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Question: How to solve this BVP?

Question:How to solve this BVP?

Aisyah 5 Maple
ode dsolve numeric differential-equations bvp
December 17 2019
1

I have problem on RKF45 with result on 

"unable to convert to an explicit first order system"


 

``

> 

restart
> 

with(plots):
> 

vf := .3:
> 

kf := .2:
> 

c := .2:
> 

`Τt` := .3:
> 

N := .1:
> 

`φd` := .2:
> 

`ρs` := 8933:
> 

`ρf` := 997.1:
> 

y := .1:
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alpha := .5:
> 

Pr := 6.2:
> 

beta := .5:
> 

`βT` := .2:
> 

`ε` := .1:
> 

ks := 400:
> 

kf := .613:
> 

cps := 385:
> 

cpf := 4179:
> 

delta := .5:
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Ec := .2:
> 

`Τv` := 1:
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A := .5:
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B := .5:
> 

omega := 1.5:
> 

K := (ks+2*kf-2*phi*(kf-ks))/(ks+2*kf+phi*(kf-ks))

(401.226+798.774*phi)/(401.226-399.387*phi)

 (1)
> 

Eq1 := (1-`φd`)*(diff(f(eta), eta, eta, eta))/(1-phi)^2.5-(1-`φd`)*(1-phi+phi*`ρs`/`ρf`)*((diff(f(eta), eta))^2-f(eta)*(diff(f(eta), eta, eta)))+alpha*beta*(diff(G(eta), eta)-(diff(f(eta), eta))) = 0

.8*(diff(diff(diff(f(eta), eta), eta), eta))/(1-phi)^2.5-.8*(1+7.958981045*phi)*((diff(f(eta), eta))^2-f(eta)*(diff(diff(f(eta), eta), eta)))+.25*(diff(G(eta), eta))-.25*(diff(f(eta), eta)) = 0

 (2)
> 

Eq2 := (diff(G(eta), eta))^2-G(eta)*(diff(G(eta), eta, eta))-beta*(diff(f(eta), eta)-(diff(G(eta), eta))) = 0

(diff(G(eta), eta))^2-G(eta)*(diff(diff(G(eta), eta), eta))-.5*(diff(f(eta), eta))+.5*(diff(G(eta), eta)) = 0

 (3)
> 

Eq3 := G(eta)*(diff(G(eta), eta))+beta*(f(eta)-G(eta)) = 0

G(eta)*(diff(G(eta), eta))+.5*f(eta)-.5*G(eta) = 0

 (4)
> 

Eq4 := K*(diff(theta(eta), eta, eta))/Pr+(1-phi+phi*`ρs`*cps/(`ρf`*cpf))*(f(eta)*(diff(theta(eta), eta))-2*(diff(f(eta), eta))*theta(eta))+alpha*`βT`*(H(eta)-theta(eta))+Ec*alpha*(diff(G(eta), eta)-(diff(f(eta), eta)))^2/`Τv`+(A*(diff(f(eta), eta))+B*theta(eta))/Pr = 0

.1612903226*(401.226+798.774*phi)*(diff(diff(theta(eta), eta), eta))/(401.226-399.387*phi)+(1-.1746332371*phi)*(f(eta)*(diff(theta(eta), eta))-2*(diff(f(eta), eta))*theta(eta))+.10*H(eta)-0.1935483870e-1*theta(eta)+.10*(diff(G(eta), eta)-(diff(f(eta), eta)))^2+0.8064516130e-1*(diff(f(eta), eta)) = 0

 (5)
> 

Eq5 := 2*(diff(G(eta), eta))*H(eta)-G(eta)*(diff(H(eta), eta))+y*`βT`*(H(eta)-theta(eta)) = 0

2*(diff(G(eta), eta))*H(eta)-G(eta)*(diff(H(eta), eta))+0.2e-1*H(eta)-0.2e-1*theta(eta) = 0

 (6)
> 

bcs1 := (D(f))(0) = `ε`+delta*((D@@2)(f))(0), f(0) = 0, (D(f))(10) = 0;

(D(f))(0) = .1+.5*((D@@2)(f))(0), f(0) = 0, (D(f))(10) = 0

 (7)
> 

bcs2 := (D(G))(10) = 0, G(10) = f(10);

(D(G))(10) = 0, G(10) = f(10)

 (8)
> 

bcs3 := theta(0) = 1+omega*(D(theta))(0), theta(10) = 0;

theta(0) = 1+1.5*(D(theta))(0), theta(10) = 0

 (9)
> 

bcs4 := H(10) = 0;

H(10) = 0

 (10)
> 

L := [0., 0.5e-1, .1, .15, .2]:
> 

for k to 5 do R := dsolve(eval({Eq1, Eq2, Eq3, Eq4, Eq5, bcs1, bcs2, bcs3, bcs4}, phi = L[k]), [f(eta), G(eta), theta(eta), H(eta)], numeric, output = listprocedure); Y || k := rhs(R[3]) end do

Error, (in dsolve/numeric/bvp/convertsys) unable to convert to an explicit first-order system

  
> 

plot([Y || (1 .. 5)], 0 .. 10)

Error, (in plot) procedure expected, as range contains no plotting variable

  
> 

``

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Download RKF45_A.mw

 

thank you in advance for your help and kindness.
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