tomleslie

7060 Reputation

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10 years, 284 days

MaplePrimes Activity


These are answers submitted by tomleslie

over 2D-input and 2D-math formats

Is the attached what you expected (For some reason this wont display inline  maybe the animations make the file size too big?)

anim.mw

@AHSAN 

the attached, maybe?

restart

A := beta*(-60*lambda*(1+(1/2)*x^2)^4+(-63*k^3+99*k^2-99*k+63)*(1+(1/2)*x^2)^3-288*Q*(k^2-(11/8)*k+1)*(1+(1/2)*x^2)^2-486*Q^2*(k-1)*(1+(1/2)*x^2)-324*Q^3)/(20*(1+(1/2)*x^2)^7)-(3*(k*(1+(1/2)*x^2)+2*Q-1-(1/2)*x^2))/(2*(1+(1/2)*x^2)^3)

opts := [[beta = .1, Q = .5516, lambda = -0.396e-1, k = .1], [beta = .2, Q = .4903, lambda = -0.411e-1, k = .2], [beta = .3, Q = .4290, lambda = -0.419e-1, k = .3], [beta = .4, Q = .3677, lambda = -0.417e-1, k = .4]]

yVals := [-.1, -.12, -.14, -.16]; N := plot([seq(eval(A, opts[j]), j = 1 .. 4)], x = -2 .. 2, color = [black, red, blue, green], thickness = 2); B := plot(yVals, x = -1 .. -.7, color = [black, red, blue, green], thickness = 3); T := plots:-textplot([seq([-1.1, yVals[j], typeset(k = rhs(opts[j, 3]), ", ", Q = rhs(opts[j, 2]))], j = 1 .. 4)], align = left)

plots:-display(N, B, T, size = [900, 700], gridlines = false)

 

 


 

Download legend2.mw

@user-JG 

as in

ifactor(725);
ifactor(1125);
ifactor(2048);

``(5)^2*``(29)

 

``(3)^2*``(5)^3

 

``(2)^11

(1)

 


 

Download fac.mw

the randomize() command with no arguments. This resets the "seed" for Maple's random number generators, based on the system clock, so the seed will be different each time your worksheet is executed.

Without using "randomize()", you will get the same random numbers every time you execute a worksheet as you have observed. This behaviour is by design (and very useful for "debug" purposes!)

which will return the "next" line as a string in the input file.

What you want to do with this string is up to you - parse() maybe?

Since you now have a string up to a linebreak, how you "signify" the linebreaks is also up to you

Solution to your problem depends very much on how you want "linebreaks" to be signified in the output - eg a new row in a matrix?, a new entry in a table? whatever? Ypu are going to have to get a lot more specific on

  • what to do with each line
  • what to do with the linebreak

working out your problem!

Is this a 1D math input versus 2D math input issue?

See the attached for a couple of solutions in both input formats - which of these is difficult to understand?

I have taken into account the remark by CarlLove that the VectorField() command is a bit "redundant" - so, in the attached, there are execution groups which use it, and execution groups which don't - doesn't affect the answer.

Which of these don't you understand?

restart; with(Student:-VectorCalculus)

g := x^2+y^2+z^2
NULL

x^2+y^2+z^2

(1)

G2D := unapply('evalVF(VectorField(Gradient(z)), p)', [z, p]); G2D(g, `<,>`(1, 1, 1))

G2D := proc (z, p) options operator, arrow; evalVF(VectorField(Gradient(z)), p) end proc

 

Vector[column](%id = 18446744074370475182)

(2)

G2D := unapply('evalVF(Gradient(z), p)', [z, p]); G2D(g, `<,>`(1, 1, 1))

G2D := proc (z, p) options operator, arrow; evalVF(Gradient(z), p) end proc

 

Vector[column](%id = 18446744074370472406)

(3)

  G1D:= unapply
        ( 'evalVF
           ( VectorField
             (  Gradient(z)
             ),
             p
           )',
           [z,p]
         );
  G1D( g, <1,1,1> );

G1D := proc (z, p) options operator, arrow; evalVF(VectorField(Gradient(z)), p) end proc

 

Vector[column](%id = 18446744074370467230)

(4)

  G1D:= unapply
        ( 'evalVF
          (  Gradient(z),
             p
          )',
          [z,p]
        );
  G1D( g, <1,1,1> );

G1D := proc (z, p) options operator, arrow; evalVF(Gradient(z), p) end proc

 

Vector[column](%id = 18446744074370461814)

(5)

 

NULL

Download SVC2.mw

  1. You had a couple of typos where you had written, for example,  diff*(v(z),z), rather than diff(v(z),z)
  2. Why are you using pdsolve() to solve a system of ODEs??? Changed this to dsolve().

Maple now reports no errors, although it does not povide a solution. You may have to consider generating a 'numerical' solutiion.

See the attache
 

restart

NULL

D21 := 3; D31 := 14; D41 := 12; D51 := 23; D61 := 12; D71 := 14; D81 := 14

D32 := 5; D42 := 12; D52 := 23; D62 := 123; D72 := 16; D82 := 24

D43 := 8; D53 := 15; D63 := 34; D73 := 18; D83 := 79

D54 := 9; D64 := 81; D74 := 20; D84 := 17

D65 := 10; D75 := 2; D85 := 48; D86 := 6; D87 := 13; D88 := 19

D11 := 1; D22 := 2; D33 := 4; D44 := 5; D55 := 7; D66 := 10; D77 := 21

D76 := 14

D12 := D21; D13 := D31; D14 := D41; D15 := D51; D16 := D61; D17 := D71; D18 := D81

D23 := D32; D24 := D42; D25 := D52; D26 := D62; D27 := D72; D28 := D82

D34 := D43; D35 := D53; D36 := D63; D37 := D73; D38 := D83

D45 := D54; D46 := D64; D47 := D74; D48 := D84

D56 := D65; D57 := D75; D58 := D85

D67 := D76; D68 := D86

D78 := D87

xs := 0; ys := 0

NULL

e1 := diff(w(z), z)+1/2*((diff(u(z), z))^2+(diff(v(z), z))^2)+(diff(f(z), z))*(ys*((diff(u(z), z))*cos(f(z))+(diff(v(z), z))*sin(f(z)))+xs*((diff(u(z), z))*sin(f(z))-(diff(v(z), z))*cos(f(z))))

diff(w(z), z)+(1/2)*(diff(u(z), z))^2+(1/2)*(diff(v(z), z))^2

(1)

NULL

e2 := diff(tx(z), z)-(diff(f(z), z))*((diff(u(z), z))*cos(f(z))+(diff(v(z), z))*sin(f(z)))

diff(tx(z), z)-(diff(f(z), z))*((diff(u(z), z))*cos(f(z))+(diff(v(z), z))*sin(f(z)))

(2)

NULL

e3 := diff(ty(z), z)+(diff(f(z), z))*((diff(u(z), z))*sin(f(z))-(diff(v(z), z))*cos(f(z)))

diff(ty(z), z)+(diff(f(z), z))*((diff(u(z), z))*sin(f(z))-(diff(v(z), z))*cos(f(z)))

(3)

NULL

e4 := dr*(diff(f(z), z, z))

dr*(diff(diff(f(z), z), z))

(4)

NULL

e7 := -ty(z)+(diff(v(z), z))*sin(f(z))+(diff(u(z), z))*cos(f(z))

-ty(z)+(diff(v(z), z))*sin(f(z))+(diff(u(z), z))*cos(f(z))

(5)

NULL

e6 := tx(z)-(diff(u(z), z))*sin(f(z))+(diff(v(z), z))*cos(f(x))

tx(z)-(diff(u(z), z))*sin(f(z))+(diff(v(z), z))*cos(f(x))

(6)

NULL

e8 := diff(f(z), z)

diff(f(z), z)

(7)

NULL

e5 := (1/2)*(diff(f(z), z))^2

(1/2)*(diff(f(z), z))^2

(8)

``

DD := matrix([[D11, D12, D13, D14, D15, D16, D17, D18], [D21, D22, D23, D24, D25, D26, D27, D28], [D31, D32, D33, D34, D35, D36, D37, D38], [D41, D42, D43, D44, D45, D46, D47, D48], [D51, D52, D53, D54, D55, D56, D57, D58], [D61, D62, D63, D64, D65, D66, D67, D68], [D71, D72, D73, D74, D75, D76, D77, D78], [D81, D82, D83, D84, D85, D86, D87, D88]])

array( 1 .. 8, 1 .. 8, [( 3, 6 ) = (34), ( 2, 8 ) = (24), ( 7, 5 ) = (2), ( 4, 4 ) = (5), ( 6, 7 ) = (14), ( 6, 6 ) = (10), ( 5, 8 ) = (48), ( 8, 3 ) = (79), ( 7, 4 ) = (20), ( 6, 1 ) = (12), ( 8, 8 ) = (19), ( 1, 5 ) = (23), ( 3, 4 ) = (8), ( 1, 2 ) = (3), ( 6, 5 ) = (10), ( 1, 3 ) = (14), ( 5, 7 ) = (2), ( 7, 6 ) = (14), ( 6, 8 ) = (6), ( 1, 4 ) = (12), ( 3, 5 ) = (15), ( 2, 7 ) = (16), ( 5, 1 ) = (23), ( 8, 2 ) = (24), ( 1, 1 ) = (1), ( 1, 6 ) = (12), ( 3, 2 ) = (5), ( 5, 3 ) = (15), ( 2, 1 ) = (3), ( 6, 4 ) = (81), ( 8, 7 ) = (13), ( 4, 6 ) = (81), ( 7, 1 ) = (14), ( 2, 3 ) = (5), ( 2, 5 ) = (23), ( 1, 7 ) = (14), ( 4, 5 ) = (9), ( 3, 3 ) = (4), ( 4, 1 ) = (12), ( 7, 7 ) = (21), ( 5, 6 ) = (10), ( 7, 2 ) = (16), ( 2, 4 ) = (12), ( 4, 7 ) = (20), ( 8, 6 ) = (6), ( 2, 2 ) = (2), ( 3, 8 ) = (79), ( 6, 3 ) = (34), ( 6, 2 ) = (123), ( 5, 4 ) = (9), ( 8, 1 ) = (14), ( 7, 3 ) = (18), ( 8, 4 ) = (17), ( 2, 6 ) = (123), ( 1, 8 ) = (14), ( 4, 3 ) = (8), ( 4, 2 ) = (12), ( 8, 5 ) = (48), ( 3, 7 ) = (18), ( 4, 8 ) = (17), ( 3, 1 ) = (14), ( 5, 5 ) = (7), ( 5, 2 ) = (23), ( 7, 8 ) = (13)  ] )

(9)

F := DD.e

e := Vector([e1, e2, e3, e4, e5, e6, e7, e8])

Vector[column](%id = 18446744074565103238)

(10)

N := F[1]; Mx := F[2]; My := F[3]; Mw := F[4]; gT := F[5]; Vy := F[6]; Vx := F[7]; Ts := F[8]

NULL

diff(N*(diff(u(z), z)), z)

eq1 := diff(N*(`*`(diff(u(z), z)+(diff(f(z), z))*(ys*cos(f(z))+xs*sin(f(z))))), z)+diff((Vx-Mx*(diff(f(z), z)))*cos(f(z))+(My*(diff(f(z), z))-Vy)*sin(f(z)), z) = 0

eq2 := diff(N*(`*`(diff(v(z), z)+(diff(f(z), z))*(ys*sin(f(z))-xs*sin(f(z))))), z)+diff((Vx-Mx*(diff(f(z), z)))*sin(f(z))+(Vy-My*(diff(f(z), z)))*cos(f(z)), z) = 0

eq3 := diff(N, z) = 0

eq4 := diff(Mx, z)-Vy = 0

eq5 := diff(My, z)+Vx = 0

eq6 := diff(N*(cos(f(z))*((diff(u(z), z))*ys-(diff(v(z), z))*xs)+sin(f(z))*((diff(v(z), z))*ys+(diff(u(z), z))*xs)), z)-N*(diff(f(z), z))*(cos(f(z))*((diff(v(z), z))*ys+(diff(u(z), z))*xs)+sin(f(z))*((diff(v(z), z))*xs-(diff(u(z), z))*ys))-(diff(cos(f(z))*(Mx*(diff(u(z), z))+My*(diff(v(z), z)))+sin(f(z))*(Mx*(diff(v(z), z))-My*(diff(u(z), z))), z))+cos(f(z))*(Mx*(diff(v(z), z))*(diff(f(z), z))-My*(diff(u(z), z))*(diff(f(z), z))+Vy*(diff(u(z), z))-Vx*(diff(v(z), z)))+sin(f(z))*(-Mx*(diff(u(z), z))*(diff(f(z), z))-My*(diff(v(z), z))*(diff(f(z), z))+Vy*(diff(v(z), z))+Vx*(diff(u(z), z)))-dr*(diff(Mw, z, z))+diff(gT*(diff(f(z), z)), z)+diff(Ts, z) = 0

NULL

NULL

NULL

NULL

NULL

NULL

ode := eq1, eq2, eq3, eq4, eq5, eq6

 

sol := dsolve([ode], {f(z), tx(z), ty(z), u(z), v(z), w(z)})

(11)

sol := dsolve({eq1, eq2, eq3, eq4, eq5, eq6, f(0) = f1, f(L) = f2, tx(0) = tx1, tx(L) = tx2, ty(0) = ty1, ty(L) = ty2, u(0) = u1, u(L) = u2, v(0) = v1, v(L) = v2, w(0) = w1, w(L) = w2}, {f(z), tx(z), ty(z), u(z), v(z), w(z)})

(12)

``

NULL


 

Download diffProb.mw

d

Just cheat using the 'view' option, whihc I have (sort of) generalized in the attached

  restart;
  xlim:=10:
  ylim:=10:
  expr:=4*x^2+9*y^2:
  plot3d( expr,
          x=-xlim..xlim,
          y=-ylim..ylim,
          view=[ -xlim..xlim,
                 -ylim..ylim,
                  0..min( [ eval(expr,[x=0, y=ylim]),
                            eval(expr,[y=0, x=xlim])
                          ]
                        )
               ],
         style=surface,
         colorscheme=["zcoloring", z->z]
       );

 

 


 

Download paraPlot.mw

 

 

The a, b, and c values in the above coordinate transformations can be given using the coordinate specification as a function, e.g., conical(a,b) or ellcylindrical(2). The values a, b, and c if necessary, should be specified.  If not specified, the default values used are a = 1, b = 1/2, and c = 1/3.

 

The toolbar on a Maple help page has navigation icons at the top left. The second thrid and fourth buttons (from the left) will allow you to navigate backwards, forwards, or upwards to the parent topic.

Are you suggesting that these buttons don't work?

Also not clear what your problem is in creating a "function". See the attached for a "toy" example

  restart;
  alias(SVC=Student[VectorCalculus]):
#
# Define a function which takes an expression and a point.
# It creates a vector field from the gradient of the
# supplied expression and then evaluates the vector field
# at the supplied point
#
  f:= (z,p)-> SVC:-evalVF
              ( SVC:-VectorField
                (  SVC:-Gradient(z)
                ),
                p
              );

 f( exp(x*y^3*z^2), <1,1,1> );

f := proc (z, p) options operator, arrow; SVC:-evalVF(SVC:-VectorField(SVC:-Gradient(z)), p) end proc

 

Vector[column](%id = 18446744074397177606)

(1)

 

 


 

Download SVC.mw

 

Your ODE system is generating complex numbers - is this expected behaviour?

The Maple help states

All IVP methods can be used for complex-valued IVPs with a real-valued independent variable (though for the default stiff and nonstiff IVP methods, it may be necessary to specify that the problem is complex via the complex option). None of the BVP or DAE methods can currently be used for complex-valued problems, requiring the system be converted to a real system before calling dsolve.

