Numbertheory

Numbertheory - Research Sudarshan Balakrishnan June 28,...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Research Sudarshan Balakrishnan June 28, 2011 Linearly dispersive wave equation The purpose of this write-up is to examine the solution to the linearly dis- persive wave equation at different times and observe any number-theoretic results that might arise. Using the program qd.m, we notice the following results: Solution profile t x u ( t,x ) t = < x < 1 < x < 2 t = 2 < x < 2 1 2 < x < 3 2 < x < 2 1 t = 3 < x < 3 1 3 < x < 2 3 2 3 < x < < x < 4 3 4 3 < x < 5 3 1 5 3 < x < 2 1 t = 2 3 < x < 3 1 3 < x < 2 3 1 2 3 < x < < x < 4 3 4 3 < x < 5 3 5 3 < x < 2 1 An immediate observation here is for every t = p q , the solution u ( t,x ) is constant for every subinterval q < x < + q . In other words, for the interval of length from q to + q the solution u ( t,x ) is constant. The 1 next step was to observe if this holds good for further intervals, we proceed similarly: Solution profile t x u ( t,x ) t = 4 < x < 4 1 4 < x < 2 2 < x < 3 4 3 4 < x < < x < 5 4 5 4 < x < 6 4 1 6 4 < x < 7 4 1 7 4 < x < 2 1 As we notice, the pattern exists for 4 . For 3 4 , we notice the same : Solution profile t x u ( t,x ) t = 3 4 < x < 4 1 4 < x < 2 1 2 < x < 3 4 1 3 4 < x < < x < 5 4 5 4 < x < 6 4 6 4 < x < 7 4 7 4 < x < 2 1 However, we dont notice the same pattern for t = p 5 which only takes identical values for intervals 4 5- 6 5 . There might be an interesting result for p 5 , but I cant seem to notice it. But the earlier pattern remains for t = 6 , as we observe the results : 2 Solution profile t x u ( t,x ) t = 6 < x < 6 1 6 < x < 2 6 2 6 < x < 3 6 3 6 < x < 4 6 4 6 < x < 5 6 5 6 < x < < x < 7 6 7 6 < x < 8 6 1 8 6 < x < 9 6 1 9 6 < x < 10 6 1 10 6 < x < 11 6 1 11 6 < x < 2 1 We notice the same at t = 2 6 = 3 , t = 3 6 = 2 and t = 4 6 = 2 3 . Now, at t = 5 6 we get: Solution profile t x u ( t,x ) t = 6 < x < 6 1 6 < x < 2 6 1 2 6 < x < 3 6 1 3 6 < x <...
View Full Document

This note was uploaded on 11/19/2011 for the course MATH 101 taught by Professor Wormer during the Spring '08 term at UCSD.

Page1 / 12

Numbertheory - Research Sudarshan Balakrishnan June 28,...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online