faisal

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6 years, 164 days

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These are questions asked by faisal

Hello Dr/Pof/Colleague

Please help me on the ODE BVP problem

my target i want RUN double loop


 

restart

with(plots)

b := 0; c := 0; k[1] := 1; k[2] := 0; Un := 0; d := 0

Eq1 := (101-100*lambda)*(1+a*phi(eta))*(diff(f(eta), `$`(eta, 3)))+(diff(f(eta), `$`(eta, 2)))*(f(eta)+g(eta)+a*(diff(phi(eta), eta)))-(diff(f(eta), eta))^2+k[2]+Un*(k[2]-(1/2)*eta*(diff(f(eta), `$`(eta, 2)))-(diff(f(eta), eta))) = 0

(101-100*lambda)*(1+a*phi(eta))*(diff(diff(diff(f(eta), eta), eta), eta))+(diff(diff(f(eta), eta), eta))*(f(eta)+g(eta)+a*(diff(phi(eta), eta)))-(diff(f(eta), eta))^2 = 0

(1)

Eq2 := (101-100*lambda)*(1+a*phi(eta))*(diff(g(eta), `$`(eta, 3)))+(diff(g(eta), `$`(eta, 2)))*(f(eta)+g(eta)+a*(diff(phi(eta), eta)))-(diff(g(eta), eta))^2+k[2]+Un*(k[2]-(1/2)*eta*(diff(g(eta), `$`(eta, 2)))-(diff(g(eta), eta))) = 0

(101-100*lambda)*(1+a*phi(eta))*(diff(diff(diff(g(eta), eta), eta), eta))+(diff(diff(g(eta), eta), eta))*(f(eta)+g(eta)+a*(diff(phi(eta), eta)))-(diff(g(eta), eta))^2 = 0

(2)

NULL

Valpha := [0, .1, .2]; Va := [0, 0]

etainf := 100

bcs := (D(f))(0) = k[1], (D(g))(0) = alpha, f(0) = 0, g(0) = 0, (D(f))(etainf) = k[2], (D(g))(etainf) = k[2]

(D(f))(0) = 1, (D(g))(0) = alpha, f(0) = 0, g(0) = 0, (D(f))(100) = 0, (D(g))(100) = 0

(3)

dsys := {Eq1, Eq2, bcs}

for j to 2 do for i to 3 do a := Va[j]; alpha := Valpha[i]; dsol[j][i] := dsolve(dsys, numeric, continuation = lambda); print(alpha); print(a); print(dsol[j][i](0)) end do end do

Error, incorrect number of arguments to _Inert_FORFROM

"for j from 1 to 2 do   for i from 1 to 3 do a:=Va[j]; alpha:=Valpha[i];  dsol[j][i]:=dsolve(dsys,numeric, continuation=lambda);  print(alpha);    print(a);  print(dsol[j][i](0));  od  end"

 

NULL

 


 

Download 3DAKc2alpha.mw


 

restart

with(PDEtools)

with(plots)

P__r := .71; lambda := 1.0; K__r := 1.0; S__r := .5; m := .5; M := sqrt(10.0); `ϰ` := .5; Omega := sqrt(5.0); Gr := 6.0; Gm := 5.0; S__c := .22

PDE := {diff(phi(x, t), t) = (diff(phi(x, t), x, x))/S__c-K__r*phi(x, t)+S__r*(diff(theta(x, t), x, x)), diff(theta(x, t), t) = lambda*(diff(theta(x, t), x, x))/P__r, diff(u(x, t), t) = diff(u(x, t), x, x)-M^2*(u(x, t)-m*w(x, t))/(m^2+1)-u(x, t)/`ϰ`-2*Omega^2*w(x, t)+Gr*theta(x, t)+Gm*phi(x, t), diff(w(x, t), t) = diff(w(x, t), x, x)+M^2*(m*u(x, t)-w(x, t))/(m^2+1)-w(x, t)/`ϰ`+2*Omega^2*u(x, t)}

{diff(phi(x, t), t) = 4.545454545*(diff(diff(phi(x, t), x), x))-1.0*phi(x, t)+.5*(diff(diff(theta(x, t), x), x)), diff(theta(x, t), t) = 1.408450704*(diff(diff(theta(x, t), x), x)), diff(u(x, t), t) = diff(diff(u(x, t), x), x)-9.999999999*u(x, t)-5.999999996*w(x, t)+6.0*theta(x, t)+5.0*phi(x, t), diff(w(x, t), t) = diff(diff(w(x, t), x), x)+14.00000000*u(x, t)-9.999999999*w(x, t)}

(1)

``

IBC := {phi(0, t) = 1, phi(9, t) = 0, phi(x, 0) = 0, theta(0, t) = 1, theta(9, t) = 0, theta(x, 0) = 0, u(0, t) = t, u(9, t) = 0, u(x, 0) = 0, w(0, t) = 0, w(9, t) = 0, w(x, 0) = 0}

sol := pdsolve(PDE, IBC, numeric, spacestep = 0.1e-1)

