kill(all)$

biotech :appendix 3, limits of fisher equation (-∞,0,∞),version 1.0
peter.vlasschaert@gmail.com,10/5/2020

rem :first model epidemics ( with extra term of diffusion)
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new: 21/04/2020:maxima : biotech : numerical model (rk4),epidemic: one ode, version 1.0a.
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reason : wave  propagation , z = x-c*t, t =time , x = space ( =distance)

p1:'diff(u,z,2)+2*'diff(u,z)+u*(1-u)=0;

from : biotech ,appendix  1 :  2 before ('diff(u,z,1))  = > limiting value
When D=1 →  c = 2

p1a:ode2(p1,u,z);

u^2 << u

p1b:args(lhs(p1));

→  linearized: version , 'p1'
=====================

p1c:p1b[1]+p1b[2]+u=0;

p1c1:ode2(p1c,u,z);

limit , z →∞,u(z)
============

p1c1a:'limit(lhs(p1c1),z,inf) ='limit(rhs(p1c1),z,inf);

p1c1a1:ev(p1c1a,nouns);

check value :inf = 10^20

p1c1b:expand(subst([u=0,z=10^20],p1c1));

p1c1c:float(p1c1b);

find max:
=======
use : p1c1

u' = diff(u,z) =0

p2:factor(diff(rhs(p1c1),z));

p2a:(%k2*z-%k2+%k1) =0;

p2b:collectterms(lhs(p2a),%k2)=0;

both:{%k1,%k2} ≠0

p2b1:coeff(lhs(p2b),%k1)=0;
p2b2:coeff(lhs(p2b),%k2)=0;
p2b2a:max_or_minz =solve(p2b2,z)[1];

find :sign(u'')
==========
use: p2  @ z=1,need  %k1 =0, contradiction  (1=0)

p2c:factor(diff(p2,z));

p2c1:float(subst([z=1,%k1=0],p2c));

 %k2 > 0 => max  (1)
%k2 <  0 => min   (2)

we want find :limit u for z=0
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use :p1c1,rhs

p2c1a:limit(rhs(p1c1),z,0,minus);

p2c1a1:subst([%k2=0],limit(rhs(p1c1),z,-0.001));

p2c1b:limit(rhs(p1c1),z,0,plus);

p2c1b1:subst([%k2=0],limit(rhs(p1c1),z,0.001));

from biotech : appendix  2, from BC condition 
u(0) =g[0](0) =1/2
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reason : u(z),when z=0  →u(0) =1/2

use :'p1c1'

we find : %k1

p3:p1c1;

BC : z=0,u(0) = 1/2

p3a:'limit(lhs(p1c1),z,0) ='limit(rhs(p1c1),z,0);

p3b:ev(subst([z=0,u=1/2],p3a),nouns);

p3b1:rhs(p3b) =lhs(p3b);

use: p3

p3c:subst([z=0,p3b1],p3);

final :result : linearized pde
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z  → ∞ , u=0     ;linearized pde , steady state value
z →   0, u=1/2  ,linearized  pde

final result : 'p1' = pde
=================
u=1 ,steady state ,see above when : %k2 >0,see 'above' = max value
suspect :sigmoid curve
→  u = 1  →   (z → -∞)
study arround:0.
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values :-0.01 ;use ,'p2c1a1', 'p3b1'

p4a:subst([p3b1],p2c1a1);

values :0.01 ;use ,'p2c1a1', 'p3b1'

p4b:subst([p3b1],p2c1b1);

second : way : sigmoid curve, by study around point  'z=0'