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Madhukesh J K

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unable to convert to an explicit first-o...

Madhukesh J K 140 Maple 17
ode dsolve numeric
November 26 2020
1 2


I m Stucked with this step

> 

restart; with(plots)
> 

eq1 := 2*n*4^n*eta^((n+1)*(1/2))*(diff(f(eta), `$`(eta, 2)))^(n-1)*(diff(f(eta), `$`(eta, 3)))+4^n*(n+1)*eta^((n-1)*(1/2))*(diff(f(eta), `$`(eta, 2)))^n+4*f(eta)*(diff(f(eta), `$`(eta, 2)))-4*m*(diff(f(eta), eta))^2+m-2*M*(diff(f(eta), eta)) = 0;

2*n*4^n*eta^((1/2)*n+1/2)*(diff(diff(f(eta), eta), eta))^(n-1)*(diff(diff(diff(f(eta), eta), eta), eta))+4^n*(n+1)*eta^((1/2)*n-1/2)*(diff(diff(f(eta), eta), eta))^n+4*f(eta)*(diff(diff(f(eta), eta), eta))-4*m*(diff(f(eta), eta))^2+m-2*M*(diff(f(eta), eta)) = 0

 (1)
> 

eq2 := 2*eta*(diff(theta(eta), `$`(eta, 2)))+2*(diff(theta(eta), eta))+Pr*(f(eta)*(diff(theta(eta), eta))-s*(diff(f(eta), eta))*theta(eta))+Nb*(diff(theta(eta), eta))*(diff(phi(eta), eta))+Nt*(diff(theta(eta), `$`(eta, 2)))^2 = 0;

2*eta*(diff(diff(theta(eta), eta), eta))+2*(diff(theta(eta), eta))+Pr*(f(eta)*(diff(theta(eta), eta))-s*(diff(f(eta), eta))*theta(eta))+Nb*(diff(theta(eta), eta))*(diff(phi(eta), eta))+Nt*(diff(diff(theta(eta), eta), eta))^2 = 0

 (2)
> 

eq3 := 2*eta*(diff(phi(eta), `$`(eta, 2)))+2*(diff(phi(eta), eta))+Sc*(f(eta)*(diff(phi(eta), eta))-s*(diff(f(eta), eta))*phi(eta))+Nb*(2*eta*(diff(theta(eta), `$`(eta, 2)))+2*(diff(theta(eta), eta)))/Nt = 0;
``

2*eta*(diff(diff(phi(eta), eta), eta))+2*(diff(phi(eta), eta))+Sc*(f(eta)*(diff(phi(eta), eta))-s*(diff(f(eta), eta))*phi(eta))+Nb*(2*eta*(diff(diff(theta(eta), eta), eta))+2*(diff(theta(eta), eta)))/Nt = 0

 (3)
> 

eq4 := 2*eta*(diff(chi(eta), `$`(eta, 2)))+2*(diff(chi(eta), eta))+Lb*(f(eta)*(diff(chi(eta), eta))-s*(diff(f(eta), eta))*chi(eta))-Pe*(2*eta*chi(eta)*(diff(phi(eta), `$`(eta, 2)))+2*chi(eta)*(diff(phi(eta), eta))+2*eta*(diff(chi(eta), `$`(eta, 2)))*(diff(phi(eta), `$`(eta, 2)))) = 0;

2*eta*(diff(diff(chi(eta), eta), eta))+2*(diff(chi(eta), eta))+Lb*(f(eta)*(diff(chi(eta), eta))-s*(diff(f(eta), eta))*chi(eta))-Pe*(2*eta*chi(eta)*(diff(diff(phi(eta), eta), eta))+2*chi(eta)*(diff(phi(eta), eta))+2*eta*(diff(diff(chi(eta), eta), eta))*(diff(diff(phi(eta), eta), eta))) = 0

 (4)
> 

bcs := (D(f))(a) = 0, f(a) = 2*s*a*(D(phi))(a)/Sc, theta(a) = 1, phi(a) = 1, chi(a) = 1, (D(f))(10) = 1/2, theta(10) = 0, phi(10) = 0, chi(10) = 0;

(D(f))(a) = 0, f(a) = 2*s*a*(D(phi))(a)/Sc, theta(a) = 1, phi(a) = 1, chi(a) = 1, (D(f))(10) = 1/2, theta(10) = 0, phi(10) = 0, chi(10) = 0

 (5)
> 

params := {Lb = .1, M = .1, Nb = .6, Nt = .2, Pe = 5, Pr = 6.2, Sc = .1, a = 0.1e-1, m = 1/3, n = 1, s = .1};

{Lb = .1, M = .1, Nb = .6, Nt = .2, Pe = 5, Pr = 6.2, Sc = .1, a = 0.1e-1, m = 1/3, n = 1, s = .1}

 (6)
> 

sol := dsolve(eval([eq1, eq2, eq3, eq4, bcs], params), numeric, output = listprocedure, maxmesh = 1024)

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

  
> 

odeplot(sol, [[eta, f(eta)][eta, phi(eta)], [eta, theta(eta)], [eta, chi(eta)]], eta = eval(c, params) .. 10)

Error, invalid subscript selector

  
> 

NULL


 

Download Pblm2.mw

system is singular at left endpoint, use...

