- #1

kaizen.moto

- 98

- 0

Please help me to prove the Fourier expansion for three different cases as follows. I need some help to show that L.H.S = R.H.S

Case1: when m not equal to 0 and n not equal to 0.

dU^0/dz (1- x/a) - dU^a/dz (x/a) = -(2/m^2 Pi^2)(1- Cos[m*Pi]) [dUn^0/dz - dUn^a/dz]

Case2: when m = 0 and n not equal to 0.

dU^0/dz (1- x/a) - dU^a/dz (x/a) = -1/2 [dUn^0/dz + dUn^a/dz]

Case3: when m not equal to 0 and n = 0.

dU^0/dz (1- x/a) - dU^a/dz (x/a) = 0

where

dU^0/dz = dUn^0/dz Sin[ n*Pi*x/a] and dU^a/dz = dUn^a/dz Sin[ n*Pi*x/a].

The idea is that firstly using Fourier expansion, to expand (1 - x/a), then multiply with dU^0/dz, similarly, apply Fourier expansion for (x/a), then multiply with dU^a/dz.

Thanks for any help.