Fork me on GitHub
Math for the people, by the people.

User login

Perturbation in PDE

Primary tabs

Perturbation in PDE

Hey,

I really need help. Can anyone explain me the way to solve this:

I think this is ment to be solved using Lindstedt method.

Consider the PDE u_tt-u_xx+k^2*u+eps*u^3=0.

a) We assume k>0, and look for a solution u(t,x)=U(w(eps)t,x), where U=U0+eps*U1+eps^2*U2+O(eps^2), and w(eps)=1+eps*w1+eps^2*w2+O(eps^3).
We assume u(0,x)=A*cos(x)+eps*B*cos(3x)+O(eps^2), and u_t(0,x)=0. A~=0, and U_j is bounded for all j.

We look for a positive value k0 of k, for which not all resonant terms can be eliminated from the equation for U2. Show that for k~=k0 all resonant terms have been eliminated from the equations for U1 and U2.

b) Again assume that k>0. Let u(t,x,eps) be a solution of the PDE that is periodic in x with period 2*pi, periodic in t with minimal period P(eps) and is even in x and in t. Assume that as eps -> 0 the solution u(t,x,eps) converges to a nonzero function u0(t,x) and its period P(eps) converges to a nonzero P0. Show that there is a countable set E of positive real numbers such that if k is not in E then there exist a nonzero A and an integer n such that u0(t,x)=A*cos(sqrt(n^2+k^2)*t)*cos(n*x).

c) Is k0 from (a) is in E from (b)?

This would mean the world if anyone would help me!
You can also contact me by mail:
w.jhon1984@gmail.com


I’m think k0 come from the cutoff frequency designing , the media . Hence, k0 from (b) process !.

Happy to see Pahio is again active on this site!

Happy to see Pahio is again active on this site!

Happy to see Pahio is again active on this site!

Happy to see Pahio is again active on this site!

Join maths corner on facebook. Procedure: join fb and I can add you as member - your contributions are welcome.

Fermat’s theorem works even if the base is a Gausssian integer subject to a) the prime under consideration is of shape 4m+1 and b) the exponent and base are co-prime.

((2+3*I)^16-1)/17

Fermat’s theorem works even if the base is a Gausssian integer subject to a) the prime under consideration is of shape 4m+1 and b) the exponent and base are co-prime.

((2+3*I)^16-1)/17

Fermat’s theorem works even if the base is a Gausssian integer subject to a) the prime under consideration is of shape 4m+1 and b) the exponent and base are co-prime.

((2+3*I)^16-1)/17 = -47977440 - 803040*I

This works even when the base is a Gaussian integer:

Reading GPRC: gprc.txt …Done.

GP/PARI CALCULATOR Version 2.6.1 (alpha) i686 running mingw (ix86/GMP-5.0.1 kernel) 32-bit version compiled: Sep 20 2013, gcc version 4.6.3 (GCC) (readline v6.2 enabled, extended help enabled)

Copyright (C) 2000-2013 The PARI Group

PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER.

Type ? for help, \qto quit. Type ?12 for how to get moral (and possibly technical) support.

parisize = 4000000, primelimit = 500000 (17:50) gp ¿ ((14+15*I)^104-1)/105 (17:51) gp ¿

This works even when the base is a Gaussian integer:

Reading GPRC: gprc.txt …Done.

GP/PARI CALCULATOR Version 2.6.1 (alpha) i686 running mingw (ix86/GMP-5.0.1 kernel) 32-bit version compiled: Sep 20 2013, gcc version 4.6.3 (GCC) (readline v6.2 enabled, extended help enabled)

Copyright (C) 2000-2013 The PARI Group

PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER.

Type ? for help, \qto quit. Type ?12 for how to get moral (and possibly technical) support.

parisize = 4000000, primelimit = 500000 (17:50) gp ¿ ((14+15*I)^104-1)/105 = -249662525598174865517621222098021785366399633335910441957688800663877876192221716937263714468906280908614454012799368615180549371243472 - 118511838209654103558982122027130965758920275429164915998560474682902951765213030198935065103035392002339412087987613469408163154998032*I (17:51) gp ¿

What puzzles me is that the theorem works when the base is a prime in the ring of Gaussian integers and the exponent is a prime of shape 4m + 1 but does not work when the exponent is a prime of shape 4m+3.Can any one throw some light on this?

Hi Deva, perhaps the entry ’theorem on sums of two squares by Fermat’ may explain it or help this problem,

Jussi

Hi Jussi. Thanks will try.

Hi Jussi. Thanks will try.

Subscribe to Comments for "Perturbation in PDE"