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Question: Verification problem for power series solutions for some PDE

Question:Verification problem for power series solutions for some PDE

dds 10 Maple 2022
pde differential-equations rifsimp
April 06 2024
2

Dear Maple community,

I am facing a little problem verifying a power series solution obtained with Maple since the direct substitution back into the PDEs does not seem to be conclusive because the residual contains the powers of independent variables to almost any order.

Please see the enclosed Maple file with a minimal working example:

> 

restart:
> 

with(DETools):
> 

PDE1 := diff(eta(t,x),t) + 1/2*diff(u(t,x),x) + 1/2*eta(t,x)*diff(u(t,x),x) - 1/48*diff(u(t,x),x$3) + diff(eta(t,x),x)*u(t,x);

diff(eta(t, x), t)+(1/2)*(diff(u(t, x), x))+(1/2)*eta(t, x)*(diff(u(t, x), x))-(1/48)*(diff(diff(diff(u(t, x), x), x), x))+(diff(eta(t, x), x))*u(t, x)

 (1)
> 

PDE2 := diff(u(t,x),t) + u(t,x)*diff(u(t,x),x) + diff(eta(t,x),x,t,t) + diff(eta(t,x),x) - 1/6*diff(u(t,x),x,x,t);

diff(u(t, x), t)+u(t, x)*(diff(u(t, x), x))+diff(diff(diff(eta(t, x), t), t), x)+diff(eta(t, x), x)-(1/6)*(diff(diff(diff(u(t, x), t), x), x))

 (2)
> 

sys := rifsimp([PDE1, PDE2]);

table( [( Solved ) = [diff(diff(diff(eta(t, x), t), t), x) = -u(t, x)*(diff(u(t, x), x))-(diff(u(t, x), t))+(1/6)*(diff(diff(diff(u(t, x), t), x), x))-(diff(eta(t, x), x)), diff(diff(diff(u(t, x), x), x), x) = 24*eta(t, x)*(diff(u(t, x), x))+48*(diff(eta(t, x), x))*u(t, x)+48*(diff(eta(t, x), t))+24*(diff(u(t, x), x))] ] )

 (3)
> 

id := initialdata(sys[Solved]);

table( [( Finite ) = [], ( Infinite ) = [eta(t, x[0]) = _F1(t), (D[2](eta))(t[0], x) = _F2(x), (D[1, 2](eta))(t[0], x) = _F3(x), u(t, x[0]) = _F4(t), (D[2](u))(t, x[0]) = _F5(t), (D[2, 2](u))(t, x[0]) = _F6(t)] ] )

 (4)
> 

sols := rtaylor(sys[Solved], id, point=[t = 0, x = 0], order = 3);

[eta(t, x) = _F1(0)+(D(_F1))(0)*t+_F2(0)*x+(1/2)*((D@@2)(_F1))(0)*t^2+_F3(0)*t*x+(1/2)*(D(_F2))(0)*x^2+(1/6)*((D@@3)(_F1))(0)*t^3+(1/2)*(-_F4(0)*_F5(0)-(D(_F4))(0)+(1/6)*(D(_F6))(0)-_F2(0))*t^2*x+(1/2)*(D(_F3))(0)*t*x^2+(1/6)*((D@@2)(_F2))(0)*x^3, u(t, x) = _F4(0)+(D(_F4))(0)*t+_F5(0)*x+(1/2)*((D@@2)(_F4))(0)*t^2+(D(_F5))(0)*t*x+(1/2)*_F6(0)*x^2+(1/6)*((D@@3)(_F4))(0)*t^3+(1/2)*((D@@2)(_F5))(0)*t^2*x+(1/2)*(D(_F6))(0)*t*x^2+(1/6)*(24*_F1(0)*_F5(0)+48*_F2(0)*_F4(0)+48*(D(_F1))(0)+24*_F5(0))*x^3]

 (5)
> 

assign(sols);
> 

simplify(PDE1);

