restart;
  inf:=10:
  pdes:= diff(u(y,t),t)-xi*diff(u(y,t),y)=diff(u(y,t),y$2)/(1+lambda__t)+Gr*theta(y,t)+Gc*C(y,t)-M*u(y,t)-K*u(y,t),
         diff(theta(y,t),t)-xi*diff(theta(y,t),y)=1/Pr*diff(theta(y,t),y$2)+phi*theta(y,t),
         diff(C(y,t),t)-xi*diff(C(y,t),y)=1/Sc*diff(C(y,t),y$2)-delta*C(y,t)+nu*theta(y,t):
  conds:= u(y,0)=0, theta(y,0)=0, C(y,0)=0,
          u(0,t)=0, D[1](theta)(0,t)=-1, D[1](C)(0,t)=-1,
          u(inf,t)=0, theta(inf,t)=0, C(inf,t)=0:
  pars:= { Gr=1, Gc=1, M=1, nu=1, lambda__t=0.5,
           Sc=0.78, delta=0.1, phi=0.5, K=0.5, xi=0.5
         }        

  PrVals:=[0.71, 1.00, 3.00, 7.00]:
  colors:=[red, green, blue, black]:
  for j from 1 to numelems(PrVals) do
      pars1:=`union`( pars, {Pr=PrVals[j]}):
      pdSol:= pdsolve( eval([pdes], pars1),
                       eval([conds], pars1),
                       numeric
                     );
      plt[j]:=pdSol:-plot( diff(u(y,t),y), y=0, t=0..2, numpoints=200, color=colors[j]);
  od:
  plots:-display( [seq(plt[j], j=1..numelems(PrVals))]);

restart:

ODE:=[(a2+K)*(diff(U0(eta), eta, eta))/a1-Ra*(diff(U0(eta), eta))+lambda0/a1-a5*M1^2*U0(eta)/a1+K*(diff(N0(eta), eta))/a1+la*Ra*Theta0(eta)*(1+Qc*Theta0(eta)), (a2+K)*(diff(U1(eta), eta, eta))/a1-H^2*l1*U1(eta)-Ra*(diff(U1(eta), eta))+lambda1/a1-a5*M1^2*U1(eta)/a1+K*(diff(N1(eta), eta))/a1+la*Ra*(Theta1(eta))(1+2*Qc*Theta0(eta)), diff(N0(eta), eta, eta)-Ra*a1*Pj*(diff(N0(eta), eta))-2*n1*N0(eta)-n1*(diff(U0(eta), eta)), diff(N1(eta), eta, eta)-Ra*a1*Pj*(diff(N1(eta), eta))-2*n1*N1(eta)-n1*(diff(U1(eta), eta))-H^2*a1*Pj*l1*N1(eta), (a4/(a3*Pr)-delta*Ra^2/H^2+4*Rd*(1+(Tp-1)^3*Theta0(eta)^3+3*(Tp-1)^2*Theta0(eta)^2+(3*(Tp-1))*Theta0(eta))/(3*a3*Pr))*(diff(Theta0(eta), eta, eta))-Ra*(diff(Theta0(eta), eta))+a5*Ec*M1^2*U0(eta)^2/a3+(a2+K)*Ec*(diff(U0(eta), eta))^2/a1+Q*Theta0(eta)/a3+4*(diff(Theta0(eta), eta))^2*Rd*(3*(Tp-1)+6*(Tp-1)^2*Theta0(eta)+3*(Tp-1)^3*Theta0(eta)^2)/(3*a3*Pr), (a4/(a3*Pr)-delta*Ra^2/H^2+4*Rd*(1+(Tp-1)^3*Theta0(eta)^3+3*(Tp-1)^2*Theta0(eta)^2+(3*(Tp-1))*Theta0(eta))/(3*a3*Pr))*(diff(Theta1(eta), eta, eta))-(H^2*l1+2*Ra*delta*l1+Ra)*(diff(Theta1(eta), eta))+(Q/a3-delta*H^2*l1^2)*Theta1(eta)+2*(a2+K)*Ec*(diff(U0(eta), eta))*(diff(U1(eta), eta))/a1+2*a5*Ec*M^2*U0(eta)*U1(eta)/a3+4*(diff(Theta0(eta), eta, eta))*Theta1(eta)*Rd*(3*(Tp-1)+6*(Tp-1)^2*Theta0(eta)+3*(Tp-1)^3*Theta0(eta)^2)/(3*a3*Pr)+4*Rd*(diff(Theta0(eta), eta))^2*(6*(Tp-1)^2*Theta1(eta)+6*(Tp-1)^3*Theta0(eta)*Theta1(eta))/(3*a3*Pr)+4*Rd*(diff(Theta1(eta), eta))*(diff(Theta0(eta), eta))*(6*(Tp-1)+6*(Tp-1)^3*Theta0(eta)^2+12*(Tp-1)^2*Theta0(eta))/(3*a3*Pr)]:

