Ch_06 - PDE - Transform methods

# Ch_06 - PDE - Transform methods - Partial Differential...

This preview shows pages 1–2. Sign up to view the full content.

Partial Differential Equations 2. 1-D Heat Equation in Cartesian Coordinates 2.1E Finite Fourier Cosine and Sine Transform Method Finite Fourier Cosine Transform Method 2 2 uu t x  ( 0 ) I.C.: (,0 ) () ux f x for 0 x B.C’s: (0, ) 0 (,) 0 x ut uLt for 0 t Finite Fourier Cosine Transform with respect to x : Integral Transform:  0 ˆ (, ) ( , ) ( , ) c o s L cm m C u x t u t d   Derivative Formula:     (,) ( 0 ,) (,)c o s xm Cu xt Suxt u t m      2 ( 0 o s xx m x x Cuxt u t u Lt m Inversion Formula: 1 1 12 ˆˆ ˆ ( , ) ) ( 0 , ) ) c o c m m u x t C u t x LL  Find the eigenfunction and eigenvalue from the Sturm-Liouville equation: 2 2 2 0 dX X dx ( 0 x L ) B.C’s: (0) 0 X , 0 XL Eingenfunction: ) c o s mm X xx Eingenvalue: cos 0 m L (1 / 2 ) m m L Finite Fourier cosine transformation with respect to x :      2 ˆ ( 0 , ) ( , ) c o s tx x m c x x Cu u u t m  Apply B.C’s: 2 ˆ ˆ (0, ) c mc x du t dt  x uL t cos m ODE: 2 ˆ ˆ c du u dt Transformed solution: 2 ˆ ) m t m a e Apply I.C: 0

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

Ch_06 - PDE - Transform methods - Partial Differential...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online