# Problem 2, Inhomogeneous version of Laplace’s Equation with inhomogeneous BCs. We have found in clas

Problem 2, Inhomogeneous version of Laplace’s Equation with inhomogeneous BCs.
We have found in class that Laplace’s Equation can be solved by SoV when exactly one BC is inhomogeneous. Here we consider a generalisation of Laplace’s Equation that contains five inhomogeneous terms, including one in the PDE itself. (The inhomogeneous Laplace equation is called the Poisson equation, after another 19th-century French mathematician.)
Consider the following PDE for u(x, y):
2 + 2 = − 2 2
in the rectangle 0
The BCs are: ( ,0) = 1; (0, ) = 1; ( , ) = ; ( , ) = ( ).
Note: As discussed in class, when a limit and a derivative occur together without a specified order, the derivative must be taken first, and the limit next. Thus, ( , 0) = 1 means lim [ ( , )] =
1 and NOT [lim ( , )] = 1. →0
SoV will fail for two reasons: the PDE is inhomogeneous, and there are two inhomogeneous BCs. As in Problem 1, decompose the problem as follows: ( , ) = ( , ) + ( , ). We
want to choose the function ( , ) judiciously, so that it not only satisfies the Poisson equation, but also as many of the inhomogeneous BCs as possible. This will leave ( , ) to solve a homogeneous PDE with as few inhomogeneous BCs as possible; and we hope to solve for ( , ) by SoV. Note that ( , ) is not the steady-state component of the solution: Laplace’s Equation is already a steady-state description. We simply choose ( , ) to simplify the problem for ( , ).
(a) Find the most general form for ( , ) that is a solution of the inhomogeneous PDE. There are two possibilities; choose the one that has a higher power of than of .
[Hint: there should be five or six terms]. [10 marks]
(b) By choosing the coefficients in the form for ( , ) produced in part (a), ensure that the description for ( , ) consists of a homogeneous PDE and three homogeneous BCs (i.e., only one BC is inhomogeneous). Do NOT solve explicitly for ( , ).
[Hint: Try to simplify the least complicated BC’s first; also consider the BCs at = 0 and = 0 earlier than the others, because setting = 0 or = 0 might provide some simplification. It should be clear that the most complicated BC is at = .] [20 marks

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