem to calculate the work done by −→F (x, y, z) = ex2ˆı − 2xzˆj + xˆk in moving a particle aroundthe closed path determined by the intersection of positively oriented surface S : x + 4y + 2z = 4 and the coordinate planes.