Question

Once the temperature in an object reaches a steady state, the heat equation becomes the Laplace...

Once the temperature in an object reaches a steady state, the heat equation becomes the Laplace equation. Use separation of variables to derive the steady-state solution to the heat equation on the rectangle R = [0, 1] × [0, 1] with the following Dirichlet boundary conditions: u = 0 on the left and right sides; u = f(x) on the bottom; u = g(x) on the top. That is, solve uxx + uyy = 0 subject to u(0, y) = u(1, y) = 0, u(x, 0) = f(x), and u(x, 1) = g(x). You may assume the separation constant is negative: F''/F = −k, for k > 0. Extra credit: Plot u(x, y) when f(x) = sin (πx) and g(x) = sin (2πx).

Homework Answers

Answer #1

Doubt in this then comment below...i will explain you..

.

Please thumbs up for this solution..thanks..

.

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Use the Fourier sine transform to derive the solution formula for the heat equation ut =...
Use the Fourier sine transform to derive the solution formula for the heat equation ut = c2 uxx on the half-infinite bar (0 ≤ x < ∞) with Dirichlet boundary condition u(0, t) = a, for some constant a, and initial condition u(x, 0) = f(x).
Solve the heat equation and find the steady state solution: uxx = ut, 0 < x...
Solve the heat equation and find the steady state solution: uxx = ut, 0 < x < 1, t > 0, u(0,t) = T1, u(1,t) = T2, where T1 and T2 are distinct constants, and u(x,0) = 0
Find the solution formula for the heat equation ut = c2 uxx on the half-infinite bar...
Find the solution formula for the heat equation ut = c2 uxx on the half-infinite bar (0 ≤ x < ∞) with Dirichlet boundary condition u(0, t) = a, for some constant a, and initial condition u(x, 0) = f(x) using the Fourier sine transform.
Consider Dirichlet-Neumann problem uxx + uyy = 0,    −∞ < x < ∞, 0 <...
Consider Dirichlet-Neumann problem uxx + uyy = 0,    −∞ < x < ∞, 0 < y < 1 u|y=0 = f(x) uy|y=1 = g(x) Make Fourier transform by x, solve problem for ODE for uˆ(k, y) which you get as a result and write u(x, y) as a Fourier integral.
To write Laplace’s equation, Uxx + Uyy = 0, in polar coordinates, we begin with Ux...
To write Laplace’s equation, Uxx + Uyy = 0, in polar coordinates, we begin with Ux = (∂U/∂r)(∂r/∂x) + (∂U/∂θ)(∂θ/∂x) where r = √(x2+y2), θ = arctan (y/x), x = r cos θ, y = r sin θ. We get Ux = (cos θ) Ur – (1/r)(sin θ) Uθ , Uxx = [∂(Ux)/∂r] (∂r/∂x) + [∂(Ux)/∂θ](∂θ/∂x) Carry out this computation, as well as that for Uyy. Since Uxx and Uyy are both expressed in polar coordinates, their sum gives Laplace...
Solve the heat equation and find the steady state solution : uxx=ut 0<x<1, t>0, u(0,t)=T1, u(1,t)=T2,...
Solve the heat equation and find the steady state solution : uxx=ut 0<x<1, t>0, u(0,t)=T1, u(1,t)=T2, where T1 and T2 are distinct constants, and u(x,0)=0
Use the seperation of variables to find all the solutions on the form F(x)G(y) to the...
Use the seperation of variables to find all the solutions on the form F(x)G(y) to the boundary value problems u(x,0)=0 and u(x,1)=0 Where uxx=uyy
Solve the heat equation ut = k uxx, 0 < x < L, t > 0...
Solve the heat equation ut = k uxx, 0 < x < L, t > 0 u(0, t) = u(L, t) = 0, t > 0 u(x, 0) = f(x), 0 < x < L a) f(x) = 6 sin 9πx L b) f(x) = 1 if 0 < x ≤ L/2 2 if L/2 < x < L
In each of Problems 1 through 8, find the steady-state solution of the heat conduction equation...
In each of Problems 1 through 8, find the steady-state solution of the heat conduction equation α2uxx = ut that satisfies the given set of boundary conditions. 1. ux (0, t) = 0, u( L, t) = 0 2. u(0, t) = 0, ux ( L, t) = 0
In a typical steady-state heat flow in a heated plate, the following statements are true except...
In a typical steady-state heat flow in a heated plate, the following statements are true except >The difference equation is dependent on the constant h. >The difference equation is independent of the coefficient of thermal diffusivity, k. >The insulation and the thinness of the plate mean that heat transfer is limited to the x and y dimensions. >The steady-state, boundary-value problems are characterized by elliptic PDE.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT