1 years, 29 days

## Apply Keller Box method for ODE...

Maple 2018

Dear sir ,

I have implemented Dsolve method the code was executed, but i need to apply Kellor Box method to solve the ODES

Please can any one help how to implement?

because there is no post regarding the Kellor box method.

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 > with(plots):  cols := [red, blue, black,green]:  plotA:= display   ( [ seq       ( odeplot         ( Ans[k],[eta,(f(eta))],           eta=0..1,           color=cols[k]         ),         k=1..numelems(KpVals)       )     ],linestyle = "solid",     'axes'= 'boxed',labels=[eta,'f(eta)'],labelfont=[TIMES,BOLD,16]   );
 > with(plots):   cols := [red, blue, black,green]: plotB:= display( [ seq( odeplot         ( Ans[k],[eta,Theta(eta)],           eta=0..1,           color=cols[k]         ),         k=1..numelems(KpVals)       )     ],linestyle = "solid",     'axes'= 'boxed',labels=[eta,'Phi(eta)'],labelfont=[TIMES,BOLD,16]   );
 > with(plots):   cols := [red, blue, black,green]: plotC:= display( [ seq( odeplot         ( Ans[k],[eta,Phi(eta)],           eta=0..1,           color=cols[k]         ),         k=1..numelems(KpVals)       )     ],linestyle = "solid",     'axes'= 'boxed',labels=[eta,'Phi(eta)'],labelfont=[TIMES,BOLD,16]   );
 > with(plots):  cols := [red, blue, black,green]:  plotA:= display   ( [ seq       ( odeplot         ( Ans[k],[eta,(diff(f(eta),eta))],           eta=0..1,           color=cols[k]         ),         k=1..numelems(KpVals)       )     ],linestyle = "solid",     'axes'= 'boxed',labels=[eta,"f '(eta)"],labelfont=[TIMES,BOLD,16]   );
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## plot error in 3d plots ...

Maple 2018

Dear sir, there is something missing why it is not able to evaluate?

By reference of some posts I have implemented to my ODE but not getting the graph.

what is the mistake in both files?

Maple 2018
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## first and second order derivative plot ...

Maple 2018

How to plot the second order derivative and first oder derivatives plot in time dependent pde and vector plot of  theta(y,t), u(y,t) at y=0..10 and t=0..1

nowhere i found a vector plot of time-dependent pde

how to plot give me suggestions.

in vector plots, flow patterns should show with arrow marks

 > 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          }
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 > 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))]);
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## unable to plot 3d and 2d plots...

Maple 2018

I am unable to draw both 3d plots sowing error please help me to solve

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