Perturbation in PDE

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# 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

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## I’m think k0 come from the

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

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## Fermat's theorem

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

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

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

## Euler's generalisation of Fermat's theorem .......

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 ¿

## Euler's generalisation of Fermat's theorem .......

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 ¿