 

use of the rand() function

See the attached - although the various dependent functions don't seem to be doing anything interesting (at least at t=0)

restart;
mu := 0;
beta := 0.1;
alpha := 1;
Gamma1 := 0.1;
with(plots):
pde1 := {diff(u(x, t), t) = -Gamma1*diff(u(x, t), x) + mu*(w(x, t) - u(x, t)) - 0.5*beta*v(x, t)*(w(x, t) - u(x, t)) - beta*u(x, t)*v(x, t), diff(v(x, t), t) = beta*v(x, t)*(w(x, t) + u(x, t)) - alpha*v(x, t), diff(w(x, t), t) = Gamma1*diff(w(x, t), x) + mu(u(x, t) - w(x, t)) - 0.5*beta*v(x, t)*(u(x, t) - w(x, t)) - beta*w(x, t)*v(x, t)};


rr:=rand(0..10):
IBC1 := {u(0, t) = 0, u(x, 0) = 0.5 + rr(), v(x, 0) = 0, w(1, t) = 0, w(x, 0) = 0.5 +rr()};

pds1 := pdsolve(pde1, IBC1, numeric, spacestep = 0.001);

p1u := pds1:-plot(u, t = 0, numpoints = 100, legend = ["u(0)"], linestyle = 1):
p1w := pds1:-plot(w, t = 0, color = blue, numpoints = 100, legend = ["w(0)"], linestyle = 1):
p1v := pds1:-plot(v, t = 0, color = green, numpoints = 100, legend = ["v(0)"], linestyle = 1):

plots[display]({p1u, p1v, p1w}, axes = boxed);

0

 

.1

 

1

 

.1

 

{diff(u(x, t), t) = -.1*(diff(u(x, t), x))-0.5e-1*v(x, t)*(w(x, t)-u(x, t))-.1*u(x, t)*v(x, t), diff(v(x, t), t) = .1*v(x, t)*(w(x, t)+u(x, t))-v(x, t), diff(w(x, t), t) = .1*(diff(w(x, t), x))-0.5e-1*v(x, t)*(u(x, t)-w(x, t))-.1*w(x, t)*v(x, t)}

 

{u(0, t) = 0, u(x, 0) = 6.5, v(x, 0) = 0, w(1, t) = 0, w(x, 0) = 9.5}

 

_m699568992

 

 

 


 

Download pdeSol2.mw

but you need a **lot** of frames to see the point move smoothly along the spacecurve. In the attached I have used 500 frames, and changed a couple of other "cosmetic" things to make the visualisation better(?)

I have no idea why yiu are getting the message

And "Error, (in plots/pointplot3d) incorrect first argument" BUT if i use classic functions (sin,cos,sqrt) instead of b,c,d - all is well.

You wil have to post the code whihc produced it - upload the worksheet using the bi green up-arrow in the Mapleprimes toolbar.

I would normally upload an inline worksheet so that you can see this working - but this site is giving me a weird security-related messa ge which I don't understand! So just code below
 

 restart:
 with(plots):
 a:=0.2: b:=0.2: c:=5.7:
 ode1:=diff(x(t),t)=-y(t)-z(t):
 ode2:=diff(y(t),t)=x(t)+a*y(t):
 ode3:=diff(z(t),t)=b+z(t)*(x(t)-c):
 ic1:=x(0)=-0.7:
 ic2:=y(0)=-0.7:
 ic3:=z(0)=1:

 sol:=dsolve( {ode1,ode2,ode3,ic1,ic2,ic3},
              {x(t),y(t),z(t)},
              numeric,
              method=classical[rk4],
              stepsize=0.10,
              output=listprocedure
           ):

 b:=rhs(sol[2]):
 c:=rhs(sol[3]):
 d:=rhs(sol[4]):

 animate( pointplot3d,
          [ [b(t),c(t),d(t)],
            symbol=solidsphere,
            symbolsize=20,
            color=blue
          ],
          t=15..100,
          background = odeplot( sol,
                                [x(t),y(t),z(t)],
                                t=15..100,
                                color=red,
                                numpoints=7000
                              ),
          frames=500
        );

 

 

but maybe it is addressed in the attached


 

  restart;
#
# Check the exports of the 'plots()' package
#
  exports(plots);
#
# Hmmmm, 'display3d' exists What does it do?
#
  showstat(plots:-display3d);
#
# Just calls plots/display(), with identical arguments
# so it is either an 'alias' or (maybe) a deprecated
# command
#

animate, animatecurve, animate3d, arrow, changecoords, complexplot, complexplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, densityplot, display, dualaxisplot, fieldplot, fieldplot3d, gradplot, gradplot3d, implicitplot, implicitplot3d, inequal, interactive, interactiveparams, intersectplot, listcontplot, listcontplot3d, listdensityplot, listplot, listplot3d, loglogplot, logplot, matrixplot, multiple, odeplot, pareto, plotcompare, pointplot, pointplot3d, polarplot, polygonplot, polygonplot3d, polyhedraplot, polyhedra_supported, rootlocus, semilogplot, setcolors, setoptions, setoptions3d, shadebetween, spacecurve, sparsematrixplot, surfdata, textplot, textplot3d, tubeplot, cylinderplot, graphplot3d, replot, sphereplot, Interactive, display3d, setcolours, dualaxesplot, init, _pexports

 


plots:-display3d := proc()
   1   `plots/display`(_passed)
end proc
 

 

#
# Rewriting diskmethod so that it "works"
#
  diskmethod:= proc(f,a,b,N)
                    local X,Y,s,x,dX,i,disk;
                  #
                  # Ensure all commands from the 'plots' package
                  # are available
                  #
                    uses plots:
                  #
                  # Should be using 'Array()' rather than the
                  # deprecated 'array()'
                  #
                    X:= Array(0..N);
                    Y:= Array(1..N);
                    disk:= Array(1..N);
                    dX:= evalf((b-a)/N);
                    X[0]:= a;
                    s:= 0;
                    for i to N do
                        X[i]:= a+i*dX;
                        Y[i]:= evalf(f(X[i]));
                        s:= s+evalf(Pi*Y[i]^2*dX);
                        disk[i]:= plot3d( [ x,
                                            Y[i]*cos(theta),
                                            Y[i]*sin(theta)
                                          ],
                                          x=X[i-1]..X[i],
                                          theta=0..2*Pi,
                                          grid=[2,51]
                                        ):
                    end do;
                    print(`The approximate volume using`,N,`disks is V =`, s);
                  #
                  # keep the 'display3d()' command although the
                  # 'display()' command would work equally well
                  #
                    return  display3d( {seq(disk[i],i=1..N)},
                                       axes=box,
                                       projection=0.8,
                                       orientation=[-60,70]
                                     );

               end proc:
 

#
# Apply the diskmethod() procedure to an arbitrarily
# chosen function
#
   diskmethod(z->exp(z)*sin(z),0,2,20);

`The approximate volume using`, 20, `disks is V =`, 79.94886772

 

 

 


 

Download upDaye.mw

I have no ide a ow you plan to get a 3D plot from a system which contains only one independent variable. A 3D plot kind of implies that you have two independent variables.

The attached shows the solution of the system you have supplied.

You will note that I have deleted the equations chi(0) = 1, chi(10) = 0  from your bcs/ics since the dependent variable chi() occurs nowhere in your system, so why would bcs involving chi() be of any interest whatsoever?

restart:

ODES := diff(f(eta), eta$4)+(2*f(eta)*diff(f(eta), eta$3)+2*g(eta)*diff(g(eta), eta))*(1-phi)^2.5*(1-phi+phi*rhos/rhof)-sigmanf*M*(diff(f(eta), eta$2))*(1-phi)^2.5/sigmaf = 0,
        diff(g(eta), eta$2)-(1-phi)^2.5*(1-phi+phi*rhos/rhof)*(2*(diff(f(eta), eta))*g(eta)-2*(diff(g(eta), eta))*f(eta))-sigmanf*M*g(eta)*(1-phi)^2.5/sigmaf = 0,
         k[nf]*diff(theta(eta), eta$2)/(Pr*k[f])+((1-phi+phi*rhos*cps/(rhof*cpf))*2)*f(eta)*(diff(theta(eta), eta))-4*lambda*(1-phi+phi*rhos*cps/(rhof*cpf))*(f(eta)^2*diff(theta(eta), eta$2)+f(eta)*diff(f(eta), eta)*diff(theta(eta), eta))+sigmanf*M*Ec*(diff(f(eta), eta)^2+g(eta)^2)/sigmaf = 0;

diff(diff(diff(diff(f(eta), eta), eta), eta), eta)+(2*f(eta)*(diff(diff(diff(f(eta), eta), eta), eta))+2*g(eta)*(diff(g(eta), eta)))*(1-phi)^2.5*(1-phi+phi*rhos/rhof)-sigmanf*M*(diff(diff(f(eta), eta), eta))*(1-phi)^2.5/sigmaf = 0, diff(diff(g(eta), eta), eta)-(1-phi)^2.5*(1-phi+phi*rhos/rhof)*(2*(diff(f(eta), eta))*g(eta)-2*(diff(g(eta), eta))*f(eta))-sigmanf*M*g(eta)*(1-phi)^2.5/sigmaf = 0, k[nf]*(diff(diff(theta(eta), eta), eta))/(Pr*k[f])+2*(1-phi+phi*rhos*cps/(rhof*cpf))*f(eta)*(diff(theta(eta), eta))-4*lambda*(1-phi+phi*rhos*cps/(rhof*cpf))*(f(eta)^2*(diff(diff(theta(eta), eta), eta))+f(eta)*(diff(f(eta), eta))*(diff(theta(eta), eta)))+sigmanf*M*Ec*((diff(f(eta), eta))^2+g(eta)^2)/sigmaf = 0

