2. Consider the following 1-D wave equation with initial condition u (x, 0)- F (x) where F(x) is a given function. a) Show that u (x, t)-F (x - t) is a solution to the given PDE. b) If the function F...
1. Wave equation. Consider the wave equation on the finite interval (0, L) PDE BC where Neumann boundary conditions are specified Physically, with Neumann boundary conditions, u(r, t) could represent the height of a fluid that sloshes between two walls. (a) Find the general Fourier series solution by repeating the derivation from class now considering Neumann instead of Dirichlet boundary conditions. Your final solution should be (b) Consider the following general initial conditions u(x, 0)x) IC IC Derive formulas that...
14 points Consider the following equation : PDE: u+ 0 ,0<x <1, 0<y <1 BCs: u(0, y)= 0, u (1, y ) = 0 ,0<y <1 ICs: u (x,0)=0, u (x,1)=2 ,0<x <1 a) Using the PDE and the boundary conditions write the form of the solution u (x ,t) b) Now apply the initial condition to solve for the unknown coefficients in the solution from part (a) 14 points Consider the following equation : PDE: u+ 0 ,0
Question 2 ul lu (a) Find the solution u(x,t) for the 1-D wave equationfor -oo < x < oo with initial conditions u (x,0)-A(x) , where A(x) s presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and t, somewhat similar to f(x) on page 85 of the Notes Part 2. di+10 dı+15di+20 (b) Check for the wave equation in (a) that if f(xtct) (use appropriate value...
Problem 1 (20 points) Consider the PDE for the function u(x, t) e 0<x<T, t> 0 with the boundary conditions n(0, t) 0, u(T, t) 0, t> 0 and the initial condition 0 u(x, 0) 1+cos(2a), (a) Give a one-sentence physical interpretation of this problem. (b) Find the solution u(x, t) using a Fourier cosine series representation An (t) cos(nax) u(x,t)= Ao(t) + n=1
Consider the partial differential equation, with the initial condition: 1 2 cup +3cºu, = 9x²y?, u(x,0) = x3 +1 Find the characteristic curves and the orthogonal trajectories and sketch both on the same graph. Find a solution of the partial differential equation with the given initial con- dition valid in the first quadrant of the (x, y)-plane. Is this solution unique? Explain.
PDE question Consider the one dimensional wave equation on the half line: Ut(x,0) = g(x) Utt - Uzx= 0 0 < < u(0,t) = 0 u(x,0) = f(x) (a) What is the solution? (b) For the particular initial conditions 12 - 2 25254 f(x) = { 6- 4<r<6 otherwise g(x) = 0 sketch the solution u(x, t) for t = 0, 2, 4, 6.
d1=8 d2=9 lu for Find the solution u(x,t) for the l-D wave equation-=- Qx2 25 at2 (a) oo < x < oo with initial conditions u(x,0)-A(x) , where A(x) Is presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and t, somewhat similar to f(x) on page 85 of the Notes Part 2. d2+5 di+10 di+15dı+20 (b) Check for the wave equation in (a) that if (x...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive integer, show that the function , sin(x), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is Write down (but do not evaluate) an integral that can be used to...
Consider the partial differential equation, with the initial condition: 1 2yuz + 3x?uy = 9x?y?, u(x,0) = x3 + 1 Find the characteristic curves and the orthogonal trajectories and sketch both on the same graph. Find a solution of the partial differential equation with the given initial con- dition valid in the first quadrant of the (x, y)-plane. Is this solution unique? Explain.
Problem 2 (10 points). Consider the wave equation for a vibrating string of infinite length with the initial conditions where the initial displacement f(x) is specified as 0, if 21 Determine the expression of the function u(, 0.5) that represents the spatial profle of the string at timet 0.5. Provide the graph of this function Problem 2 (10 points). Consider the wave equation for a vibrating string of infinite length with the initial conditions where the initial displacement f(x) is...