## Solution to BVP , pls help...

Am trying to valid a research work done by kuiken(1968)

Kuiken_(1968).pdf

where we have this two eauations:

restart;
Digits := 35;
with(ODETools);
with(student);
with(plots);
inf := 4;
equ1 := diff(f[0](eta), `\$`(eta, 3))+theta[0](eta);
equ2 := diff(theta[0](eta), `\$`(eta, 2))+3*f[0](eta)*(diff(theta[0](eta), eta));
Bcs1 := f[0](0) = 0, (D(f[0]))(0) = 0, theta[0](0) = 1, theta[0](inf) = 0, (D(D(f[0])))(inf) = 0;
S1 := dsolve({Bcs1, equ1, equ2}, {f[0](eta), theta[0](eta)}, type = numeric, method = bvp[midrich]);
proc(x_bvp)  ...  end;
S1(0);
[                            d
[eta = 0., f[0](eta) = 0., ----- f[0](eta) = 0.,
[                           deta

d   /  d            \
----- |----- f[0](eta)| = 0.82449782146165697398999365896678734,
deta \ deta          /

theta[0](eta) = 1.0000000000000000000000000000000000,

d                                                         ]
----- theta[0](eta) = -0.71098574970825563256340736114251047]
deta                                                       ]
S1(inf);
[
[eta = 4., f[0](eta) = 1.7815670728545914261072119522795076,
[

d
----- f[0](eta) = 0.51061876174095320088291844433043562,
deta

d   /  d            \
----- |----- f[0](eta)| = 0., theta[0](eta) = 0.,
deta \ deta          /

d
----- theta[0](eta) = -0.000054818176138173095945902421930470836
deta

]
]
]

Pls, I need to find the function of the limit of f[0](eta) at eta tend to infinity. checked equation 45 of the attached document and for the two equation pls checked equation 36 and 37 for the ODE equation solved above.

Kuiken_solution for equation 36 and 37.pdf

## combuned solutions...

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in this program im trying to combine the result, but it showing some error can help me please

## not executing for three values...

http://www.sciencedirect.com/science/article/pii/S100757041300508X> restart;
> l := 1; p := 1; A := .5; B := .5; pr := 1; n := [.5, 1, 1.5]; M := 0; b := .5; L := 0; s := .5; K := [blue, green];
1
[0.5, 1, 1.5]
[blue, green]

> for j to nops(n) do R1 := 2*n[j]/(n[j]+1); R2 := 2*p/(n[j]+1); R3 := 2/(n[j]+1);

sol1 := dsolve([diff(diff(diff(f(eta), eta), eta), eta)+f(eta)*(diff(diff(f(eta), eta), eta))+R1*(1-(diff(f(eta), eta))^2)-M*(diff(f(eta), eta)) = 0, diff(diff(theta(eta), eta), eta)+pr*f(eta)*(diff(theta(eta), eta))-R2*pr*(diff(f(eta), eta))*theta(eta)+R3*(A*(diff(f(eta), eta))+B*theta(eta)) = 0, f(0) = 1, (D(f))(0) = L+b*((D@@2)(f))(0), (D(f))(7) = 1, theta(0) = 1+s*(D(theta))(0), theta(7) = 0], numeric, method = bvp);

plots[odeplot](sol1, [eta, ((D@@2)(f))(eta)], color = red);

fplt[j] := plots[odeplot](sol1, [eta, diff(diff(f(eta), eta), eta)], color = K[j], axes = boxed); fplt[j] := plots[odeplot](sol1, [eta, f(eta)], color = K[j], axes = boxed);

tplt[j] := plots[odeplot](sol1, [[eta, theta(eta)]], color = K[j], axes = normal) end do;

plots:-display([seq(fplt[j], j = 1 .. nops(n))]);

plots:-display([seq(fplt[j], j = 1 .. nops(n))]);

plots:-display([seq(tplt[j], j = 1 .. nops(n))]);
http://www.sciencedirect.com/science/article/pii/S100757041300508X