Madhukesh J K 140 Maple 17
ode dsolve numeric
November 26 2020
1 1

When I try to solve this problem above error displays

Here is the file
 

Download Main.mw

Error, (in dsolve/numeric/process_input)...

Madhukesh J K 140 Maple 17
ode syntax 2dmath
November 25 2020
2 6


When i try to solve the problem i got error like this 

Please help me get the solution

> 

restart
> 

with(plots):
> 

eq1 := (2*(eta*(diff(f(eta), `$`(eta, 3)))+diff(f(eta), `$`(eta, 2))))/((1-phi[1])^2.5*(1-phi[2])^2.5)+((1-phi[2])*(1-phi[1]+phi[1]*rho[s1]/rho[f])+phi[2]*rho[s2]/rho[f])*f(eta)*(diff(f(eta), `$`(eta, 2)))-sigma[hnf]*M*(diff(f(eta), eta))/(sigma[f]*(m^2+1)) = 0;

2*(eta*(diff(diff(diff(f(eta), eta), eta), eta))+diff(diff(f(eta), eta), eta))/((1-phi[1])^2.5*(1-phi[2])^2.5)+((1-phi[2])*(1-phi[1]+phi[1]*rho[s1]/rho[f])+phi[2]*rho[s2]/rho[f])*f(eta)*(diff(diff(f(eta), eta), eta))-sigma[hnf]*M*(diff(f(eta), eta))/(sigma[f]*(m^2+1)) = 0

 (1)
> 

eq2 := k[hnf]*(eta*(diff(theta(eta), `$`(eta, 2)))+diff(theta(eta), eta))/k[f]+(1/2)*Pr*((1-phi[2])*(1-phi[1]+phi[1]*`ρC`[p][s1]/`ρC`[p][f])+phi[2]*`ρC`[p][s2]/`ρC`[p][f])*f(eta)*(diff(theta(eta), eta)) = 0;

k[hnf]*(eta*(diff(diff(theta(eta), eta), eta))+diff(theta(eta), eta))/k[f]+(1/2)*Pr*((1-phi[2])*(1-phi[1]+phi[1]*`ρC`[p][s1]/`ρC`[p][f])+phi[2]*`ρC`[p][s2]/`ρC`[p][f])*f(eta)*(diff(theta(eta), eta)) = 0

 (2)
> 

eq3 := 2*(1-phi[1])^2.5*(1-phi[2])^2.5*(eta*(diff(chi(eta), `$`(eta, 2)))+diff(chi(eta), eta))+Sc*f(eta)*(diff(chi(eta), eta))-beta*Sc*(1+delta*theta(eta))^n*exp^(-E/(1+delta*theta(eta)))*chi(eta) = 0;

2*(1-phi[1])^2.5*(1-phi[2])^2.5*(eta*(diff(diff(chi(eta), eta), eta))+diff(chi(eta), eta))+Sc*f(eta)*(diff(chi(eta), eta))-beta*Sc*(1+delta*theta(eta))^n*exp^(-E/(1+delta*theta(eta)))*chi(eta) = 0

 (3)
> 

bcs := f(c) = (1/2)*lambda*c, (D(f))(c) = (1/2)*lambda, theta(c) = 1, chi(c) = 1, (D(f))(10) = (1-lambda)*(1/2), theta(10) = 0, chi(10) = 0;

f(c) = (1/2)*lambda*c, (D(f))(c) = (1/2)*lambda, theta(c) = 1, chi(c) = 1, (D(f))(10) = 1/2-(1/2)*lambda, theta(10) = 0, chi(10) = 0

 (4)
> 

params := {E = .5, M = .1, Pr = 6.2, Sc = .5, beta = .5, c = .1, delta = .5, lambda = -1, m = .5, n = .1, k[f] = .6130, k[hnf] = 1.039626683, phi[1] = .1, phi[2] = .1, rho[f] = 997.1, rho[s1] = 5180, rho[s2] = 1800, sigma[f] = 0.5e-1, sigma[hnf] = 0.8888888337e-1, `ρC`[p][f] = 4.1668809*10^6, `ρC`[p][s1] = 3470600, `ρC`[p][s2] = 1290600};

