## 4 Reputation

17 years, 351 days

## Equivalent system...

eq0 := [diff(s(x, t), t) = -a*i(x, t)*s(x, t), diff(i(x, t), t) = a*i(x, t)*s(x, t)-b*i(x, t)+c*(diff(i(x, t), x))];

then it is equivalent to the following splited system

sys:=[i(x,t) = -diff(s(x,t),t)/a/s(x,t),S(x)+ (s(x,t)^2*a-s(x,t)*b*ln(s(x,t))-diff(s(x,t),t)+diff(s(x,t),x)*c)/c/s(x,t)];

where S(x) is an arbitrary function.

I think that the presence of ln(s(x,t)) item lead to main problems in (numeric) solving process.

I'd advise you to try numerics in the cases when exact solutions exist, e.g.

si_ans1:= {i(x,t) = m*(RootOf(Int(1/(_f*(-_f*a+b*ln(_f))),_f = `` .. _Z)*m-Int(1/(_f*(-_f*a+b*ln(_f))),_f = `` .. _Z)*c+t*m+x+_C1*m)*a-b*ln(RootOf(Int(1/(_f*(-_f*a+b*ln(_f))),_f = `` .. _Z)*m-Int(1/(_f*(-_f*a+b*ln(_f))),_f = `` .. _Z)*c+t*m+x+_C1*m)))/(-m+c)/a, s(x,t) = RootOf(-Int((-m+c)/_f/m/(-_f*a+b*ln(_f)),_f = `` .. _Z)*m+t*m+x+_C1*m)};

or

si_ans2:=[i(x,t) = -D(_F1)((x+t*c)/c)*(-c*k-RootOf(x-Int(1/(-c*k-_a*a+b*ln(_a))/_a*c,_a = `` .. _Z)+_F1((x+t*c)/c))*a+b*ln(RootOf(x-Int(1/(-c*k-_a*a+b*ln(_a))/_a*c,_a = `` .. _Z)+_F1((x+t*c)/c))))/c/a,s(x,t) = RootOf(x-Int(1/(-c*k-_a*a+b*ln(_a))/_a*c,_a = `` .. _Z)+_F1((x+t*c)/c))];

## Solving a system of pdes via general sol...

Your PDEs system can be solved exactly with help of Maple by the following scheme ( I leave out  bulky outputs).

with(PDETools);

sys:=[diff(v(s,n),n) + diff(u(s,n),s) +diff(xi(s,n),s) + A*n*diff(c(s),s) = 0,

A1*diff(xi(s,n),n) + diff(v(s,n),s) -c(s)+A2*v(s,n) + A3* c(s) = 0,

diff(u(s,n),s) + 2*A2* u(s,n)=A2*(xi(s,n) + A*n*c(s)) -A1*diff(xi(s,n),s)-A2*n*c(s)];

Here, as I understand,  c(s) is a parameter.

From the beginning we need to split the system

cas:=[casesplit(sys,[xi,u,v])];

As a result we obtain system of 4 equations, which is equivalent to initial one. First of all we have to solve the last equation

pp := diff(v(s,n),`\$`(s,3)) = (diff(c(s),`\$`(s,2))*A1-3*A1*A2*A*diff(c(s),s)-A1*diff(v(s,n),`\$`(n,2),s)-2*A2*A1*diff(v(s,n),`\$`(n,2))-A1*A*diff(c(s),`\$`(s,2))+A1*A2*diff(c(s),s)-A3*diff(c(s),`\$`(s,2))*A1-A2*diff(v(s,n),`\$`(s,2))*A1+4*A2*diff(v(s,n),`\$`(s,2))-diff(c(s),`\$`(s,2))+3*A2*A3*diff(c(s),s)+3*A2^2*diff(v(s,n),s)-3*A2*diff(c(s),s)+A3*diff(c(s),`\$`(s,2)))/(-1+A1);

for unknown function v(s,n). It is linear non-homogeneous third-order PDE (so, generally speaking, we need  three initial conditions !). One of exact solutions of this PDE can be obtained

an0:=subs({_c=0,_C1=0,_C2=0,_C3=0,_C4=0,_C5=0

},pdsolve(pp,v, INTEGRATE, build));

Then we will seek the solution in the familiar form

an_W := v(s,n) = w(s,n) + rhs(an0);

which lead to the homogeneous linear PDE with constant coefficients  for w(s,n)

ppp:=collect(numer(factor(pdetest(an_W,pp))),diff);

ppp := -3*A2^2*diff(w(s,n),s)+2*A2*A1*diff(w(s,n),`\$`(n,2))+(A1*A2-4*A2)*diff(w(s,n),`\$`(s,2))+A1*diff(w(s,n),`\$`(n,2),s)+(-1+A1)*diff(w(s,n),`\$`(s,3));

This equation can be solved by Fourier method or by separation of variables (with similar result). Suppose that

an_w:=w(s,n)=Int(Phi(s,omega)*exp(I*n*omega),omega=-infinity..infinity);

and from

pdetest(an_w,ppp);

after some manipulation (by hand) we find that Phi(s,omega) must fit the following ODE

WW := (-A1*omega^2-3*A2^2)*diff(Phi(s,omega),s)+(A1*A2-4*A2)*diff(Phi(s,omega),`\$`(s,2))+(-1+A1)*diff(Phi(s,omega),`\$`(s,3))-2*A2*A1*Phi(s,omega)*omega^2;

which can be solved

ans_w:=pdsolve(WW);

Substituting it to an_W we obtain the general solution for  v(s,n). Now to produce a complete solution we need to specify 3 arbitrary functions {_F1(omega), _F2(omega), _F3(omega)}  in above solution by standard way from general solution and the set of initial conditions .

Then substituting it to cas  we get the split system for u and xi, which can be easily solved too .

## An operator solution to nonlinear differ...

It seems to me that your method of attack the DE problems is very close to the operator method. It has a long history. See, please

A.A.Agrachev, R.V.Gamkrelidze, The exponential representation of flows and the chronological calculus. Matem. sbornik,1978, v.107. (in Russian)

A.A.Agrachev, S.A.Vakhrameev, Chronological series and Cauchy-Kovalevska theorem.Itogi nauki. VINITI. Problemy geometrii,1981, v.12 (in Russian)

and (unfortunately without references on above papers)

http://arxiv.org/abs/math-ph/0409035

Yuri

## Cauchy meaning...

What does Cauchy mean in this sense?

Here the Cauchy prolem has  standard meanins - roughly speaking when PDE (system) is supplemented with conditions on one point of one independent variable.

Yuri

## PDE problems with newton iteration -> Fo...

Dear Jurjen,

I think that procedure  "Formal series solutions to non-linear DE (ODE or PDE) or systems of them (Cauchy problem)"

http://www.maplesoft.com/applications/app_center_view.aspx?AID=1906