(LB,UB):= (0,1):


BCs:= [
  
  U0(0) = 0, U1(0) = 0, N0(0) = 0, N1(0) = 0, Theta0(0) = 0, Theta1(0) = 0, U0(1) = 0, U1(1) = 0, N0(1) = 0, N1(1) = 0, Theta0(1) = 1, Theta1(1) = 0
]:

Params:= Record(
   
   M1=  1.2, Rd=0.8,la=0.8,n1=1.2,Q=0.2,Pj=0.001,Ra=0.8,Ec=1,    Pr= 21,   delta= 0.2,    t1= (1/4)*Pi, lambda0=2,lambda1=3,   Qc= 0.1,    l1= 1,K=0.4,H=3 ,deltat=0.05  ):
   

NBVs:= [   
 
a1**D(U0)(0) = `C*__f` , # Skin friction coefficient
 (a4+(4*Rd*(1/3))*(1+(Tp-1)*(Theta0(0)+0.1e-2*exp(l1*t1)*Theta1(0)))^3)*((D(Theta0))(0)+0.1e-2*exp(l1*t1)*(D(Theta1))(0)) = `Nu*`    # Nusselt number     
]:
Nu:= `Nu*`:
Cf:= `C*__f`:

 

Solve:= module()
local
   nbvs_rhs:= rhs~(:-NBVs), #just the names
   Sol, #numeric dsolve BVP solution of any 'output' form
   ModuleApply:= subs(
      _Sys= {:-ODEs[], :-BCs[], :-NBVs[]},
      proc({
          M1::realcons:=  Params:-M1,
         Pr::realcons:= Params:-Pr,
         Rd::realcons:= Params:-Rd,
         la::realcons:= Params:-la,
         Tp::realcons:= Params:-Tp,
         n1::realcons:= Params:-n1,
         Q::realcons:= Params:-Q,
         Pj::realcons:= Params:-Pj,
         Ra::realcons:= Params:-Ra,
         Ec::realcons:= Params:-Ec,
         t1::realcons:=  Params:-t1,
         delta::realcons:= Params:-delta,
         lambda0::realcons:= Params:-lambda0,
         lambda1::realcons:= Params:-lambda1,
         Qc::realcons:= Params:-Qc,
         K::realcons:= Params:-K,
         l1::realcons:= Params:-l1,
         H::realcons:= Params:-H
      })
         Sol:= dsolve(_Sys, _rest, numeric);
         AccumData(Sol, {_options});
         Sol
      end proc
   ),
   AccumData:= proc(
      Sol::{Matrix, procedure, list({name, function}= procedure)},
      params::set(name= realcons)
   )
   local n, nbvs;
      if Sol::Matrix then
         nbvs:= seq(n = Sol[2,1][1,Pos(n)], n= nbvs_rhs)
      else
         nbvs:= (nbvs_rhs =~ eval(nbvs_rhs, Sol(:-LB)))[]
      fi;
      SavedData[params]:= Record[packed](params[], nbvs)
   end proc,
   ModuleLoad:= eval(Init);
export
   SavedData, #table of Records
   Pos, #Matrix column indices of nbvs
   Init:= proc()
      Pos:= proc(n::name) option remember; local p; member(n, Sol[1,1], 'p'); p end proc;
      SavedData:= table();
      return
   end proc ;
   ModuleLoad()
end module:
 

