Boundary Value Problems/Lesson 7

Boundary Value Problems

Rectangular Domain ()Edit

 

Disk Domain (Polar)Edit

 
Disc of radius c

For a disk with a radius of "c", let the polar coordinates be  , and  
 

  , boundary condition.

  continuity of potential.

  continuity of derivative.

 Edit

The solution as a product of two independent functions. By substitution into the above PDE we have:

 

Separate,

 
 

The constant may be greater than , equal to or less than zero.

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Use the continuity conditions and try to determine something more about A, B and λ.
  thus   and  
 
 
Either   or  
Before choosing, apply the second boundary condition:

The continuity of the derivative provides a second condition:
  thus  
 
 
 
Either   or  
If either A or B are zero then   also must hold. So all we need is   which implies  . Remember  

Example of Potential equation on semi-annulus.Edit

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