Solving 2D Laplace Equation with Boundary Conditions

Click For Summary
SUMMARY

The forum discussion centers on solving the 2D Laplace equation, represented as Uxx + Uyy = 0, with specified boundary conditions: U(x,0) = x, U(0,y) = 0, U(1,y) = 1, and U(x,1) = x. A user sought an analytical solution while also having solved the equation numerically using a finite difference method. The discussion emphasizes the method of separation of variables as a common approach to derive the solution, and the user ultimately identified an error in their MATLAB program that affected their graphical representation of the solution.

PREREQUISITES
  • Understanding of the 2D Laplace equation
  • Familiarity with boundary value problems
  • Knowledge of separation of variables technique
  • Proficiency in MATLAB for numerical simulations
NEXT STEPS
  • Study the method of separation of variables in depth
  • Explore MATLAB's capabilities for solving partial differential equations
  • Learn about finite difference methods for numerical solutions
  • Investigate common pitfalls in programming numerical simulations
USEFUL FOR

Mathematicians, engineers, and students involved in computational mathematics or numerical analysis, particularly those working with partial differential equations and boundary value problems.

dzi
Messages
2
Reaction score
0
Hi,

My 2D L.e.:

Uxx + Uyy = 0

with boundary conditions:

U (x,0) =x
U (0,y) =0
U (1,y) =1
U (x,1) =x

Please, need help with analytical sollution!

i'm trying to do smth like it is described in attachment pic, but i it's not working..

Thanks!
 

Attachments

  • laplace.png
    laplace.png
    25.1 KB · Views: 583
Physics news on Phys.org
Try separation of variables:
U(x, y) = \xi(x) \eta(y)
then find trial solutions for both, work out the product and impose the boundary conditions.

That is the (most common, probably not the only) way to do it. If you want us to help you locate your error or give you a complete answer, you'll have to show some work.
 
dear CompuChip..

i solved equation numerical with finite diferencial method..
and also needed analytical sollution for comparison..

finaly few minutes ago i found my error..
huhh..
it wasn't placed in calculation like i was thinkig,
but in my Matlab program which i used for drawing sollution of equation.. :approve:

i appreciate on quick answer,
thanks..
 

Similar threads

Graduate The boundary conditions in reference to Laplace's equation
  • · Replies 22 ·
Replies
22
Views
4K
Undergrad How does one solve Uxx+Uyy+Uzz=C when C is non-zero?
  • · Replies 12 ·
Replies
12
Views
4K
Graduate Solving Laplace's equation in polar coordinates for specific boundary conditions
  • · Replies 3 ·
Replies
3
Views
3K
MHB Solve Laplace equation on unit disk
  • · Replies 33 ·
2
Replies
33
Views
5K
Graduate How to solve Diffusion Equation with time dependant conditions?
  • · Replies 13 ·
Replies
13
Views
3K
Graduate Searching for radial part solution of Klein-Gordon type equation
  • · Replies 1 ·
Replies
1
Views
2K
MHB Boundary value problem for Laplace equation
  • · Replies 16 ·
Replies
16
Views
3K
Undergrad Solving a differential equation using Laplace transform
  • · Replies 5 ·
Replies
5
Views
3K
Graduate 2D Cartesian Laplace equation with a single point diffusion
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Solution of delayed forcing function
  • · Replies 5 ·
Replies
5
Views
4K