That is u(x,0)-f(x) where f(x) (L-x)/(L-a), a<x<L. (a) State the problem including the PDE and all boundary and i...
2. In lectures we solved the heat PDE in 1 +1 dimensions with constant-temperature boundary conditions u(0,t)u(L,t) -0. If these boundary conditions change from zero temperature, we need to do a little bit more work. Consider the following initial/boundary-value problem (IBVP) 2 (PDE) (BCs) (IC) u(0,t) = a, u(x,00, u(L, t)=b, st. and let's take L = 1, a = 1, b = 2 throughout for simplicity. Solve this problem using the following tricks b and A"(x)-0 (a) Find a...
solve the PDE +u= at2 on 3 € (0,L), t > 0, with boundary conditions au 2x2 u(0,t) = 0, u(L, t) = 0 au and initial condition u(x,0) = f(x), at (x,0) = g(x) following the steps below. (a) Separate the variables and write differential equations for the functions (x) and h(t); pick the separation constant so that we recover a problem already studied. (b) Find the eigenfunctions and eigenvalues. (c) Write the general solution for this problem. (d)...
Find solution to the IBVP PDE BCs Ic u(0, t)-0, 0<oo l u(1, t) 0, 0<t< oo u(z,0)=x-x2ババ1
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 is given as 1; x< 10 x > 10 u(x, 0) = F(x) = use part (a) to write the solution u(x, t) c) Sketch u(x,0) and u(x,1) on the same u-versus-x graph d) Explain in your own...
Mark which statements below are true, using the following: Consider the diffusion problem au Ou u(0, t) = 0, u(L, t) = 50 u(x,0-fx where FER is a constant, forcing term. Any attempt to solve this using separation of variables fails. This is because the PDE is not homogeneous. A more fruitful approach arises from splitting the solution into the sum of two parts, taking into account that all change eventually dies out. That is there is a transient part...
Problem # 3 [20 Points] Solve PDE: ut = uxx - u, 0 < x < 1, 0 < t < ∞ BCs: u(0, t)=0 u(1, t)=0 0 < t < ∞ IC: u(x, 0) = sin(πx), 0 ≤ x ≤ 1 directly by separation of variables without making any preliminary trans- formation. Does your solution agree with the solution you would obtain if transformation u(x, t)= e(caret)(-t) w(x, t) were made in advance?
Problem 1. Consider the nonhomogeneous heat equation for u,t) ut = uzz + sin(2x), 0<x<π, t>0 subject to the nonhomogeneous boundary conditions u(0, t) t > 0 u(n, t) = 0, 1, - and the initial condition Lee) Find the solution u(z, t) by completing each of the following steps: (a) Find the equilibrium temperature distribution ue(x). (b) Denote v(x, t) u(a, t) - e(). Derive the IBVP for the function v(x,t). (c) Find v(x, t) (d) Find u(, t)...
ut = Kuzz-cr(z-L) where u = u(x, t) for 0 L and t 0 a(0,t) = 1 (a(L, t) = 1 where к.с > 0 are constants. For all plots in this lab, we will take c-2, к-3. L-1, but L will otherwise be left unspecified We were unable to transcribe this image ut = Kuzz-cr(z-L) where u = u(x, t) for 0 L and t 0 a(0,t) = 1 (a(L, t) = 1 where к.с > 0 are constants....
PDE questions. Please show all steps in detail. 2. Consider the initial-boundary value problem 0
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...