kill(all)$

biotech : find U(z) ? , wavefront propagation from FE ,part 3 ,version 1.0
peter.vlasschaert@gmail.com,13/05/2020

load packages
============

load(lrats)$

see website :appendix 2:
biotech :appendix 2:asymptotic solution(second order diff. equation),p1
--------------------------------------------------------------------------------------------------
z = x-c*t

p1:pp:ε*'diff(ξ,z,2)+'diff(g,ξ,1)+g*(1-g) =0;

result :asymptotic expansion ,
----------------------------------------

p1a:g(ξ,ε) :=-(%e^ξ*(log((%e^(-2*ξ)*(%e^ξ+1)^2)/4)+z)*ε)/(%e^ξ+1)^2+1/(%e^ξ+1);

p1aa:part(rhs(p1a),1);
p1ab:part(rhs(p1a),2);

relation : g(ξ,ε) → U(z)

=>  1e)  infinite limits: -∞,+∞, when ε= 0 → g(z,0)

p2a:'limit(g(ξ,0),ξ,minf) =limit(g(ξ,0),ξ,minf);

p2b:'limit(g(ξ,0),ξ,inf) =limit(g(ξ,0),ξ,inf);

=>  2e)  limit : z = 0;when ε= 0 → g(z,0)

p2c:'limit(g(ξ,0),ξ,0) =limit(g(ξ,0),ξ,0);

relation between: g(ξ,ε) → U(z)
========================
see :appendix 2, g(ξ,ε)  → U(z)

p3:[ξ =z/c,ε=1/c^2];

 operator : L(U) =0
============
see :appendix 2:method 2: I need real values

assume(c>0);

p4:'diff(U,z,2)+c*'diff(U,z)+U*(1-U^q) =0 ;

when q=1

p4a:subst([q=1],p4) =0;

find: U=U(z), use 'p3'
---------------------------
use : only two terms of asymptotic,see 'appendix'
use :p3

p4aa:U(z) = subst(p3,g(ξ,ε));

=>  1e)  infinite limits z: -∞,+∞,

p4bb:'limit(rhs(p4aa),z,minf) =limit(rhs(p4aa),z,minf);

p4cc:'limit(rhs(p4aa),z,inf) =limit(rhs(p4aa),z,inf);

=>  2e)  infinite limits :  z=0

p4dd:'limit(rhs(p4aa),z,0) =limit(rhs(p4aa),z,0);;

when  c=2 , limiting value 
-----------------------------------
use :p4

nonlinear ode :  c=2,wavefront

p5:subst([c=2],p4);

solution       :  c=2,wavefront,p4aa

p5a:subst([c=2],p4aa);

travelling wavefront ( for ,c=2 ( = wave speed))
====================================
plot :  x axis → z, y axis → U(z)

1e) plot : around   z=0
------------------------------

wxplot2d([rhs(%)], [z,-1,1])$

the limit around  (z=0, '0-,0+')

p6a:'limit(rhs(p4aa),z,0,minus) =limit(rhs(p4aa),z,0,minus);

p6b:'limit(rhs(p4aa),z,0,plus) =limit(rhs(p4aa),z,0,plus);

see : p4dd,'above'

2e)plot :general plot
----------------------------

wxplot2d([rhs(p5a)], [z,-100,100])$

   ==============================================================================================