kill(all)$

biotech :appendix 1a: How to get ε second order diff equation from the travel wave propagation ,version1
peter.vlasschaert@gmail.com,12/05/2020

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

load(lrats)$

see: website ( Fisher Equation = FE)
maxima : biotech : Fisher Equation( = pde),travelling wave propagation :part 1,version 1.0

given : step:travelling wavefront solution U(z) :FE
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p1:'diff(U,z,2)+c*'diff(U,z)+U*(1-U) = 0;

? find ε second order nonlinear ode
============================

1e step : given :first        :ζ = z/c =ξ^(1/2)
                          second : U(z) = g(ζ)

1e step :'diff(U,z) →'diff(g,ξ)
--------------------------------------

p2:[ξ = z/c ,ξ =ε^(1/2)*z];

p3:[U=g];

p3a:'diff(U,z) = 'diff(U,ξ)*'diff(ξ,z);

p3a1:lhs(p3a)=part(rhs(p3a),1)*subst([p2[2]],part(rhs(p3a),2));

p3a2:lhs(p3a) =part(rhs(p3a),1)*ev(ratsimp(('diff((z*sqrt(ε)),z,1))),diff);

use :U=g

p3a3:lhs(p3a) = subst(p3,rhs(p3a2));

2e step :'diff(U,z,2) →'diff(g,ξ,2)
-------------------------------------------

p4a1:'diff(lhs(p3a2),z)='diff(rhs(p3a3),ξ,1)*'diff(ξ,z);

p4a2:lhs(p4a1)=part(rhs(p4a1),1)*ev(ratsimp(('diff((z*sqrt(ε)),z,1))),diff);;

depends(g,ξ);

p4a3: lhs(p4a1)=ev(rhs(p4a2),diff);

3e step : subst,p3a3 & p4a3 into p1
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p5a:subst([p3a3,p4a3,p3[1]],p1);

4e step : remove c from nonlinear ode p5a
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p6a:solve(p2[2],ε^(1/2));

p6b:solve(p2[1],c);

p6c:subst(append(p6a,p6b),p5a);

5e step : replace ξ to z
--------------------------------
nonlinear ode : appendix 2 

p7:subst([ξ =z],p6c);