Explore BrainMass
Share

Explore BrainMass

    A solution of the 2d Laplace equation

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    Solve Laplace of u = 0 subject to the conditions:

    u(x,0) = f1(x)
    u(0,y) = 0
    u(x,b) = 0
    u(a,y) = 0

    0<x<a
    0<y<b

    (The question attachment contains a slightly different question. The question is restated correctly in the solution attachment)

    © BrainMass Inc. brainmass.com October 9, 2019, 9:02 pm ad1c9bdddf
    https://brainmass.com/math/calculus-and-analysis/solution-2d-laplace-equation-170570

    Attachments

    Solution Preview

    Problem #2
    Solve subject to
    Boundary conditions:
    u(x,0) = f1(x)
    u(0,y) = 0
    u(x,b) = 0
    u(a,y) = 0

    0<x<a
    0<y<b

    Use the method of separation of variables to derive u1 and An where

    u1(x,y) = ∑ Ansin(nπx/a) sinh(nπ(b-y)/a)

    and

    An = 2/a sinh(nπb/a) ∫ f1(x) sin(nπx/a) dx

    Let's look for a solution of the form . The 2d Laplace equation is , which leads to

    As usual with separation of variables, we can set both of these to a constant giving the two ODEs:

    I recall (I think!) showing you previously how to ...

    Solution Summary

    Separation of variables is an commonly used technique for solving Laplace's equation and other PDEs. The solution illustrates an example of this technique comprising two pages written in Word with equations in Mathtype. Some familiarity with the technique of separation of variables is assumed (namely, that the solutions of the resulting ODEs split into 3 cases depending on the value of the constant) but the solution is otherwise explained step by step.

    $2.19