{E = .5, M = .1, Pr = 6.2, Sc = .5, beta = .5, c = .1, delta = .5, lambda = -1, m = .5, n = .1, k[f] = .6130, k[hnf] = 1.039626683, phi[1] = .1, phi[2] = .1, rho[f] = 997.1, rho[s1] = 5180, rho[s2] = 1800, sigma[f] = 0.5e-1, sigma[hnf] = 0.8888888337e-1, `ρC`[p][f] = 4166880.900, `ρC`[p][s1] = 3470600, `ρC`[p][s2] = 1290600}

 (5)
> 

ODEs := [eq1, eq2, eq3]

[2*(eta*(diff(diff(diff(f(eta), eta), eta), eta))+diff(diff(f(eta), eta), eta))/((1-phi[1])^2.5*(1-phi[2])^2.5)+((1-phi[2])*(1-phi[1]+phi[1]*rho[s1]/rho[f])+phi[2]*rho[s2]/rho[f])*f(eta)*(diff(diff(f(eta), eta), eta))-sigma[hnf]*M*(diff(f(eta), eta))/(sigma[f]*(m^2+1)) = 0, k[hnf]*(eta*(diff(diff(theta(eta), eta), eta))+diff(theta(eta), eta))/k[f]+(1/2)*Pr*((1-phi[2])*(1-phi[1]+phi[1]*`ρC`[p][s1]/`ρC`[p][f])+phi[2]*`ρC`[p][s2]/`ρC`[p][f])*f(eta)*(diff(theta(eta), eta)) = 0, 2*(1-phi[1])^2.5*(1-phi[2])^2.5*(eta*(diff(diff(chi(eta), eta), eta))+diff(chi(eta), eta))+Sc*f(eta)*(diff(chi(eta), eta))-beta*Sc*(1+delta*theta(eta))^n*exp^(-E/(1+delta*theta(eta)))*chi(eta) = 0]

 (6)
> 

``
> 

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

Error, (in dsolve/numeric/process_input) system must be entered as a set/list of expressions/equations

  
> 

``


 

Download 1.mw
 

 

 

How to obtain multiple solution...

Madhukesh J K 140 Maple
homework numeric differential-equations
October 07 2020
0 1

How to obtain the multiple solution and graph given in the paper. 

Stefan Blowing and Slip Effects on Unsteady Nanofluid Transport Past a Shrinking Sheet: Multiple Solutions

https://doi.org/10.1002/htj.21470

 

Can anyone help to get solutions.

How to solve 2 ODE equation with Boundar...

Madhukesh J K 140 Maple
ode dsolve numeric
September 22 2020
2 0

Equations

 

ODES := (diff(f(eta), `$`(eta, 4)))/((1-phi1)^2.5*(1-phi2)^2.5*((1-phi2)*(1-phi1+phi1*rhos1/rhosf)+phi2*rhos2/rhosf))+S*(f(eta)*(diff(f(eta), `$`(eta, 3)))-3*(diff(f(eta), `$`(eta, 2)))-eta*(diff(f(eta), `$`(eta, 3)))-(diff(f(eta), eta))*(diff(f(eta), `$`(eta, 2)))) = 0,

(khnf/kf+(4/3)*R)*(diff(theta(eta), `$`(eta, 2)))/((1-phi2)*(1-phi1+phi1*rhos1*cp1/(rhosf*cpf))+phi2*rhos2*cp2/(rhosf*cpf))+S*Pr*(f(eta)*(diff(theta(eta), eta))-eta*(diff(theta(eta), eta))-gamma*(eta^2*(diff(theta(eta), `$`(eta, 2)))-2*eta*f(eta)*(diff(theta(eta), `$`(eta, 2)))-eta*(diff(f(eta), eta))*(diff(theta(eta), eta))+f(eta)*(diff(f(eta), eta))*(diff(theta(eta), eta))+f(eta)^2*(diff(theta(eta), `$`(eta, 2))))) = 0

Boundary Conditions

 f(0) = 0, ((D^2)(f))(0) = 0, (D(theta))(0) = 0, f(1) = 0, (D(f))(1) = 0, theta(1) = 1

 

phi1 = .1, phi2 = .1, rhos1 = 2720, rhos2 = 2810, rhosf = 997.1, khnf = 1.083061737, kf = .613, cp1 = 893, cp2 = 960, cpf = 4179, Pr = 6.2, knf = .8154646474., S=0.5,R=0.5, gamma=0.5

I am tried to solve it showing following error

Error, (in dsolve/numeric/bvp) matrix is singular

 

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