Dear sir

I am trying to plot the following link paper graphs for practice but I getting the plots for only one set of values here in this paper they plotted many so if you dont muned can help in this case. For example in this first graph named as Fig.1. please can you do this favour... and the paper link is  http://www.sciencedirect.com/science/article/pii/S100757041300508X

## multiple assignment...

> restart;
> n := [1, 2, 3, 4, 5]; pr := .71; p := 0; q := 0; b := 0; l := 0; s := 0;
> for j to nops(n) do R1 := 2*n[j]/(1+n[j]); R2 := 2*p/(1+n); sys := diff(diff(diff(f(eta), eta), eta), eta)+f(eta)*(diff(diff(f(eta), eta), eta))+1-(diff(f(eta), eta))^2 = 0, (diff(diff(theta(eta), eta), eta))/pr+f(eta)*(diff(theta(eta), eta))-R2*(diff(f(eta), eta))*theta(eta) = 0; bcs := f(0) = 0, (D(f))(0) = l+b*((D@@2)(f))(0), (D(f))(-.5) = 1, theta(0) = 1+s*(D(theta))(0), theta(-.5) = 0; proc (f1, th1, { output::name := 'number' }) local res1, fvals, thvals, res2; option remember; res1 := dsolve({sys, f(0) = 0, theta(0) = 1+th1, (D(f))(-2) = f1, (D(theta))(-2) = th1, ((D@@2)(f))(0) = f1-1}, numeric, :-output = listprocedure); fvals := (subs(res1, [seq(diff(f(eta), [`\$`(eta, i)]), i = 0 .. 2)]))(0); thvals := (subs(res1, [seq(diff(theta(eta), [`\$`(eta, i)]), i = 0 .. 1)]))(0); res2 := dsolve({sys, f(0) = fvals[1], theta(0) = thvals[1], theta(1) = 0, (D(f))(0) = fvals[2], (D(f))(1) = 0}, numeric, :-output = listprocedure); if output = 'number' then [fvals[3]-(subs(res2, diff(f(eta), `\$`(eta, 2))))(0), thvals[2]-(subs(res2, diff(theta(eta), eta)))(0)] else res1, res2 end if end proc; p1 := proc (f1, th1) p(args)[1] end proc; p2 := proc (f1, th1) p(args)[2] end proc; p(.3, -.2); par := fsolve([p1, p2], [.3, -.2]); res1, res2 := p(op(par), output = xxx); plots:-display(plots:-odeplot(res1, [[eta, f(eta)], [eta, theta(eta)]]), plots:-odeplot(res2, [[eta, f(eta)], [eta, theta(eta)]])); plots:-display(plots:-odeplot(res1, [[eta, diff(f(eta), eta)], [eta, diff(theta(eta), eta)]]), plots:-odeplot(res2, [[eta, diff(f(eta), eta)], [eta, diff(theta(eta), eta)]])); plots:-display(plots:-odeplot(res1, [[eta, diff(f(eta), eta, eta)]]), plots:-odeplot(res2, [[eta, diff(f(eta), eta, eta)]])); fplt[j] := plots[odeplot](sol1, [eta, diff(diff(f(eta), eta), eta)], color = L[j], axes = boxed); tplt[j] := plots[odeplot](sol1, [[eta, theta(eta)]], color = L[j], axes = boxed) end do;

Dear Sir

In this above problem it showing that error as  Error, cannot split rhs for multiple assignment please can you tell why it is showing like this  ?? and where i did multiple assignments ??

## can you help me regarding my ode problem...

Dear sir

In my ode problem i do not know that how to set range (eta) from -2 to 2 please can  you help me.

