Laplace's equation on a rectangle with mixed b.c.s

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Homework Help Overview

The discussion revolves around solving Laplace's equation on a rectangle defined by the boundaries 0 < x < L and 0 < y < H, with mixed boundary conditions. Participants are exploring how to apply these conditions effectively in their solution attempts.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to apply separation of variables but expresses uncertainty about how to handle the last boundary condition. Some participants suggest considering a specific function to meet the boundary conditions, while others inquire about the assumptions regarding the separation of variables approach.

Discussion Status

The discussion is ongoing, with participants exploring different functions and their implications on the boundary conditions. There is an exchange of ideas regarding the application of separation of variables and the nature of the functions needed to satisfy the given conditions.

Contextual Notes

Participants are grappling with the implications of mixed boundary conditions and the potential use of superposition. There is a specific mention of constraints related to the boundary conditions that may affect the solution approach.

sarideli18
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Homework Statement



Solve Laplace's equation on the rectangle 0< x< L, 0< y< H with the boundary conditions du/dx(0, y) = 0, du/dx(L, y)=y, du/dy(x, 0)=0, U(x, H)=x.

Homework Equations





The Attempt at a Solution



I would be able to solve it by separation of variables if the last boundary condition were du/dx(x,H)=x. How can I apply the last boundary condition? Does principle of superposition work?
 
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You need a function that has derivative 0 at y= 0 and value x at y= H. The first condition makes me think of a quadratic. Look at [itex]v(x, y)= (x/H^2)y^2[/itex]. And then change your function to w(x,y)= u(x,y)- v(x,y).
 
Then how can I go further with separation of variables?
 
By the way, Can I assume U=X(x)Y(y) and say Y(H)=1 and Xx(L)=1 ?
 

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