Ch_06 - PDE - Transform methods

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

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

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