## nested ode programs...

> restart;
> with(plots);
> n := [0, .5, 1, 5]; pr := .71; p := 0; l := [1, 2, 3]; b := 0; s := 0; L := [green, blue, black, gold];
[green, blue, black, gold]
> R1 := 2*n/(n+1);
2 [0, 0.5, 1, 5]
------------------
[0, 0.5, 1, 5] + 1
for j from 1 to nops(l) do; for j from 1 to nops(n) do        R1 := 2*n[j]/(1+n[j]);        R2 := 2*p/(1+n[j]); sol1 := dsolve([diff(diff(diff(f(eta),eta),eta),eta)+f(eta)*diff(diff(f(eta),eta),eta)+R1*(1-diff(f(eta),eta)^2) = 0, (1/pr)*diff(diff(theta(eta),eta),eta)+f(eta)*diff(theta(eta),eta)-R2*diff(f(eta),eta)*theta(eta) = 0, f(0) = 0, (D(f))(0) = l+b*((D@@2)(f))(0), (D(f))(-2) =1, theta(0) = 1+s*(D(theta))(0), theta(-2) = 0], numeric, method = bvp); fplt[j]:= plots[odeplot](sol1,[eta,diff(diff(f(eta),eta),eta)],color=["blue","black","orange"]);         tplt[j]:= plots[odeplot](sol1, [eta,theta(eta)],color=L[j]); fplt[j]:= plots[odeplot](sol1,[eta,diff(f(eta),eta)],color=L[j]);      od:od:

Error, (in dsolve/numeric/bvp) unable to store 'Limit([eta, 2*eta, 3*eta]+eta^2*[.250000000000000, .500000000000000, .750000000000000]-.250000000000000*eta^2, eta = -2., left)' when datatype=float[8]
> plots:-display([seq(fplt[j], j = 1 .. nops(n))], color = [green, red], [seq(fplt[j], j = 1 .. nops(l))]);

> sol1(0);

Dear sir

In the  above problem i tried to write a nested program but its not executing and showing the error as Error, (in dsolve/numeric/bvp) unable to store 'Limit([eta, 2*eta, 3*eta]+eta^2*, i want the plot range from -2 to 2 but taking only 0 to -2 ,and -2.5 to 3 but taking only 0 to 1

## set the color for ode problems...

> restart;
> with(plots);
> pr := .72; p := 0; n := [2, 3, 4, 5]; s := 1; a := .2; b := 1;
> R1 := 2*n/(n+1);
2 [2, 3, 4, 5]
----------------
[2, 3, 4, 5] + 1
> R2 := 2*p/(n+1);
0
>
>
> for j to nops(n) do R1 := 2*n[j]/(1+n[j]); R2 := 2*p/(1+n[j]); sol1 := dsolve([diff(diff(diff(f(eta), eta), eta), eta)+f(eta)*(diff(diff(f(eta), eta), eta))+R1*(1-(diff(f(eta), eta))^2) = 0, diff(diff(theta(eta), eta), eta)+pr*s^f(eta)*(diff(theta(eta), eta))+R2*pr*s*(diff(f(eta), eta))*theta(eta)+2*(a*(diff(f(eta), eta))+b*theta(eta))/(n[j]-1) = 0, f(0) = 0, (D(f))(0) = 1+b*((D@@2)(f))(0), (D(f))(5) = 0, theta(0) = 1+s*(D(theta))(0), theta(5) = 0], numeric, method = bvp); fplt[j] := plots[odeplot](sol1, [eta, diff(diff(f(eta), eta), eta)], axes = boxed); tplt[j] := plots[odeplot](sol1, [[eta, theta(eta)]], axes = boxed) end do;
>
> plots:-display([seq(fplt[j], j = 1 .. nops(n))]);

> plots:-display([seq(tplt[j], j = 1 .. nops(n))]);

Dear sir

In the above problem graph, i am getting all the lines are in same color then how to identify the lines of different values like n=2,3,4,5,6(or can we set different color for different values of n for each line)

## Error, Initial newton iteration is not converging...

For the ODE system with boundary conditions, I was able to obtain solutions for n=0, but not for n>0. I obtained the error, Initial newton iteration is not converging. Anyone knows the solution for this? I am open to all suggestions and any help would be greatly appreciated:)

ODE_solution.mw

## solve ODE equation by dsolve ...

I have tried to solve the following ode equation, but I have got error. What is the potencial problem?

http://i65.tinypic.com/xdcl8p.jpg

## MAPLE To MATLAB...

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Paramétres

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 (1.1)
 > beta := 30;
 (1.2)
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 (1.3)

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 (1.4)
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Résolution & plots

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 (2.1)
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 (2.2)
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ET

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Hi evey ones ;

how can i convert these code below TO matlab

thank a lot Maple_to_MATLAB.mw

## error for dsolve differential equation.......

hi..i have a problem for solving this nonlinear differential equationerror.mw

 (1)

 (2)

############################################################CHANGE OF VARIABLE:::           x=y*L

thanks...

