Differential Equations Lecture Work Solutions 305

Differential Equations Lecture Work Solutions 305 - Now...

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

View Full Document Right Arrow Icon
Now collect terms to compute the terms on the right at time n +1 u n +1 j +1 2 u n +1 j + u n +1 j 1 =0 · u +0 · u t · u x · u tt · u tx +( x ) 2 u xx · u ttt · u ttx + t 2 (∆ x ) 2 u txx · u xxx t 2 ) 2 (∆ x ) 2 2 u ttxx + (∆ x ) 4 12 u xxxx + ··· Divide by 2(∆ x ) 2 we get u n +1 j +1 2 u n +1 j + u n +1 j 1 2(∆ x ) 2 = 1 2 u xx + t 4 u txx + (∆ t ) 2 16 u ttxx + (∆ x ) 2 24 u xxxx + Now to terms at time n u n j +1 = u t 2 u t +∆ xu x + 1 2 ( t 2 ) 2 u tt t 2 xu tx + 1 2 (∆ x ) 2 u xx 1 6 ( t 2 ) 3 u ttt +3( t 2 ) 2 x 6 u ttx 3 t 2 (∆ x ) 2 6 u txx + (∆ x ) 3 6 u xxx + 1 24 ( t 2 ) 4 u tttt 4( t 2 ) 3 x 24 u tttx +6( t 2 ) 2 (∆ x ) 2 24 u ttxx 4 t 2 (∆ x ) 3 24 u txxx + (∆ x ) 4 24 u xxxx + u n j 1 = u t 2 u t xu x + 1 2 ( t 2 ) 2 u tt + t 2 xu tx + 1 2 (∆ x ) 2 u xx 1 6 ( t 2 ) 3 u ttt 3( t 2 ) 2 x 6 u ttx 3 t 2 (∆ x ) 2 6 u txx (∆ x ) 3 6 u xxx + 1 24 ( t 2 ) 4 u tttt +4( t 2 ) 3 x 24 u tttx t 2 ) 2 (∆ x ) 2 24 u ttxx +4 t 2 (∆ x ) 3 24 u txxx +
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