#procedure that generates 3-D plots (dropped-shadow contour + surface) of an expression


ParamPlot3d:= proc(
   Z::{procedure, `module`}, #procedure that extracts z-value from Solve's dsolve solution
   X::name= range(realcons), #x-axis-parameter range
   Y::name= range(realcons), #y-axis-parameter range
   FP::list(name= realcons), #fixed values of other parameters
   {
      #fraction of empty space above and below plot (larger "below"
      #value improves view of dropped-shadow contourplot):
      zmargin::[realcons,realcons]:= [.05,0.15],
      eta::realcons:= :-LB, #independent variable value
      dsolveopts::list({name, name= anything}):= [],
      contouropts::list({name, name= anything}):= [],
      surfaceopts::list({name, name= anything}):=[]    
   }
)
local
   LX:= lhs(X), RX:= rhs(X), LY:= lhs(Y), RY:= rhs(Y),
   Zremember:= proc(x,y)
   option remember; #Used because 'grid' should be the same for both plots.
      Z(
         Solve(
            LX= x, LY= y, FP[],
            #Default dsolve options can be changed by setting 'dsolveopts':
            'abserr'= 0.5e-7, 'interpolant'= false, 'output'= Array([eta]),  
            dsolveopts[]
         )
      )
   end proc,
   plotspec:= (Zremember, RX, RY),
   C:= plots:-contourplot(
      plotspec,
      #These default plot options can be changed by setting 'contouropts':
      'grid'= [25,25], 'contours'= 5, 'filled',
      'coloring'= ['yellow', 'orange'], 'color'= 'green',
      contouropts[]
   ),
   P:= plot3d(
      plotspec,
      #These default plot options can be changed by setting 'surfaceopts':
      'grid'= [25,25], 'style'= 'surfacecontour', 'contours'= 6,
      surfaceopts[]
   ),
   U, L #z-axis endpoints after margin adjustment
;
   #Stretch z-axis to include margins:
   (U,L):= ((Um,Lm,M,m)-> (M*(Lm-1)+m*Um, M*Lm+m*(Um-1)) /~ (Um+Lm-1))(
      zmargin[],
      (max,min)(op(3, indets(P, 'specfunc'('GRID'))[])) #actual z-axis range
   );
   plots:-display(
      [
         plots:-spacecurve(
            {
               [[lhs(RX),rhs(RY),U],[rhs(RX),rhs(RY),U],[rhs(RX),rhs(RY),L]], #yz backwall
               [[rhs(RX),rhs(RY),U],[rhs(RX),lhs(RY),U],[rhs(RX),lhs(RY),L]]  #xz backwall
            },
            'color'= 'grey', 'thickness'= 0
         ),
         plottools:-transform((x,y)-> [x,y,L])(C), #dropped-shadow contours
         P
      ],
      #These default plot options can be changed simply by putting the option in the
      #ParamPlot3d call:
      'view'= ['DEFAULT', 'DEFAULT', L..U], 'orientation'= [-135, 75], 'axes'= 'frame',
      'labels'= [lhs(X), lhs(Y), Z], 'labelfont'= ['TIMES', 'BOLDOBLIQUE', 16],
      'caption'= nprintf(cat("%a = %4.2f, "$nops(FP)-1, "%a = %4.2f"), (lhs,rhs)~(FP)[]),
      'captionfont'= ['TIMES', 14],
      'projection'= 2/3,   
      _rest
   )
end proc:

ParamPlot3d(
   GetNu,Q= 0..5, Rd= 0..5, [
   
   Pr= 21   ],
   labels= [Q, gamma, Nu]
);

Error, (in plot/iplot2d/levelcurve) could not evaluate expression

Error, invalid input: eval received NULL, which is not valid for its 2nd argument, eqns

Error, invalid input: eval received NULL, which is not valid for its 2nd argument, eqns

Error, invalid input: eval received NULL, which is not valid for its 2nd argument, eqns

Error, invalid input: eval received NULL, which is not valid for its 2nd argument, eqns

Error, (in plot) unexpected options: [9.1682928*Y(Y)*b[3](Y)+3.0560976*b[2](Y)+2.433571428*Y(Y)^2*b[3](Y)+1.622380952*Y(Y)*b[2](Y)+.8111904760*b[1](Y)+1-1.346703274*b[3](Y)*Y(Y)^3-1.346703274*b[2](Y)*Y(Y)^2-1.346703274*b[1](Y)*Y(Y)-1.346703274*b[0](Y)-.5*(b[3](Y)*Y(Y)^3+b[2](Y)*Y(Y)^2+b[1](Y)*Y(Y)+b[0](Y))^2+.8111904760*c[3](Y)*Y(Y)^3+.8111904760*c[2](Y)*Y(Y)^2+.8111904760*c[1](Y)*Y(Y)+.8111904760*c[0](Y)+0.5e-1*d[2](Y)*Y(Y)^2+0.5e-1*d[1](Y)*Y(Y)+0.5e-1*d[0](Y) = 0, Y = 0 .. 1]

Error, (in plot) unexpected options: [.205*c[1](Y)*Y(Y)+.205*c[2](Y)*Y(Y)^2+.205*c[3](Y)*Y(Y)^3-.525*d[1](Y)*Y(Y)-.525*d[2](Y)*Y(Y)^2+.525*b[0](Y)+.525*b[1](Y)*Y(Y)+.525*b[2](Y)*Y(Y)^2+.525*b[3](Y)*Y(Y)^3+21.63764058*(b[3](Y)*Y(Y)^3+b[2](Y)*Y(Y)^2+b[1](Y)*Y(Y)+b[0](Y))^2-.105*h[0](Y)+6.799682664*Y(Y)*c[3](Y)+30.71903373*Y(Y)^2*c[3](Y)+20.47935582*Y(Y)*c[2](Y)-.105*Y(Y)^2*h[2](Y)-.105*Y(Y)*h[1](Y)-.525*d[0](Y)+.205*c[0](Y)+10.23967791*c[1](Y)+2.266560888*c[2](Y)+16.04451240*(3*Y(Y)^2*b[3](Y)+2*Y(Y)*b[2](Y)+b[1](Y))^2+5.25*(b[3](Y)*Y(Y)^3+b[2](Y)*Y(Y)^2+b[1](Y)*Y(Y)+b[0](Y))^3 = 0, Y = 0 .. 1]

Error, (in plot) unexpected options: [1.0*Y(Y)*d[2](Y)+.5*d[1](Y)+.5*b[3](Y)*Y(Y)^3+.5*b[2](Y)*Y(Y)^2-.5*d[2](Y)*Y(Y)^2+.5*b[1](Y)*Y(Y)-.5*d[1](Y)*Y(Y)+.5*b[0](Y)-.5*d[0](Y) = 0, Y = 0 .. 1]

Error, (in plot) unexpected options: [1.0*Y(Y)*h[2](Y)+.5*h[1](Y)+.5*c[3](Y)*Y(Y)^3+.5*c[2](Y)*Y(Y)^2-.5*Y(Y)^2*h[2](Y)+.5*c[1](Y)*Y(Y)-.5*Y(Y)*h[1](Y)+.5*c[0](Y)-.5*h[0](Y) = 0, Y = 0 .. 1]