## time dependent variable...

> restart;
> libname = [shootlib, libname];
> with(shoot);
Error, invalid input: with expects its 1st argument, pname, to be of type {`module`, package}, but received shoot
> with(plots);
Pr := 10; s = -.1; lambda := 0; Gr := 1.0; Gm := 1.0; beta := -1.20;
10
s = -0.1
0
1.0
1.0
-1.20
> M := 0.; z := .1; Xi := .5; Nt := .5; Nb := .2; l := 5; Nr := .5; epsilon1 := .2; epsilon2 := .2;
0.
0.1
0.5
0.5
0.2
5
0.5
0.2
0.2
> Prff := Pr/(1+4.*N*(1/3));
10
-----------------
1 + 1.333333333 N
> FNS := {f(eta), h(eta), r(eta), u(eta), v(eta), theta(eta), `&varphi;`(eta)};
{f(eta), h(eta), r(eta), u(eta), v(eta), theta(eta), &varphi;(eta)}
> ODE := {diff(h(eta), eta)+.75*l*f(eta)*h(eta)-(1/4)*l*u(eta)*epsilon2-Nt*(.75*f(eta)*r(eta)-(1/4)*u(eta)*epsilon1+Nb*r(eta)*h(eta)+Nt*r(eta)*r(eta))/Nb = 0, .75*f(eta)*r(eta)+diff(r(eta), eta)-(1/4)*u(eta)*epsilon1+Nb*r(eta)*h(eta)+Nt*r(eta)*r(eta) = 0, diff(v(eta), eta)+3*(f(eta)*v(eta)-u(eta)*u(eta))/(4*Pr)-(M+lambda)*u(eta)+theta(eta)-Nr*`&varphi;`(eta) = 0, diff(f(eta), eta) = u(eta), diff(u(eta), eta) = v(eta), diff(theta(eta), eta) = r(eta), diff(`&varphi;`(eta), eta) = h(eta)};
/ / d \
{ 0.75 f(eta) r(eta) + |----- r(eta)| - 0.05000000000 u(eta)
\ \ deta /

2 / d \ 3
+ 0.2 r(eta) h(eta) + 0.5 r(eta) = 0, |----- v(eta)| + -- f(eta) v(eta)
\ deta / 40

3 2 / d \
- -- u(eta) + theta(eta) - 0.5 &varphi;(eta) = 0, |----- h(eta)|
40 \ deta /

+ 3.75 f(eta) h(eta) - 0.1250000000 u(eta) - 1.875000000 f(eta) r(eta)

2
- 0.5000000000 r(eta) h(eta) - 1.250000000 r(eta) = 0,

d d d
----- f(eta) = u(eta), ----- u(eta) = v(eta), ----- theta(eta) = r(eta),
deta deta deta

d \
----- &varphi;(eta) = h(eta) }
deta /
> IC := {f(0) = s, h(0) = xi, r(0) = tau, u(0) = 0, v(0) = alpha(0), theta(0) = 1-(1/4)*epsilon1, `&varphi;`(0) = (1/4)*epsilon2};
{f(0) = s, h(0) = xi, r(0) = tau, u(0) = 0, v(0) = alpha(0),

theta(0) = 0.9500000000, &varphi;(0) = 0.05000000000}
> L := 2;
2
> BC = {u(L) = 0, theta(L) = 0, `&varphi;`(L) = 0};
BC = {u(2) = 0, theta(2) = 0, &varphi;(2) = 0}
> S := Shoot(ODE, IC, BC, FNS, [alpha = .42453091564332, tau = -.21166705749821127, xi = -.4944583739651814]);
/ / / d \
Shoot|{ 0.75 f(eta) r(eta) + |----- r(eta)| - 0.05000000000 u(eta)
\ \ \ deta /

