We can expect the solution u(x,y) to be in the form X(x)Y(y).
or
I believe that these are the correct forms of X(x) and Y(y).
We can expect the solution u(x,y) to be in the form X(x)Y(y). or I believe that...
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...
Thank you.
5. Find the solution u(x, y) of Laplace's equation in the rectangle 0<<a, 0<y<b that satisfies the boundary conditions u(0, y = 0, u(a, y) = 0,uy 3,0) = 0, 2,b) = g(1), where J2 0<x<a/2 g(x) = - 0<a/2 <<a
1. (a) Derive the solution u(x, y) of Laplace's equation in the rectangle 0 < x <a, 0 <y <b, that satisfies the boundary conditions u(0,y) = 0, u(a, y) = 0, u(x,0) = 0, u(x,b) = g(x), 0 0 0 < a. (b) Find the solution if a = 4, b = 2, and g(x) = 0 <r <a/2, a-r, a/2 < x <a.
3. (20 points) Denote u(ar, y) the steady-state temperature in a rectangle area 0 z 10, 0yS 1. Find the temperature in the rectangle if the temperature on the up side is kept at 0°, the lower side at 10° while the temperature on the left side is S0)= sin(y) and the right side is insulated. Answer the following questions. (a) (10 points) Write the Dirichlet problem including the Laplace's equation in two dimensions and the boundary conditions. (b) (10...
Please show all work and provide and an original solution.
We can apply the Method of Separation of Variables to obtain a representation for the solution u u(, t) for the following partial differential equation (PDE) on a bounded domain with homogeneous boundary conditions. The PDE model is given by: u(r, 0) 0, (2,0) = 4. u(0,t)0, t 0 t 0 (a) (20 points) Assume that the solution to this PDE model has the form u(x,t) -X (r) T(t). State...
For : U(x,0) = Sin(ax) a=
2.6
using the Explicit Forward Euler and Crank-Nicholson
methods.
Example 92. One-Dimensional Parabolic PDE: Heat Flow Equation. Consider the parabolic PDE d-u(x, t) du(x, t) 0t with the initial condition and the boundary conditions (E9.2.2) We were unable to transcribe this image
Example 92. One-Dimensional Parabolic PDE: Heat Flow Equation. Consider the parabolic PDE d-u(x, t) du(x, t) 0t with the initial condition and the boundary conditions (E9.2.2)
Partial Differential Equations. Let be the upper half of a disk of radius 1. Solve the Dirichlet problem for the Laplace equation: in for -1 < x <1 and y = 0 for We were unable to transcribe this imageu : We were unable to transcribe this imageWe were unable to transcribe this imageu = y We were unable to transcribe this image u : u = y
Consider the Laplace equation v²u(x, y, z) = 0 in 3D and find its most general solution for a parallelepiped defend by planar as (see the figure below) (x = 0,X = 2, y = 0, y = 2, (z=0,2 = 5, with the Dirichlet boundary conditions as u(0, y, z) = u(2, y, z) = 0 u(x, 0, 2) = u(x, 2, z) = 0 (u(x, y,0) = 0, u(x,y,5) = 17
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
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...