_m2167514531200

(2)

p1 := sol:-plot(t = .3, color = red); p2 := sol:-plot(t = .5, color = gold); p3 := sol:-plot(t = .7, color = purple); p4 := sol:-plot(t = 1., color = green); plots[display]({p1, p2, p3, p4})

 

q1, q2, q3, q4 := seq(eval(u(x, t), sol:-value(t = t0, output = listprocedure)), t0 = [.3, .5, .7, 1]); plot([q1, q2, q3, q4], 0 .. 10, color = [red, gold, purple, green])

 

p1 := sol:-plot(t = 1, S__c = .1, color = red); p2 := sol:-plot(t = 1, S__c = .2, color = gold); p3 := sol:-plot(t = 1, S__c = .3, color = purple); p4 := sol:-plot(t = 1, S__c = .4, color = green); plots[display]({p1, p2, p3, p4})

Error, (in plot/options2d) unexpected option: .22 = .1

 

Error, (in plot/options2d) unexpected option: .22 = .2

 

Error, (in plot/options2d) unexpected option: .22 = .3

 

Error, (in plot/options2d) unexpected option: .22 = .4

 

 

q1, q2, q3, q4 := seq(eval(diff(u(0, t), t), sol:-value(t = t0, output = listprocedure)), t0 = [.3, .5, .7, 1]); plot([q1, q2, q3, q4], 0 .. 10, color = [red, gold, purple, green])

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

 

``


 

Download pde_baru.mwpde_baru.mw

Dear Prof DRs ,Please see the attachments

how to PLOT PDE IBCS for different  Sc , Pr, Gr, Gm at fixed t? Also for Nusselt (theta prime)  ,skin friction (f double prime)?


 

NULL

restart

with(plots)

Pr := 0.1e-1

0.1e-1

(1)

Eq1 := (101-100*d)*(diff(h(eta), eta))+2*f(eta) = 0; Eq2 := (101-100*d)*(diff(f(eta), `$`(eta, 2)))-h(eta)*(diff(f(eta), eta))-f(eta)^2+g(eta)^2-beta*(f(eta)+(1/2)*eta*(diff(f(eta), eta))) = 0; Eq3 := diff(g(eta), `$`(eta, 2))-h(eta)*(diff(g(eta), eta))-2*f(eta)*g(eta)-beta*(g(eta)+(1/2)*eta*(diff(g(eta), eta))) = 0; Eq4 := diff(p(eta), eta)-2*(diff(f(eta), eta))+beta*(h(eta)+eta*(diff(h(eta), eta))) = 0; Eq5 := diff(theta(eta), `$`(eta, 2))-Pr*h(eta)*(diff(theta(eta), eta))-Pr*beta*(3*theta(eta)+eta*(diff(theta(eta), eta))) = 0

(101-100*d)*(diff(h(eta), eta))+2*f(eta) = 0

 

(101-100*d)*(diff(diff(f(eta), eta), eta))-h(eta)*(diff(f(eta), eta))-f(eta)^2+g(eta)^2-beta*(f(eta)+(1/2)*eta*(diff(f(eta), eta))) = 0

 

diff(diff(g(eta), eta), eta)-h(eta)*(diff(g(eta), eta))-2*f(eta)*g(eta)-beta*(g(eta)+(1/2)*eta*(diff(g(eta), eta))) = 0

 

diff(p(eta), eta)-2*(diff(f(eta), eta))+beta*(h(eta)+eta*(diff(h(eta), eta))) = 0

 

diff(diff(theta(eta), eta), eta)-0.1e-1*h(eta)*(diff(theta(eta), eta))-0.1e-1*beta*(3*theta(eta)+eta*(diff(theta(eta), eta))) = 0

(2)

NULL

NULL

`Vλ` := [.5, 1, 1.5]; `Vβ` := [0.5e-1, .1, .2, .4, .7, 1, 1.50, 2]

etainf := 1

bcs := h(0) = 0, p(0) = 0, theta(0) = 1, (D(f))(0) = lambda*f(0)^(4/3)/(f(0)^2+(1-g(0))^2)^(1/3), (D(g))(0) = -lambda*f(0)^(1/3)*(1-g(0)).(1/(f(0)^2+(1-g(0))^2)^(1/3)), f(etainf) = 0, g(etainf) = 0, theta(etainf) = 0

h(0) = 0, p(0) = 0, theta(0) = 1, (D(f))(0) = lambda*f(0)^(4/3)/(f(0)^2+(1-g(0))^2)^(1/3), (D(g))(0) = -f(0)^(1/3)*(1-g(0))*lambda/(f(0)^2+(1-g(0))^2)^(1/3), f(1) = 0, g(1) = 0, theta(1) = 0