2 / d \ 3
+ 0.2 r(eta) h(eta) + 0.5 r(eta) = 0, |----- v(eta)| + -- f(eta) v(eta)
\ deta / 40

3 2 / d \
- -- u(eta) + theta(eta) - 0.5 &varphi;(eta) = 0, |----- h(eta)|
40 \ deta /

+ 3.75 f(eta) h(eta) - 0.1250000000 u(eta) - 1.875000000 f(eta) r(eta)

2
- 0.5000000000 r(eta) h(eta) - 1.250000000 r(eta) = 0,

d d d
----- f(eta) = u(eta), ----- u(eta) = v(eta), ----- theta(eta) = r(eta),
deta deta deta

d \
----- &varphi;(eta) = h(eta) }, {f(0) = s, h(0) = xi, r(0) = tau, u(0) = 0,
deta /

v(0) = alpha(0), theta(0) = 0.9500000000, &varphi;(0) = 0.05000000000}, BC,

{f(eta), h(eta), r(eta), u(eta), v(eta), theta(eta), &varphi;(eta)}, [

alpha = 0.42453091564332, tau = -0.21166705749821127,

\
xi = -0.4944583739651814]|
/
RungeKutta(ODE, BC, alpha = .42453091564332, tau = -.21166705749821127, xi = -.4944583739651814, output=plot);
/ / / d \
RungeKutta|{ 0.75 f(eta) r(eta) + |----- r(eta)| - 0.05000000000 u(eta)
\ \ \ deta /

2 / d \ 3
+ 0.2 r(eta) h(eta) + 0.5 r(eta) = 0, |----- v(eta)| + -- f(eta) v(eta)
\ deta / 40

3 2 / d \
- -- u(eta) + theta(eta) - 0.5 &varphi;(eta) = 0, |----- h(eta)|
40 \ deta /

+ 3.75 f(eta) h(eta) - 0.1250000000 u(eta) - 1.875000000 f(eta) r(eta)

2
- 0.5000000000 r(eta) h(eta) - 1.250000000 r(eta) = 0,

d d d
----- f(eta) = u(eta), ----- u(eta) = v(eta), ----- theta(eta) = r(eta),
deta deta deta

d \
----- &varphi;(eta) = h(eta) }, BC, alpha = 0.42453091564332,
deta /

\
tau = -0.21166705749821127, xi = -0.4944583739651814, output = plot|
/
>

Dear sir

in the above problem im geiitng the problem with , with(shoot) command and even it is not executing at

S := Shoot(ODE, IC, BC, FNS, [alpha = .42453091564332, tau = -.21166705749821127, xi = -.4944583739651814]) this command, here alpha,tau and zi variable should change.

## shooting method...

> restart;
> with(plots);
> Eql := diff(f(eta), eta, eta, eta)+.5*f(eta)*(diff(f(eta), eta, eta)) = 0;
/ d / d / d \\\ / d / d \\
|----- |----- |----- f(eta)||| + 0.5 f(eta) |----- |----- f(eta)|| = 0
\ deta \ deta \ deta /// \ deta \ deta //
> blt := 10;
10
> bcs1 := f(0) = f0, (D(f))(0) = 0, (D(f))(blt) = 1;
f(0) = f0, D(f)(0) = 0, D(f)(10) = 1
> L := [0];
[0]
> for k to 1 do R := dsolve(eval({Eql, bcs1}, f0 = L[k]), f(eta), numeric, output = listprocedure); X1 || k := rhs(R[3]); X2 || k := rhs(R[4]) end do;
[
[eta = proc(eta) ... end;, f(eta) = proc(eta) ... end;,
[

d
----- f(eta) = proc(eta) ... end;,
deta

d / d \ ]
----- |----- f(eta)| = proc(eta) ... end;]
deta \ deta / ]
proc(eta) ... end;
proc(eta) ... end;
> print([X2], [1 .. 1, 0]);

in the above problem i should get the asnser (at print line) but its not getting so please can you tell me why it is not getting.

## cubic b spline to solve ode...

I need to solve an ode of the type ay''+by'+cy=f(x) using cubic b spline.

can any one help me with the code or algorithm. Thanks