(1)

bcs:= f(0) = 0,
      D(f)(0) = A1+gamma1*((D@@2)(f))(0),
      f(10) = 0,
      D(f)(10) = 0,
      g(0) = 1+gamma2*(D(g))(0),
      g(10) = 0,
      theta(0) = 1+gamma3*(D(theta))(0),
      theta(10) = 0;

f(0) = 0, (D(f))(0) = A1+gamma1*((D@@2)(f))(0), f(10) = 0, (D(f))(10) = 0, g(0) = 1+gamma2*(D(g))(0), g(10) = 0, theta(0) = 1+gamma3*(D(theta))(0), theta(10) = 0

(2)

params:=[ lambda = 0.1e-1, sigma = .1, Ec = .2, E = .1, M = 5,
          delta = .1, n = .1, Sc = 3, A1 = .5, gamma1 = .5,
          gamma2 = .5, gamma3 = .5, Pr = 6.2, phi = 0.1e-1,
          rhos = 5.06*10^3, rhof = 997, cps = 397.21, cpf = 4179,
          k[nf] = .6358521729, k[f] = .613, sigmanf = 0.5654049962e-5,
          sigmaf = 5.5*10^(-6)
        ];

[lambda = 0.1e-1, sigma = .1, Ec = .2, E = .1, M = 5, delta = .1, n = .1, Sc = 3, A1 = .5, gamma1 = .5, gamma2 = .5, gamma3 = .5, Pr = 6.2, phi = 0.1e-1, rhos = 5060.00, rhof = 997, cps = 397.21, cpf = 4179, k[nf] = .6358521729, k[f] = .613, sigmanf = 0.5654049962e-5, sigmaf = 0.5500000000e-5]

(3)

sol:=dsolve( eval([ODES, bcs], params), numeric);

