Solve Laplace’s equation inside a rectangle 0 = x = L, 0 = y = H, with the following boundary...

Question

Solve Laplace’s equation inside a rectangle 0 = x = L, 0 = y = H, with the following boundary conditions [Hint: Separate variables. If there are two homogeneous boundary conditions in y, let u(x,y) = h(x) f(y), and if there are two homogeneous boundary conditions in x, let u(x,y) = f(x) h(y).]:

(b) ?u/?x (0,y) = g(y), ?u/?x (L,y) = 0, u(x,0) = 0, u(x,H)=0

David · Accepted Answer

Solve Laplace’s equation inside a rectangle 0 ≤ x ≤ L, 0 ≤ y ≤ H, with the following boundary 
conditions [Hint: Separate variables. If there are two homogeneous boundary conditions in y, let 
u(x,y) = h(x) φ(y), and if there are two homogeneous boundary conditions in x, let u(x,y) = φ(x) h(y).]: 
(a) ∂u/∂x (0,y) = g(y), ∂u/∂x (L,y) = 0, u(x,0) = 0, u(x,H)=0
Solution: 
We search for the solution of the boundary value problem as a superposition of solutions 
                                                                 u(x, y) = φ(x)h(y)  
With separated variables of Laplace’s equation that satisfy the three homogeneous boundary 
conditions.  
Substituting u(x, y) = φ(x) h(y) into Laplace’s equation
   

   

We obtain 
   ( ) ( )     ( ) ( )    
   ( )
 ( )
 
   ( )
 ( )
 
Since the left-hand side does not depend on y while the right-hand side does not depend on x,

Solve Laplace’s equation inside a rectangle 0 = x = L, 0 = y = H, with the following boundary conditions [Hint: Separate variables. If there are two homogeneous boundary conditions in y, let u(x,y) =...

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