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Chem Differential Eq HW Solutions Fall 2011 115

Chem Differential Eq HW Solutions Fall 2011 115 - Section...

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Section 7.3 The Fourier Transform Method 115 and d dt u ( w, 0) = g ( w ) - cw 2 A ( w ) + cw 2 B ( w ) = g ( w ) - A ( w ) + B ( w ) = g ( w ) cw 2 - f ( w ) + 2 B ( w ) = g ( w ) cw 2 B ( w ) = 1 2 f ( w ) + g ( w ) cw 2 ; A ( w ) = 1 2 f ( w ) - g ( w ) cw 2 . Hence u ( w,t ) = 1 2 f ( w ) - g ( w ) cw 2 e - cw 2 t + 1 2 f ( w ) + g ( w ) cw 2 e cw 2 t . = f ( w ) ( e cw 2 t + e - cw 2 t ) 2 + g ( w ) cw 2 ( e cw 2 t - e - cw 2 t ) 2 = f ( w ) cosh( cw 2 t ) + g ( w ) cw 2 sinh( cw 2 t ) Taking inverse Fourier transforms, we get
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