proc (x_bvp) local res, data, solnproc, _ndsol, outpoint, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then outpoint := evalf[_EnvDSNumericSaveDigits](x_bvp) else outpoint := evalf(x_bvp) end if; data := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(30, {(1) = .0, (2) = .3115883682169779, (3) = .6252703963731303, (4) = .9433773563232228, (5) = 1.2680729118280434, (6) = 1.600281886516404, (7) = 1.9393162978590073, (8) = 2.2835156081889263, (9) = 2.6311011542021747, (10) = 2.9806940207148984, (11) = 3.331447607599669, (12) = 3.6829002252546412, (13) = 4.034807188531737, (14) = 4.386996935591378, (15) = 4.73932872237401, (16) = 5.0917150564146505, (17) = 5.444119100021418, (18) = 5.796534378827212, (19) = 6.14896147215437, (20) = 6.501398314085222, (21) = 6.8538379593121155, (22) = 7.206269623503382, (23) = 7.558681172777766, (24) = 7.911069970557719, (25) = 8.263448176851682, (26) = 8.615834971676511, (27) = 8.968244186479987, (28) = 9.320681820649735, (29) = 9.672590968929137, (30) = 10.0}, datatype = float[8], order = C_order); Y := Matrix(30, 8, {(1, 1) = .0, (1, 2) = .23340239204225743, (1, 3) = -.5331952159154854, (1, 4) = 1.0666085359590798, (1, 5) = .46065338069468376, (1, 6) = -1.078693238610632, (1, 7) = .8364902331198539, (1, 8) = -.3270195337602919, (2, 1) = 0.5158502793783559e-1, (2, 2) = .1109879856055622, (2, 3) = -.27668306416758, (2, 4) = .6044763196631948, (2, 5) = .22213904293970846, (2, 6) = -.5196819027931153, (2, 7) = .6885240156543468, (2, 8) = -.5415121349318274, (3, 1) = 0.7545420180139172e-1, (3, 2) = 0.4843316407335183e-1, (3, 3) = -.13743108602223522, (3, 4) = .31140452186526585, (3, 5) = .10669558168830623, (3, 6) = -.24924695608308325, (3, 7) = .5255895465339944, (3, 8) = -.47755030372360524, (4, 1) = 0.8532150106540043e-1, (4, 2) = 0.1734020508150595e-1, (4, 3) = -0.664061716186653e-1, (4, 4) = .15286409656975167, (4, 5) = 0.5077745895666975e-1, (4, 6) = -.11843000308023045, (4, 7) = .3919781336678781, (4, 8) = -.3623991577992995, (5, 1) = 0.8818327733552848e-1, (5, 2) = 0.21669075489643436e-2, (5, 3) = -0.3135964126186165e-1, (5, 4) = 0.7266056738654511e-1, (5, 5) = 0.2382609523268939e-1, (5, 6) = -0.5547821753784222e-1, (5, 7) = .2914824056491054, (5, 8) = -.2610029523584158, (6, 1) = 0.8754308880649408e-1, (6, 2) = -0.50970399710089966e-2, (6, 3) = -0.14509049773109677e-1, (6, 4) = 0.33689834209551746e-1, (6, 5) = 0.1100010120847999e-1, (6, 6) = -0.25569638336403393e-1, (6, 7) = .2181802258302483, (6, 8) = -.1846813863050687, (7, 1) = 0.8516290087644296e-1, (7, 2) = -0.850170573923812e-2, (7, 3) = -0.6602354622485683e-2, (7, 4) = 0.15331378501474606e-1, (7, 5) = 0.5005359954317998e-2, (7, 6) = -0.11614621967582133e-1, (7, 7) = .16533613982623024, (7, 8) = -.13043730206507836, (8, 1) = 0.8193181516914232e-1, (8, 2) = -0.10066600435936227e-1, (8, 3) = -0.2969795158033158e-2, (8, 4) = 0.6889665326704975e-2, (8, 5) = 0.22535893940688217e-2, (8, 6) = -0.5220047360349346e-2, (8, 7) = .12733587324756138, (8, 8) = -0.9276269749510344e-1, (9, 1) = 0.7829328021105589e-1, (9, 2) = -0.10775285776671337e-1, (9, 3) = -0.13266725973456585e-2, (9, 4) = 0.30734234225339046e-2, (9, 5) = 0.10081565475017406e-2, (9, 6) = -0.2331040156855463e-2, (9, 7) = 0.9989421892581841e-1, (9, 8) = -0.6677313096888332e-1, (10, 1) = 0.7446332748908925e-1, (10, 2) = -0.11093223857671173e-1, (10, 3) = -0.5906331487901736e-3, (10, 4) = 0.13660736436957612e-2, (10, 5) = 0.44956688186038114e-3, (10, 6) = -0.10376087463598667e-2, (10, 7) = 0.7988529624275498e-1, (10, 8) = -0.4880242646206497e-1, (11, 1) = 0.705441211168358e-1, (11, 2) = -0.11235156259279292e-1, (11, 3) = -0.26259587767334694e-3, (11, 4) = 0.6063455840099236e-3, (11, 5) = 0.2002307022510662e-3, (11, 6) = -0.4613014836786184e-3, (11, 7) = 0.6509452375662474e-1, (11, 8) = -0.362821006967305e-1, (12, 1) = 0.6658290111930798e-1, (12, 2) = -0.11298377411755068e-1, (12, 3) = -0.11671367942552407e-3, (12, 4) = 0.26909965532350734e-3, (12, 5) = 0.891668364505363e-4, (12, 6) = -0.2050552019113641e-3, (12, 7) = 0.5398097032001109e-1, (12, 8) = -0.27470295223740986e-1, (13, 1) = 0.6260131094583453e-1, (13, 2) = -0.11326512778841079e-1, (13, 3) = -0.5186493634598218e-4, (13, 4) = 0.11953975242454077e-3, (13, 5) = 0.3972454953620348e-4, (13, 6) = -0.9118819225490443e-4, (13, 7) = 0.4547890536528926e-1, (13, 8) = -0.2119966749093241e-1, (14, 1) = 0.5860973189596408e-1, (14, 2) = -0.11339018833798774e-1, (14, 3) = -0.2300049509428586e-4, (14, 4) = 0.53287832190102304e-4, (14, 5) = 0.17712081150251953e-4, (14, 6) = -0.4058455079812291e-4, (14, 7) = 0.38849887604394395e-1, (14, 8) = -0.16688439452594375e-1, (15, 1) = 0.54613529121855776e-1, (15, 2) = -0.11344545926454884e-1, (15, 3) = -0.1006977160373165e-4, (15, 4) = 0.24090702642066673e-4, (15, 5) = 0.7906338166119685e-5, (15, 6) = -0.1808330127261758e-4, (15, 7) = 0.33577848499792035e-1, (15, 8) = -0.13409641090172729e-1, (16, 1) = 0.5061538759703425e-1, (16, 2) = -0.11346915687156164e-1, (16, 3) = -0.4107238785469627e-5, (16, 4) = 0.11572563554242374e-4, (16, 5) = 0.3533968879613638e-5, (16, 6) = -0.8068189730222743e-5, (16, 7) = 0.29297770506115987e-1, (16, 8) = -0.11004706907351869e-1, (17, 1) = 0.4661650647834903e-1, (17, 2) = -0.11347768954534387e-1, (17, 3) = -0.1001614906174004e-5, (17, 4) = 0.6988010971430737e-5, (17, 5) = 0.15818568420031865e-5, (17, 6) = -0.36048864416160054e-5, (17, 7) = 0.25748420492069875e-1, (17, 8) = -0.9227474850541309e-2, (18, 1) = 0.4261736600106166e-1, (18, 2) = -0.11347707465285151e-1, (18, 3) = 0.13562072356889462e-5, (18, 4) = 0.7082712647990626e-5, (18, 5) = 0.7090780445924124e-6, (18, 6) = -0.16129820537020167e-5, (18, 7) = 0.22740562612709105e-1, (18, 8) = -0.7907506996885659e-2, (19, 1) = 0.3861826800619717e-1, (19, 2) = -0.11346720170368266e-1, (19, 3) = 0.4519365253222092e-5, (19, 4) = 0.11771845229883779e-4, (19, 5) = 0.31830356968205614e-6, (19, 6) = -0.7227503107876116e-6, (19, 7) = 0.20135279160158028e-1, (19, 8) = -0.69258279305857255e-2, (20, 1) = 0.3461964925151819e-1, (20, 2) = -0.11344195107036112e-1, (20, 3) = 0.10520087384895672e-4, (20, 4) = 0.23985583904104035e-4, (20, 5) = 0.14309031022091713e-6, (20, 6) = -0.3243163355613911e-6, (20, 7) = 0.17828986408851104e-1, (20, 8) = -0.6198747542466478e-2, (21, 1) = 0.30622372406040512e-1, (21, 2) = -0.11338536900897415e-1, (21, 3) = 0.2318958219703342e-4, (21, 4) = 0.5151495460511566e-4, (21, 5) = 0.644176893923037e-7, (21, 6) = -0.1457394223666943e-6, (21, 7) = 0.15743043694089114e-1, (21, 8) = -0.5667032245809346e-2, (22, 1) = 0.26628214747996367e-1, (22, 2) = -0.11326149022374868e-1, (22, 3) = 0.5062865573991706e-4, (22, 4) = 0.11202920591133182e-3, (22, 5) = 0.29042471883372703e-7, (22, 6) = -0.6558782013259308e-7, (22, 7) = 0.13816496386260714e-1, (22, 8) = -0.5288587198765507e-2, (23, 1) = 0.2264089792219356e-1, (23, 2) = -0.1129912568422263e-1, (23, 3) = 0.11043732551192093e-3, (23, 4) = 0.24452442337685054e-3, (23, 5) = 0.13112919793978457e-7, (23, 6) = -0.29561186696401987e-7, (23, 7) = 0.12000966662358389e-1, (23, 8) = -0.5033410522076286e-2, (24, 1) = 0.1866826401170284e-1, (24, 2) = -0.112401564797252e-1, (24, 3) = 0.24111499738008638e-3, (24, 4) = 0.5346976419992809e-3, (24, 5) = 0.5928961072990368e-8, (24, 6) = -0.13344220143954003e-7, (24, 7) = 0.10257018636687407e-1, (24, 8) = -0.48799355119546525e-2, (25, 1) = 0.14727247740388631e-1, (25, 2) = -0.11111322235991487e-1, (25, 3) = 