Differential Equations Lecture Work Solutions 72

Differential Equations Lecture Work Solutions 72 - 8. b...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
8. b Similar to 8.a u ( r, θ )=( A 0 + B 0 ln r ) a 0 + X n =1 ( A n r n + B n r n )( a n cos + b n sin ) To use the boundary conditions: u r ( a, θ )= f ( θ ) u r ( b, θ )= g ( θ ) We need to diferentiate u with respect to r u r ( r, θ )= B 0 r a 0 + X n =1 ( nA n r n 1 nB n r n 1 )( a n cos + b n sin ) Substitute r = a u r ( a, θ )= B 0 a a 0 + X n =1 ( nA n a n 1 nB n a n 1 )( a n cos + b n sin ) This is a Fourier series expansion o± f ( θ ) thus the coefficients are B 0 a a 0 = 1 2 π Z 2 π 0 f ( θ ) α 0 ( nA n a n 1 nB n a n 1 ) a n = 1 π Z 2 π 0 f ( θ )cos nθ dθ α n ( nA n a n 1 nB n a n 1 ) b n = 1 π Z 2 π 0 f ( θ )s in nθ dθ β n Now substitute r = b u r ( b, θ )= B 0
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.
Ask a homework question - tutors are online