Fourier analysis and PDE

1. Show the details of your work. Solve the bending beam equation ∂4u ∂2u = −c2 4 2 ∂t ∂x with boundary conditions u(0, t) = 0, ux (0, t) = 0 u(1, t)xx = 0, uxxx (1, t) = 0 and initial condition u(x, 0) = x(1 − x). Be sure to show all steps in solving the pde and explain the physical meaning of the boundary conditions. 2. Show the details of your work. Find the temperature in the rod of length 1, ∂u ∂2u = c2 2 ∂t ∂x with boundary conditions u(0, t) = 0, u(1, t) = T (constant) and initial condition u(x, 0) = f (x). To do this, first find the steady state solution us (x) and then let u(x, t) = w(x, t) + us (x). Now find w to complete the problem. Be sure to show all steps in solving the pde and explain the physical meaning of the boundary conditions. 3. Show the details of your work. Solve for the steady state heat distribution in a square plate ∂2u ∂2u + 2 =0 ∂x2 ∂y with boundary conditions u(0, y) = 0, ux (1, y) = 0 u(x, 0) = 0, u(x, 1) + ux (x, 1) = 1 Be sure to show all steps in solving the pde and explain the physical meaning of the boundary conditions. 4. Show the details of your work. Use the Fourier Cosine Transform to solve for the temperature in a semiinfinite rod ∂u ∂2u , 0