0.5270984995463362e-3, (25, 4) = 0.1170937093093958e-2, (25, 5) = 0.26838123050175895e-8, (25, 6) = -0.6034138504105973e-8, (25, 7) = 0.855170598697519e-2, (25, 8) = -0.48120360449768335e-2, (26, 1) = 0.10855008195701099e-1, (26, 2) = -0.10829450488239615e-1, (26, 3) = 0.1153894867353147e-2, (26, 4) = 0.25678763780253673e-2, (26, 5) = 0.12148554830354332e-8, (26, 6) = -0.27359561753335984e-8, (26, 7) = 0.685723214134763e-2, (26, 8) = -0.4815859442170024e-2, (27, 1) = 0.7133340023707945e-2, (27, 2) = -0.1021185617301731e-1, (27, 3) = 0.25296367642439264e-2, (27, 4) = 0.5639179648904069e-2, (27, 5) = 0.5470798807280974e-9, (27, 6) = -0.12499110098424873e-8, (27, 7) = 0.5150935662073345e-2, (27, 8) = -0.4875323842547573e-2, (28, 1) = 0.37421142469540865e-2, (28, 2) = -0.8856750974397944e-2, (28, 3) = 0.5553314223775997e-2, (28, 4) = 0.12399873013868015e-1, (28, 5) = 0.23898046352870996e-9, (28, 6) = -0.5886263239784974e-9, (28, 7) = 0.3417257705562996e-2, (28, 8) = -0.4964941256428728e-2, (29, 1) = 0.10802063238274308e-2, (29, 2) = -0.5886264243625825e-2, (29, 3) = 0.12191496724982421e-1, (29, 4) = 0.2726046003802178e-1, (29, 5) = 0.8780666620362564e-10, (29, 6) = -0.3145483024743549e-9, (29, 7) = 0.16553925271360083e-2, (29, 8) = -0.5041588407307376e-2, (30, 1) = .0, (30, 2) = .0, (30, 3) = 0.253604956754486e-1, (30, 4) = 0.5675696394163127e-1, (30, 5) = .0, (30, 6) = -0.2456041414998205e-9, (30, 7) = .0, (30, 8) = -0.5061752025331262e-2}, datatype = float[8], order = C_order); YP := Matrix(30, 8, {(1, 1) = .23340239204225743, (1, 2) = -.5331952159154854, (1, 3) = 1.0666085359590798, (1, 4) = -1.664001255855841, (1, 5) = -1.078693238610632, (1, 6) = 2.5272733396431115, (1, 7) = -.3270195337602919, (1, 8) = -1.6386284623210314, (2, 1) = .1109879856055622, (2, 2) = -.27668306416758, (2, 3) = .6044763196631948, (2, 4) = -1.215840511423586, (2, 5) = -.5196819027931153, (2, 6) = 1.2179350168284138, (2, 7) = -.5415121349318274, (2, 8) = -0.47464609900130204e-1, (3, 1) = 0.4843316407335183e-1, (3, 2) = -.13743108602223522, (3, 3) = .31140452186526585, (3, 4) = -.6825882056056622, (3, 5) = -.24924695608308325, (3, 6) = .5834767497639158, (3, 7) = -.47755030372360524, (3, 8) = .3442111640300674, (4, 1) = 0.1734020508150595e-1, (4, 2) = -0.664061716186653e-1, (4, 3) = .15286409656975167, (4, 4) = -.3471292638250128, (4, 5) = -.11843000308023045, (4, 6) = .2768205516476608, (4, 7) = -.3623991577992995, (4, 8) = .3505094459139962, (5, 1) = 0.21669075489643436e-2, (5, 2) = -0.3135964126186165e-1, (5, 3) = 0.7266056738654511e-1, (5, 4) = -.16751347787271742, (5, 5) = -0.5547821753784222e-1, (5, 6) = .12946382276355417, (5, 7) = -.2610029523584158, (5, 8) = .27069010676941024, (6, 1) = -0.50970399710089966e-2, (6, 2) = -0.14509049773109677e-1, (6, 3) = 0.33689834209551746e-1, (6, 4) = -0.7814245095776906e-1, (6, 5) = -0.25569638336403393e-1, (6, 6) = 0.5956799063497078e-1, (6, 7) = -.1846813863050687, (6, 8) = .19173848565538026, (7, 1) = -0.850170573923812e-2, (7, 2) = -0.6602354622485683e-2, (7, 3) = 0.15331378501474606e-1, (7, 4) = -0.3562665053782395e-1, (7, 5) = -0.11614621967582133e-1, (7, 6) = 0.27010822204296493e-1, (7, 7) = -.13043730206507836, (7, 8) = .13175827536203807, (8, 1) = -0.10066600435936227e-1, (8, 2) = -0.2969795158033158e-2, (8, 3) = 0.6889665326704975e-2, (8, 4) = -0.1600805749658852e-1, (8, 5) = -0.5220047360349346e-2, (8, 6) = 0.1211822723699175e-1, (8, 7) = -0.9276269749510344e-1, (8, 8) = 0.8989322565415979e-1, (9, 1) = -0.10775285776671337e-1, (9, 2) = -0.13266725973456585e-2, (9, 3) = 0.30734234225339046e-2, (9, 4) = -0.71336257921122935e-2, (9, 5) = -0.2331040156855463e-2, (9, 6) = 0.540179901547425e-2, (9, 7) = -0.6677313096888332e-1, (9, 8) = 0.6155599091580231e-1, (10, 1) = -0.11093223857671173e-1, (10, 2) = -0.5906331487901736e-3, (10, 3) = 0.13660736436957612e-2, (10, 4) = -0.3166087438640864e-2, (10, 5) = -0.10376087463598667e-2, (10, 6) = 0.24001681261683933e-2, (10, 7) = -0.4880242646206497e-1, (10, 8) = 0.42525425645775286e-1, (11, 1) = -0.11235156259279292e-1, (11, 2) = -0.26259587767334694e-3, (11, 3) = 0.6063455840099236e-3, (11, 4) = -0.14029013548588402e-2, (11, 5) = -0.4613014836786184e-3, (11, 6) = 0.1065147081067729e-2, (11, 7) = -0.362821006967305e-1, (11, 8) = 0.2970478641260874e-1, (12, 1) = -0.11298377411755068e-1, (12, 2) = -0.11671367942552407e-3, (12, 3) = 0.26909965532350734e-3, (12, 4) = -0.6213607790020075e-3, (12, 5) = -0.2050552019113641e-3, (12, 6) = 0.47261835845329093e-3, (12, 7) = -0.27470295223740986e-1, (12, 8) = 0.20994580662545045e-1, (13, 1) = -0.11326512778841079e-1, (13, 2) = -0.5186493634598218e-4, (13, 3) = 0.11953975242454077e-3, (13, 4) = -0.2751560913724046e-3, (13, 5) = -0.9118819225490443e-4, (13, 6) = 0.2097936820554086e-3, (13, 7) = -0.2119966749093241e-1, (13, 8) = 0.1501209361344297e-1, (14, 1) = -0.11339018833798774e-1, (14, 2) = -0.2300049509428586e-4, (14, 3) = 0.53287832190102304e-4, (14, 4) = -0.12162828913811559e-3, (14, 5) = -0.4058455079812291e-4, (14, 6) = 0.9320256669642998e-4, (14, 7) = -0.16688439452594375e-1, (14, 8) = 0.10853561622551052e-1, (15, 1) = -0.11344545926454884e-1, (15, 2) = -0.1006977160373165e-4, (15, 3) = 0.24090702642066673e-4, (15, 4) = -0.5314514292021953e-4, (15, 5) = -0.1808330127261758e-4, (15, 6) = 0.414531772561738e-4, (15, 7) = -0.13409641090172729e-1, (15, 8) = 0.7926225677462116e-2, (16, 1) = -0.11346915687156164e-1, (16, 2) = -0.4107238785469627e-5, (16, 3) = 0.11572563554242374e-4, (16, 4) = -0.21776491002415385e-4, (16, 5) = -0.8068189730222743e-5, (16, 6) = 0.18461586968561007e-4, (16, 7) = -0.11004706907351869e-1, (16, 8) = 0.5838126924334749e-2, (17, 1) = -0.11347768954534387e-1, (17, 2) = -0.1001614906174004e-5, (17, 3) = 0.6988010971430737e-5, (17, 4) = -0.5681828869052488e-5, (17, 5) = -0.36048864416160054e-5, (17, 6) = 0.8233742298101364e-5, (17, 7) = -0.9227474850541309e-2, (17, 8) = 0.4327726871817593e-2, (18, 1) = -0.11347707465285151e-1, (18, 2) = 0.13562072356889462e-5, (18, 3) = 0.7082712647990626e-5, (18, 4) = 0.61852946118988865e-5, (18, 5) = -0.16129820537020167e-5, (18, 6) = 0.36774596654194148e-5, (18, 7) = -0.7907506996885659e-2, (18, 8) = 0.3218787963003316e-2, (19, 1) = -0.11346720170368266e-1, (19, 2) = 0.4519365253222092e-5, (19, 3) = 0.11771845229883779e-4, (19, 4) = 0.21730558306356916e-4, (19, 5) = -0.7227503107876116e-6, (19, 6) = 0.16448235223389977e-5, (19, 7) = -0.69258279305857255e-2, (19, 8) = 0.2391268591652661e-2, (20, 1) = -0.11344195107036112e-1, (20, 2) = 0.10520087384895672e-4, (20, 3) = 0.23985583904104035e-4, (20, 4) = 0.5104646876087109e-4, (20, 5) = -0.3243163355613911e-6, (20, 6) = 0.7367368038087048e-6, (20, 7) = -0.6198747542466478e-2, (20, 8) = 0.17624361972520379e-2, (21, 1) = -0.11338536900897415e-1, (21, 2) = 0.2318958219703342e-4, (21, 3) = 0.5151495460511566e-4, (21, 4) = 0.11303581077634247e-3, (21, 5) = -0.1457394223666943e-6, (21, 6) = 0.330470490685614e-6, (21, 7) = -0.5667032245809346e-2, (21, 8) = 0.12745847793413866e-2, (22, 1) = -0.11326149022374868e-1, (22, 2) = 0.5062865573991706e-4, (22, 3) = 0.11202920591133182e-3, (22, 4) = 0.2477211210417071e-3, (22, 5) = -0.6558782013259308e-7, (22, 6) = 0.14845299463652458e-6, (22, 7) = -0.5288587198765507e-2, (22, 8) = 0.8870560358397079e-3, (23, 1) = -0.1129912568422263e-1, (23, 2) = 0.11043732551192093e-3, (23, 3) = 0.24452442337685054e-3, (23, 4) = 0.5423300064749633e-3, (23, 5) = -0.29561186696401987e-7, (23, 