Question: How do I values to constants when using Invariants for PDE

Dear Maple users,

I am progressing, but one last hitch, see below. A want the invariants of the PDE below. However, the final expression is too general to be useful. I would like to insert specific values for F1(R), F2(R), F3(R), and F4(R):

F3=F4=0; and F2=1, and F3(R)=aR, where a is a constant. R is one of my independent variables (the Reynolds number). 

 

I would like to do this at the step where the Infinitiesimals are generated by

infinies := Infinitesimals(PDE)

For example, the first entry would then be, infinies:=[ _xi[y](y, R, l, u) = y, ... ]. Then Invariants should give me much simplified expressions which I need.  How can i do this?

Thanks

Nadeem

 

with(PDEtools)

declare(u(y, R))

` u`(y, R)*`will now be displayed as`*u

(1)

declare(l(y, R))

` l`(y, R)*`will now be displayed as`*l

(2)

L := diff_table(l(y, R))

table( [(  ) = l(y, R) ] )

(3)

NULL``

U := diff_table(u(y, R))

table( [(  ) = u(y, R) ] )

(4)

DepVars := ([l, u])(y, R)

[l(y, R), u(y, R)]

(5)

PDE := U[y, y]+2*l(y, R)^2*U[y]*U[y, y]+2*l(y, R)*L[y]*U[y]^2+1/R = 0

diff(diff(u(y, R), y), y)+2*l(y, R)^2*(diff(u(y, R), y))*(diff(diff(u(y, R), y), y))+2*l(y, R)*(diff(l(y, R), y))*(diff(u(y, R), y))^2+1/R = 0

(6)

infinies := Infinitesimals(PDE)

[_xi[y](y, R, l, u) = _F2(R)*y+_F3(R), _xi[R](y, R, l, u) = _F1(R), _eta[l](y, R, l, u) = (1/2)*l*(-R*_F2(R)+_F1(R))/R, _eta[u](y, R, l, u) = (2*R*_F2(R)-_F1(R))*u/R+_F4(R)]

(7)

InfinitesimalGenerator(infinies, DepVars, prolongation = 1)

proc (f) options operator, arrow; add(xi[x[j]]*(diff(f, x[j])), j = 1 .. 2)+add(eta[u[m]]*(diff(f, u[m]))+eta[u[m], [y]]*(diff(f, u[m][y]))+eta[u[m], [R]]*(diff(f, u[m][R])), m = 1 .. 2) end proc

(8)

Phi := Invariants(infinies, DepVars)

l*exp(-(1/2)*(Int((-R*_F2(R)+_F1(R))/(R*_F1(R)), R))), -(Int(_F3(R)*exp(-(Int(_F2(R)/_F1(R), R)))/_F1(R), R))+y*exp(-(Int(_F2(R)/_F1(R), R))), u*exp(Int((-2*R*_F2(R)+_F1(R))/(R*_F1(R)), R))-(Int(_F4(R)*exp(Int((-2*R*_F2(R)+_F1(R))/(R*_F1(R)), R))/_F1(R), R)), u[y]*exp(Int((-R*_F2(R)+_F1(R))/(R*_F1(R)), R)), l[y]*exp(-(1/2)*(Int((-3*R*_F2(R)+_F1(R))/(R*_F1(R)), R))), (1/2)*Intat(-exp((1/2)*(Int((2*(diff(_F1(_j), _j))*_j+_j*_F2(_j)-_F1(_j))/(_j*_F1(_j)), _j)))*(-2*(diff(_F2(_j), _j))*_j^2*l[y]*(y*exp(-(Int(_F2(R)/_F1(R), R)))-(Int(_F3(R)*exp(-(Int(_F2(R)/_F1(R), R)))/_F1(R), R))+Int(_F3(_j)*exp(-(Int(_F2(_j)/_F1(_j), _j)))/_F1(_j), _j))*exp(Int(_F2(_j)/_F1(_j), _j)+(1/2)*(Int((-3*_j*_F2(_j)+_F1(_j))/(_j*_F1(_j)), _j))-(1/2)*(Int((-3*R*_F2(R)+_F1(R))/(R*_F1(R)), R)))+l*(-(diff(_F2(_j), _j))*_j^2+(diff(_F1(_j), _j))*_j-_F1(_j))*exp((1/2)*(Int((-_j*_F2(_j)+_F1(_j))/(_j*_F1(_j)), _j))-(1/2)*(Int((-R*_F2(R)+_F1(R))/(R*_F1(R)), R)))-2*(diff(_F3(_j), _j))*l[y]*_j^2*exp((1/2)*(Int((-3*_j*_F2(_j)+_F1(_j))/(_j*_F1(_j)), _j))-(1/2)*(Int((-3*R*_F2(R)+_F1(R))/(R*_F1(R)), R))))/(_j^2*_F1(_j)), _j = R)+l[R]*exp((1/2)*(Int((2*(diff(_F1(R), R))*R+R*_F2(R)-_F1(R))/(R*_F1(R)), R))), u[R]*exp(Int(((diff(_F1(R), R))*R-2*R*_F2(R)+_F1(R))/(R*_F1(R)), R))+Intat(-exp(Int(((diff(_F1(_k), _k))*_k-2*_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k)-(Int((-_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k))-(Int((-2*_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k)))*(-(diff(_F2(_k), _k))*_k^2*u[y]*(y*exp(-(Int(_F2(R)/_F1(R), R)))-(Int(_F3(R)*exp(-(Int(_F2(R)/_F1(R), R)))/_F1(R), R))+Int(_F3(_k)*exp(-(Int(_F2(_k)/_F1(_k), _k)))/_F1(_k), _k))*exp(Int(_F2(_k)/_F1(_k), _k)+Int((-2*_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k)+Int((-R*_F2(R)+_F1(R))/(R*_F1(R)), R))+(diff(_F4(_k), _k))*exp(Int((-_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k)+Int((-2*_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k))*_k^2-_k^2*u[y]*(diff(_F3(_k), _k))*exp(Int((-2*_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k)+Int((-R*_F2(R)+_F1(R))/(R*_F1(R)), R))+exp(Int((-_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k))*(u*exp(Int((-2*R*_F2(R)+_F1(R))/(R*_F1(R)), R))-(Int(_F4(R)*exp(Int((-2*R*_F2(R)+_F1(R))/(R*_F1(R)), R))/_F1(R), R))+Int(_F4(_k)*exp(-(Int((2*_k*_F2(_k)-_F1(_k))/(_k*_F1(_k)), _k)))/_F1(_k), _k))*(2*(diff(_F2(_k), _k))*_k^2-(diff(_F1(_k), _k))*_k+_F1(_k)))/(_k^2*_F1(_k)), _k = R)

Please Wait...