(3)

NULL

dsys := {Eq1, Eq2, Eq3, Eq4, Eq5, bcs}

for j to 3 do for i to 8 do lambda := `Vλ`[j]; beta := `Vβ`[i]; dsol[j][i] := dsolve(dsys, numeric, continuation = d); print(beta); print(lambda); print(dsol[j][i](0)) end do end do

Error, (in dsolve/numeric/bvp) singularity encountered

 

NULL

NULL

NULL


 

Download compre1try3.mw

Please help me

My Question

how may i know which is the theta prime or f double prime or theta only or f prime and theta double prime?

 i have attach double prime

tq maple
 

restart

with(student)

with(plots)

inf := 3

equ1 := (diff(f(eta), `$`(eta, 3)))/((1-`ϕ`)^2.5*(1-`ϕ`+`ϕ`*rho[s]/rho[f]))+(diff(f(eta), eta, eta))*f(eta)-(diff(f(eta), eta))^2+1+M*(1-(diff(f(eta), eta))) = 0

equ2 := (1+4/(3*N*k))*(diff(theta(eta), eta, eta))+Pr*(1-`ϕ`+`ϕ`*rho[s]*Cp[s]/(rho[f]*Cp[f]))*(diff(theta(eta), eta))*f(eta)/k+Br*(diff(f(eta), eta, eta))^2/(k*(1-`ϕ`)^2.5) = 0

Bcs := f(0) = 0, (D(f))(0) = `ε`, (D(f))(inf) = 1, theta(inf) = 0, theta(0) = 1

Pr := 6.2; Cp[s] := 385; Cp[f] := 4179; `ϕ` := .1; rho[f] := 997.1; rho[s] := 8933; k[s] := 400; k[f] := .613; k := (k[s]+2*k[f]-2*`ϕ`*(k[f]-k[s]))/(k[s]+2*k[f]+`ϕ`*(k[f]-k[s])); Br := .1; M := 1; N := 1

func := proc (v) options operator, arrow; rhs((dsolve({equ1, equ2, subs(`ε` = v, [Bcs])[]}, numeric))(0)[3])/(1-`ϕ`)^2.5 end proc; plot(func, -1 .. 1, title = typeset((diff(f(eta), eta, eta))*versus*`ε`/(1-'`ϕ`')^2.5), titlefont = [times, italic, 18])

 

func2 := proc (v) options operator, arrow; rhs((dsolve({equ1, equ2, subs(`ε` = v, [Bcs])[]}, numeric))(0)[5]) end proc; plot(func2, -1 .. 1, title = typeset((diff(theta(eta), eta))*versus*`ε`), titlefont = [times, italic, 18])

 

``


 

Download plot.mw
 

restart

with(student)

with(plots)

inf := 3

equ1 := (diff(f(eta), `$`(eta, 3)))/((1-`ϕ`)^2.5*(1-`ϕ`+`ϕ`*rho[s]/rho[f]))+(diff(f(eta), eta, eta))*f(eta)-(diff(f(eta), eta))^2+1+M*(1-(diff(f(eta), eta))) = 0

equ2 := (1+4/(3*N*k))*(diff(theta(eta), eta, eta))+Pr*(1-`ϕ`+`ϕ`*rho[s]*Cp[s]/(rho[f]*Cp[f]))*(diff(theta(eta), eta))*f(eta)/k+Br*(diff(f(eta), eta, eta))^2/(k*(1-`ϕ`)^2.5) = 0

Bcs := f(0) = 0, (D(f))(0) = `ε`, (D(f))(inf) = 1, theta(inf) = 0, theta(0) = 1

Pr := 6.2; Cp[s] := 385; Cp[f] := 4179; `ϕ` := .1; rho[f] := 997.1; rho[s] := 8933; k[s] := 400; k[f] := .613; k := (k[s]+2*k[f]-2*`ϕ`*(k[f]-k[s]))/(k[s]+2*k[f]+`ϕ`*(k[f]-k[s])); Br := .1; M := 1; N := 1

func := proc (v) options operator, arrow; rhs((dsolve({equ1, equ2, subs(`ε` = v, [Bcs])[]}, numeric))(0)[3])/(1-`ϕ`)^2.5 end proc; plot(func, -1 .. 1, title = typeset((diff(f(eta), eta, eta))*versus*`ε`/(1-'`ϕ`')^2.5), titlefont = [times, italic, 18])

 

func2 := proc (v) options operator, arrow; rhs((dsolve({equ1, equ2, subs(`ε` = v, [Bcs])[]}, numeric))(0)[5]) end proc; plot(func2, -1 .. 1, title = typeset((diff(theta(eta), eta))*versus*`ε`), titlefont = [times, italic, 18])

 

``


 

Download plot.mw

 

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