6) = 0.6678640846554114e-7, (23, 7) = -0.5033410522076286e-2, (23, 8) = 0.5711720516766994e-3, (24, 1) = -0.112401564797252e-1, (24, 2) = 0.24111499738008638e-3, (24, 3) = 0.5346976419992809e-3, (24, 4) = 0.11883286982257715e-2, (24, 5) = -0.13344220143954003e-7, (24, 6) = 0.3008934543062662e-7, (24, 7) = -0.48799355119546525e-2, (24, 8) = 0.3073581193169439e-3, (25, 1) = -0.11111322235991487e-1, (25, 2) = 0.5270984995463362e-3, (25, 3) = 0.1170937093093958e-2, (25, 4) = 0.26070803931781696e-2, (25, 5) = -0.6034138504105973e-8, (25, 6) = 0.1357248044318085e-7, (25, 7) = -0.48120360449768335e-2, (25, 8) = 0.8436782666118993e-4, (26, 1) = -0.10829450488239615e-1, (26, 2) = 0.1153894867353147e-2, (26, 3) = 0.25678763780253673e-2, (26, 4) = 0.5727324579162048e-2, (26, 5) = -0.27359561753335984e-8, (26, 6) = 0.6123050050061982e-8, (26, 7) = -0.4815859442170024e-2, (26, 8) = -0.9879395514668107e-4, (27, 1) = -0.1021185617301731e-1, (27, 2) = 0.25296367642439264e-2, (27, 3) = 0.5639179648904069e-2, (27, 4) = 0.12598167172798172e-1, (27, 5) = -0.12499110098424873e-8, (27, 6) = 0.27489995034811228e-8, (27, 7) = -0.4875323842547573e-2, (27, 8) = -0.22709985591404464e-3, (28, 1) = -0.8856750974397944e-2, (28, 2) = 0.5553314223775997e-2, (28, 3) = 0.12399873013868015e-1, (28, 4) = 0.2774183440424406e-1, (28, 5) = -0.5886263239784974e-9, (28, 6) = 0.119806591541672e-8, (28, 7) = -0.4964941256428728e-2, (28, 8) = -0.26100116905558997e-3, (29, 1) = -0.5886264243625825e-2, (29, 2) = 0.12191496724982421e-1, (29, 3) = 0.2726046003802178e-1, (29, 4) = 0.61050182760659974e-1, (29, 5) = -0.3145483024743549e-9, (29, 6) = 0.4397720401528219e-9, (29, 7) = -0.5041588407307376e-2, (29, 8) = -0.14812495993633942e-3, (30, 1) = .0, (30, 2) = 0.253604956754486e-1, (30, 3) = 0.5675696394163127e-1, (30, 4) = .12711964776797577, (30, 5) = -0.2456041414998205e-9, (30, 6) = .0, (30, 7) = -0.5061752025331262e-2, (30, 8) = .0}, datatype = float[8], order = C_order); errproc := proc (x_bvp) local outpoint, X, Y, yout, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; Digits := 15; outpoint := evalf(x_bvp); X := Vector(30, {(1) = .0, (2) = .3115883682169779, (3) = .6252703963731303, (4) = .9433773563232228, (5) = 1.2680729118280434, (6) = 1.600281886516404, (7) = 1.9393162978590073, (8) = 2.2835156081889263, (9) = 2.6311011542021747, (10) = 2.9806940207148984, (11) = 3.331447607599669, (12) = 3.6829002252546412, (13) = 4.034807188531737, (14) = 4.386996935591378, (15) = 4.73932872237401, (16) = 5.0917150564146505, (17) = 5.444119100021418, (18) = 5.796534378827212, (19) = 6.14896147215437, (20) = 6.501398314085222, (21) = 6.8538379593121155, (22) = 7.206269623503382, (23) = 7.558681172777766, (24) = 7.911069970557719, (25) = 8.263448176851682, (26) = 8.615834971676511, (27) = 8.968244186479987, (28) = 9.320681820649735, (29) = 9.672590968929137, (30) = 10.0}, datatype = float[8], order = C_order); Y := Matrix(30, 8, {(1, 1) = .0, (1, 2) = 0.10611824614708966e-9, (1, 3) = 0.21223680379586772e-9, (1, 4) = 0.1436077131623382e-8, (1, 5) = -0.27392638285861593e-10, (1, 6) = -0.54785962448851395e-10, (1, 7) = 0.22079224415980286e-8, (1, 8) = 0.441584371418012e-8, (2, 1) = -0.331703095650701e-8, (2, 2) = 0.8267000381540213e-8, (2, 3) = -0.1939085494844822e-7, (2, 4) = 0.4654504680192282e-7, (2, 5) = -0.6150008159312253e-8, (2, 6) = 0.1460991796146134e-7, (2, 7) = 0.18140088417912563e-8, (2, 8) = 0.6328810462369557e-8, (3, 1) = -0.8917210756575664e-9, (3, 2) = 0.21700229994767607e-8, (3, 3) = -0.4829794674185182e-8, (3, 4) = 0.11543854036740672e-7, (3, 5) = -0.3283658369783179e-8, (3, 6) = 0.7879139521712888e-8, (3, 7) = 0.5737694219222408e-8, (3, 8) = -0.6024855903834403e-8, (4, 1) = -0.2576580177122397e-11, (4, 2) = -0.4039409598321924e-10, (4, 3) = 0.302335916407141e-9, (4, 4) = -0.6717142178533457e-9, (4, 5) = -0.7189459892595067e-9, (4, 6) = 0.1764604350335382e-8, (4, 7) = 0.4027390273690645e-8, (4, 8) = -0.46591595151464414e-8, (5, 1) = 0.8858877918619286e-10, (5, 2) = -0.2488456456272669e-9, (5, 3) = 0.6720219476565212e-9, (5, 4) = -0.15238791179470878e-8, (5, 5) = 0.2979181135805411e-9, (5, 6) = -0.6755970313085235e-9, (5, 7) = 0.19505492150602138e-8, (5, 8) = -0.8414345678986036e-9, (6, 1) = 0.26168645454797958e-10, (6, 2) = -0.8352782342615762e-10, (6, 3) = 0.22082728044353425e-9, (6, 4) = -0.4714454264327978e-9, (6, 5) = 0.43423224026203165e-9, (6, 6) = -0.10171726124185847e-8, (6, 7) = 0.11379204944469807e-8, (6, 8) = 0.752114077749552e-9, (7, 1) = -0.11566559352127293e-10, (7, 2) = 0.14774338450891652e-10, (7, 3) = -0.30745988084052983e-10, (7, 4) = 0.9943675023847896e-10, (7, 5) = 0.28070068939194325e-9, (7, 6) = -0.6637423341934752e-9, (7, 7) = 0.10730974963934578e-8, (7, 8) = 0.8311717158773128e-9, (8, 1) = -0.16936698509707775e-10, (8, 2) = 0.30824819622681654e-10, (8, 3) = -0.7284564792463304e-10, (8, 4) = 0.1842476615538208e-9, (8, 5) = 0.11615811335162823e-9, (8, 6) = -0.2786929861516379e-9, (8, 7) = 0.1211117300283874e-8, (8, 8) = 0.43792647471687983e-9, (9, 1) = -0.10679259942837181e-10, (9, 2) = 0.17173187239000172e-10, (9, 3) = -0.4092473609612889e-10, (9, 4) = 0.10243996643978535e-9, (9, 5) = 0.18270663419312965e-10, (9, 6) = -0.48083101012017846e-10, (9, 7) = 0.13150760667327681e-8, (9, 8) = 0.4839600360840332e-10, (10, 1) = -0.4777527930244925e-11, (10, 2) = 0.368905649159761e-11, (10, 3) = -0.8710518329352822e-11, (10, 4) = 0.23811317265740257e-10, (10, 5) = -0.20504529692640966e-10, (10, 6) = 0.4430073294274799e-10, (10, 7) = 0.13322903860000354e-8, (10, 8) = -0.2124422827541839e-9, (11, 1) = -0.1984699702957749e-11, (11, 2) = -0.26110271150692904e-11, (11, 3) = 0.657906547536571e-11, (11, 4) = -0.13503464762702541e-10, (11, 5) = -0.26655845407549594e-10, (11, 6) = 0.6004031453298003e-10, (11, 7) = 0.1277853661630222e-8, (11, 8) = -0.3508423213638427e-9, (12, 1) = -0.13975097123647744e-11, (12, 2) = -0.37086858113146975e-11, (12, 3) = 0.9509535413852049e-11, (12, 4) = -0.21183298729877094e-10, (12, 5) = -0.20502257350870924e-10, (12, 6) = 0.4661455962172624e-10, (12, 7) = 0.11801568340221224e-8, (12, 8) = -0.4036141525675941e-9, (13, 1) = -0.17119807806932555e-11, (13, 2) = -0.2645192529779686e-11, (13, 3) = 0.7234950724870783e-11, (13, 4) = -0.16358125737671543e-10, (13, 5) = -0.1248494169236082e-10, (13, 6) = 0.2849744735090613e-10, (13, 7) = 0.10628832881946927e-8, (13, 8) = -0.40486162946418126e-9, (14, 1) = -0.21541805098037154e-11, (14, 2) = -0.12500804255874254e-11, (14, 3) = 0.4067123230787082e-11, (14, 4) = -0.9293133033842991e-11, (14, 5) = -0.6342172078245572e-11, (14, 6) = 0.14512819179683149e-10, (14, 7) = 0.9417690788897534e-9, (14, 8) = -0.37888494286211324e-9, (15, 1) = -0.24371771835514173e-11, (15, 2) = -0.22026352101333577e-12, (15, 3) = 0.16470646248582584e-11, (15, 4) = -0.39521655934823245e-11, (15, 5) = -0.25909209997001323e-11, (15, 6) = 0.5948986262287439e-11, (15, 7) = 0.8258808750114244e-9, (15, 8) = -0.34106202922946937e-9, (16, 1) = -0.2529466214794891e-11, (16, 2) = 0.34466509621839437e-12, (16, 3) = 0.20021919417116187e-12, (16, 4) = -0.9808780288598653e-12, (16, 5) = -0.6542113367998001e-12, (16, 6) = 0.15212927322756878e-11, (16, 7) = 0.7196639067008801e-9, (16, 8) = -0.30033109620762616e-9, (17, 1) = -0.24870943847505185e-11, (17, 2) = 0.5555991618873742e-12, (17, 3) = -0.543672064428959e-12, (17, 4) = 0.1579298109416963e-12, (17, 5) = 0.17621314637068093e-12, (17, 