((D@@2)(_F1))(0)*t+_F3(0)*x+(1/2)*((D@@3)(_F1))(0)*t^2-t*(_F4(0)*_F5(0)+_F2(0)+(D(_F4))(0)-(1/6)*(D(_F6))(0))*x+(1/2)*(D(_F3))(0)*x^2+(1/2)*(D(_F5))(0)*t+(1/2)*_F6(0)*x+(1/4)*((D@@2)(_F5))(0)*t^2+(1/2)*(D(_F6))(0)*t*x+6*((_F1(0)+1)*_F5(0)+2*_F2(0)*_F4(0)+2*(D(_F1))(0))*x^2+(1/2)*(_F1(0)+(D(_F1))(0)*t+_F2(0)*x+(1/2)*((D@@2)(_F1))(0)*t^2+_F3(0)*t*x+(1/2)*(D(_F2))(0)*x^2+(1/6)*((D@@3)(_F1))(0)*t^3+(1/2)*(-_F4(0)*_F5(0)-(D(_F4))(0)+(1/6)*(D(_F6))(0)-_F2(0))*t^2*x+(1/2)*(D(_F3))(0)*t*x^2+(1/6)*((D@@2)(_F2))(0)*x^3)*(_F5(0)+(D(_F5))(0)*t+_F6(0)*x+(1/2)*((D@@2)(_F5))(0)*t^2+(D(_F6))(0)*t*x+(1/2)*(24*_F1(0)*_F5(0)+48*_F2(0)*_F4(0)+48*(D(_F1))(0)+24*_F5(0))*x^2)-(1/2)*_F1(0)*_F5(0)-_F2(0)*_F4(0)+(_F2(0)+_F3(0)*t+(D(_F2))(0)*x+(1/2)*(-_F4(0)*_F5(0)-(D(_F4))(0)+(1/6)*(D(_F6))(0)-_F2(0))*t^2+(D(_F3))(0)*t*x+(1/2)*((D@@2)(_F2))(0)*x^2)*(_F4(0)+(D(_F4))(0)*t+_F5(0)*x+(1/2)*((D@@2)(_F4))(0)*t^2+(D(_F5))(0)*t*x+(1/2)*_F6(0)*x^2+(1/6)*((D@@3)(_F4))(0)*t^3+(1/2)*((D@@2)(_F5))(0)*t^2*x+(1/2)*(D(_F6))(0)*t*x^2+(1/6)*(24*_F1(0)*_F5(0)+48*_F2(0)*_F4(0)+48*(D(_F1))(0)+24*_F5(0))*x^3)

 (6)
> 

simplify(PDE2);

((D@@2)(_F4))(0)*t+(D(_F5))(0)*x+(1/2)*((D@@3)(_F4))(0)*t^2+((D@@2)(_F5))(0)*t*x+(1/2)*(D(_F6))(0)*x^2+(_F4(0)+(D(_F4))(0)*t+_F5(0)*x+(1/2)*((D@@2)(_F4))(0)*t^2+(D(_F5))(0)*t*x+(1/2)*_F6(0)*x^2+(1/6)*((D@@3)(_F4))(0)*t^3+(1/2)*((D@@2)(_F5))(0)*t^2*x+(1/2)*(D(_F6))(0)*t*x^2+(1/6)*(24*_F1(0)*_F5(0)+48*_F2(0)*_F4(0)+48*(D(_F1))(0)+24*_F5(0))*x^3)*(_F5(0)+(D(_F5))(0)*t+_F6(0)*x+(1/2)*((D@@2)(_F5))(0)*t^2+(D(_F6))(0)*t*x+(1/2)*(24*_F1(0)*_F5(0)+48*_F2(0)*_F4(0)+48*(D(_F1))(0)+24*_F5(0))*x^2)-_F4(0)*_F5(0)+_F3(0)*t+(D(_F2))(0)*x-(1/2)*t^2*(_F4(0)*_F5(0)+_F2(0)+(D(_F4))(0)-(1/6)*(D(_F6))(0))+(D(_F3))(0)*t*x+(1/2)*((D@@2)(_F2))(0)*x^2

 (7)

NULL

Download MinWorkingExa.mw

Thanks a lot in advance for any help or suggestions.

Kind regards,

DDe
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