kill(all)$

biotech:Fisher Equation( = pde),travelling wave propagation :part 1,version 1.0
peter.vlasschaert@gmail.com,11/05/2020

see :website
==========
biotech:Fisher Equation( = pde),intro (relation dimensional,non dimensional) ,version 1.0
FE = Fisher Equation

load package
==========

load(lrats)$

D = diffusion >0, r =rate constant > 0

FE:dimensional 
============

p1a:'diff(u,t,1)=D*('diff(u,x,2))+r*(1-u)*u;

FE:nondimensional 
===============

p1b:'diff(u,t,1)='diff(u,x,2)+(1-u)*u;

relation between : p1a & p1b , xdot=sqrt(r/D)*x,tdot=r*t

travelling wave propagation
=====================
z = x-c*t
--------------------------------------
c : wave speed
t :  time
x : space 
--------------------------------------
u(x,t) = U(z)
--------------------------------------
wave direction

assume(c>0,t>0);

p2 : z = x-c*t;

p2a:solve(subst([z=0],p2),x)[1];

→ c >0 ,t >0  wave moves in  the positiv direction.( x direction)

step1: ∂u/∂t  → ∂U/∂z, u=U
-------------------------------------

depends(z,[x,t]);

p3a:'diff(u,t) = 'diff(U,z)*'diff(z,t);

depends(U,z);

p3b:lhs(p3a)=part(rhs(p3a),1)*subst([p2],part(rhs(p3a),2));

p3c:lhs(p3b)=ev(rhs(p3b),diff);

step 2:
---------

step2a: ∂u/∂x → ∂U/∂z, u=U
-------------------------------------

p4a:'diff(u,x) = 'diff(U,z)*'diff(z,x);

p4b:lhs(p4a)= part(rhs(p3a),1)*subst([p2],part(rhs(p4a),2));

p4c:lhs(p4a)=ev(rhs(p4b),nouns);

step2b: ∂(∂u/∂x )/∂x→ ∂(∂U/∂z)/∂z, u=U
---------------------------------------------------

p4d1:'diff(lhs(p4c),x) =diff(rhs(p4a),z);

p4d2:lhs(p4d1)=part(rhs(p4d1),1)*subst([p2],part(rhs(p4d1),2));

p4d3:lhs(p4d1)=ev(rhs(p4d2),nouns);

step3 : 
---------

p5:[p3c,p4d3];

step3a : nondimensional form  FE ,'p1a', c = wave speed
-------------------------------------------------------------------------------

temp:subst(p5,p1a)$
p5a:subst([u=U],temp);
p5aa:rhs(p5a)-lhs(p5a)=0;

p5aaa:subst([r=1],p5aa);

see :website (r=1)
-------------------------
biotech :appendix 1: second order ode : coef D,c  ,version  1.0

→result 3a:nonlinear ode

step3b : dimensional FE form ,'p1b',c = wave speed
------------------------------------------------------------------------

temp1:subst(p5,p1b)$
p5b:subst([u=U],temp1);
p5bb:rhs(p5b)-lhs(p5b)=0;

→ result 3b : nonlinear ode 

see :website
------------------
biotech :appendix 3: limits of fisher equation (-∞,0,∞),version 1.0

limit :z → -∞   ,U(z) =1, one         steady state
limit :z →  ∞   ,U(z) =0 , second  steady state
--------------------------------------------------------------
use :p5bb

p6a:limit(U(z),z,minf) = 1;
p6b:limit(U(z),z,inf)    = 0;

reason : used ,phase plane analysis,use: 'p5bb',part 2(FE)