6) = -0.3801747636413731e-12, (17, 7) = 0.6246868468298555e-9, (17, 8) = -0.2614401615043155e-9, (18, 1) = -0.23694206607797214e-11, (18, 2) = 0.5572893997719535e-12, (18, 3) = -0.8896984110361459e-12, (18, 4) = 0.20376692418855247e-12, (18, 5) = 0.4301180387966875e-12, (18, 6) = -0.964388336427067e-12, (18, 7) = 0.5408832527272733e-9, (18, 8) = -0.22659645612397806e-9, (19, 1) = -0.22015667246367873e-11, (19, 2) = 0.4936107473305456e-12, (19, 3) = -0.9772762938549533e-12, (19, 4) = -0.11364003298236743e-12, (19, 5) = 0.4289557641371133e-12, (19, 6) = -0.9656110417306334e-12, (19, 7) = 0.4673534741457254e-9, (19, 8) = -0.19656059206719824e-9, (20, 1) = -0.1948954823915845e-11, (20, 2) = 0.5566017549194152e-12, (20, 3) = -0.6482895420582375e-12, (20, 4) = 0.15852857526446984e-12, (20, 5) = 0.34106747705170573e-12, (20, 6) = -0.7683242330205314e-12, (20, 7) = 0.4028438688619958e-9, (20, 8) = -0.1713240049946399e-9, (21, 1) = -0.14818839175634789e-11, (21, 2) = 0.10884161170449385e-11, (21, 3) = 0.7255753466391857e-12, (21, 4) = 0.2703562894862887e-11, (21, 5) = 0.24251995568350466e-12, (21, 6) = -0.5461744383659213e-12, (21, 7) = 0.34601596098346636e-9, (21, 8) = -0.1505060504332975e-9, (22, 1) = -0.5384936638589145e-12, (22, 2) = 0.26898991306649044e-11, (22, 3) = 0.4428361321962919e-11, (22, 4) = 0.10486521108569223e-10, (22, 5) = 0.16096311904624677e-12, (22, 6) = -0.3622620296914077e-12, (22, 7) = 0.29558474128032535e-9, (22, 8) = -0.1335761754612235e-9, (23, 1) = 0.12425625628365048e-11, (23, 2) = 0.6165991460045009e-11, (23, 3) = 0.12242088944620483e-10, (23, 4) = 0.27512363899573054e-10, (23, 5) = 0.10184745652614562e-12, (23, 6) = -0.22906492271907425e-12, (23, 7) = 0.25037341626033e-9, (23, 8) = -0.11998480263900062e-9, (24, 1) = 0.398684761093639e-11, (24, 2) = 0.11789552178764704e-10, (24, 3) = 0.24788303038910167e-10, (24, 4) = 0.5520888355181478e-10, (24, 5) = 0.6217765550826776e-13, (24, 6) = -0.13984060667479711e-12, (24, 7) = 0.2092970855722882e-9, (24, 8) = -0.10929122346811661e-9, (25, 1) = 0.6411478391263291e-11, (25, 2) = 0.16686284474819876e-10, (25, 3) = 0.3567843586703539e-10, (25, 4) = 0.7941048941516497e-10, (25, 5) = 0.3688113470566055e-13, (25, 6) = -0.8314944433378394e-13, (25, 7) = 0.17125146083733595e-9, (25, 8) = -0.10142192617208784e-9, (26, 1) = 0.30450893491740916e-11, (26, 2) = 0.8611444952827984e-11, (26, 3) = 0.17562735005836057e-10, (26, 4) = 0.39104255539945563e-10, (26, 5) = 0.2130675896479633e-13, (26, 6) = -0.48553607624828046e-13, (26, 7) = 0.13486720436642908e-9, (26, 8) = -0.972203313992699e-10, (27, 1) = -0.19462244315430928e-10, (27, 2) = -0.4232967149058138e-10, (27, 3) = -0.9651023129120386e-10, (27, 4) = -0.215626209179201e-9, (27, 5) = 0.1191566580835234e-13, (27, 6) = -0.2821269262717691e-13, (27, 7) = 0.9832182200768947e-10, (27, 8) = -0.9896817301454939e-10, (28, 1) = -0.757206056280227e-10, (28, 2) = -0.1688140924045649e-9, (28, 3) = -0.3798448848003167e-9, (28, 4) = -0.8495638973466216e-9, (28, 5) = 0.62409155932911195e-14, (28, 6) = -0.16843492236351502e-13, (28, 7) = 0.6072214888977834e-10, (28, 8) = -0.10745753913849777e-9, (29, 1) = -0.10982998868704478e-9, (29, 2) = -0.24525540538567176e-9, (29, 3) = -0.5506810406081383e-9, (29, 4) = -0.1231338645510565e-8, (29, 5) = 0.26208494832623048e-14, (29, 6) = -0.11207073751478972e-13, (29, 7) = 0.28501784984063404e-10, (29, 8) = -0.10590085540259313e-9, (30, 1) = .0, (30, 2) = .0, (30, 3) = -0.16935763523322315e-11, (30, 4) = -0.17496437072638707e-11, (30, 5) = .0, (30, 6) = -0.9583472602853035e-14, (30, 7) = .0, (30, 8) = -0.9247812763143566e-10}, datatype = float[8], order = C_order); if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "right" then return X[30] elif outpoint = "order" then return 8 elif outpoint = "error" then return HFloat(4.654504680192282e-8) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [8, 30, [f(eta), diff(f(eta), eta), diff(diff(f(eta), eta), eta), diff(diff(diff(f(eta), eta), eta), eta), g(eta), diff(g(eta), eta), theta(eta), diff(theta(eta), eta)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[30] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[30] end if; V := array([1 = 4, 2 = 0]); if Digits <= trunc(evalhf(Digits)) then L := Vector(4, 'datatype' = 'float'[8]); yout := Vector(8, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(30, 8, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(8, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(30, 8, X, Y, outpoint, yout, L, V) end if; [eta = outpoint, seq('[f(eta), diff(f(eta), eta), diff(diff(f(eta), eta), eta), diff(diff(diff(f(eta), eta), eta), eta), g(eta), diff(g(eta), eta), theta(eta), diff(theta(eta), eta)]'[i] = yout[i], i = 1 .. 8)] end proc; if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "method" then return "bvp" elif outpoint = "right" then return X[30] elif outpoint = "order" then return 8 elif outpoint = "error" then return HFloat(4.654504680192282e-8) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [8, 30, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[30] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[30] end if; if Digits <= trunc(evalhf(Digits)) and (_EnvInFsolve <> true or _EnvDSNumericSaveDigits <= trunc(evalhf(Digits))) then V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false)  ] ); L := Matrix(7, 2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0, (3, 1) = .0, (3, 2) = .0, (4, 1) = .0, (4, 2) = .0, (5, 1) = .0, (5, 2) = .0, (6, 1) = .0, (6, 2) = .0, (7, 1) = .0, (7, 2) = .0}, datatype = float[8], order = C_order); yout := Vector(8, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(30, 8, X, Y, YP, outpoint, var(yout), var(L), var(V))) else if _EnvInFsolve = true then Digits := _EnvDSNumericSaveDigits end if; V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false)  ] ); L := Matrix(7, 2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0., (3, 1) = 0., (3, 2) = 0., (4, 1) = 0., (4, 2) = 0., (5, 1) = 0., (5, 2) = 0., (6, 1) = 0., (6, 2) = 0., (7, 1) = 0., (7, 2) = 0.}, order = C_order); yout := Vector(8, {(1) = 0., (2) = 0., (3) = 0., (4) = 0., (5) = 0., (6) = 0., (7) = 0., (8) = 0.}); `dsolve/numeric/hermite`(30, 8, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 8)] end proc, (2) = Array(0..0, {}), (3) = [eta, f(eta), diff(f(eta), eta), diff(diff(f(eta), eta), eta), diff(diff(diff(f(eta), eta), eta), eta), g(eta), diff(g(eta), eta), theta(eta), diff(theta(eta), eta)], (4) = 0}); solnproc := data[1]; if not type(outpoint, 'numeric') then if outpoint = "solnprocedure" then return eval(solnproc) elif member(outpoint, ["start", "left", "right", "errorproc", "rawdata", "order", "error"]) then return solnproc(x_bvp) elif outpoint = "sysvars" then return data[3] elif procname <> unknown then return ('procname')(x_bvp) else _ndsol := pointto(data[2][0]); return ('_ndsol')(x_bvp) end if end if; try res := solnproc(outpoint); [eta = res[1], seq('[f(eta), diff(f(eta), eta), diff(diff(f(eta), eta), eta), diff(diff(diff(f(eta), eta), eta), eta), g(eta), diff(g(eta), eta), theta(eta), diff(theta(eta), eta)]'[i] = res[i+1], i = 1 .. 8)] catch: error  end try end proc

(4)

plots:-odeplot( sol,
                [ [eta, theta(eta)],
                  [eta, g(eta)],
                  [eta, f(eta)]
                ],
                eta=0..10,
                color=[red, green, blue]
              );

 

 

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