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KIRAN SAJJAN

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Can we solve nonlinear coupled Ode with...

KIRAN SAJJAN 35 Maple 2018
ode homework dsolve
February 16 2023
0 4

restart;

OdeSys := diff(U(Y), Y, Y)+Theta(Y)+N*(Theta(Y)*Theta(Y))-(M*M)*U(Y) = 0, diff(Theta(Y), Y, Y)+E*(diff(U(Y), Y))^2 = 0;

Cond := U(0) = lambda*(D(U))(0), Theta(0) = A+g*(D(Theta))(0), U(1) = 0, Theta(1) = B; sys := [OdeSys, Cond];
Ans := dsolve(sys);

To find the table values of Nusselt and ...

KIRAN SAJJAN 35 Maple 2018
February 10 2023
1 0

odeSys := {diff(Theta(x), x, x)+Pr*(R*(diff(Theta(x), x))*f(x)+Nb*(diff(Theta(x), x))*(diff(Phi(x), x))+Nt*(diff(Theta(x), x))^2), N2*(diff(G(x), x, x))-N1*(2*G(x)+diff(f(x), x, x))-N3*R*((diff(f(x), x))*G(x)-f(x)*(diff(G(x), x))), diff(Phi(x), x, x)+R*Sc*f(x)*(diff(Phi(x), x))+Nt*(diff(Theta(x), x, x))/Nb, (1+N1)*(diff(g(x), x, x))+R*((diff(g(x), x))*f(x)-g(x)*(diff(f(x), x)))-M*g(x)+2*Kr*(diff(f(x), x)), (1+N1)*(diff(f(x), x, x, x, x))-R*((diff(f(x), x))*(diff(f(x), x, x))-f(x)*(diff(f(x), x, x, x)))+N1*(diff(G(x), x, x))-M*(diff(f(x), x, x))-2*Kr*(diff(g(x), x))}; cond := f(0) = 0, (D(f))(0) = 1, g(0) = 0, Theta(0) = 1, G(0) = -n*((D@@2)(f))(0), Phi(0) = 1, f(1) = lambda, (D(f))(1) = 0, g(1) = 0, Theta(1) = 0, G(1) = n*((D@@2)(f))(1), Phi(1) = 0; ans := {};

n := .5; N1 := 0.; N2 := 1.0; N3 := .1; lambda := .1; M := .1; Kr := .1; Sc := 1.0; Nb := .1; Pr := 1.0; Nt := .1; R := .5;

ans := dsolve*{cond, eval*odeSys};

hello these are the pde and Boundary conditions  i want to calculate the value of f''(0) ,Theta(0) and  Phi(0)

what is the proper cammand to get the table values for the given equation
NBVs := [eval(ans(N1*G(x)+(1+N1)*(diff(f(x), x, x))), x = 0), eval(ans(-(diff(Theta(x), x))), x = 0), eval(ans(-(diff(Phi(x), x))), x = 0)];

FEM or FDM for the given pde can i calc...

KIRAN SAJJAN 35 Maple 2018
February 07 2023
0 1

Hi,

I want to solve system of PDE equations by maple and i dont know how can i write it codes that can solve them for me. Can you create the code for the equation

Thank you

Error, (in dsolve/numeric/bvp/convertsys...

KIRAN SAJJAN 35 Maple 2018
homework dsolve numeric
February 05 2023
1 3
> 

restart:
> 

Digits:= trunc(evalhf(Digits)); #generally a very efficient setting

15

 (1)

Setup of BVP system:

> 

#ordinary differential equations:
ODEs:= [
   #Eq 1:
   A1*(diff(f(x), x, x, x))/(A2*phi)-(diff(f(x), x))^2-M^2*(f(x))+f(x)*(diff(f(x), x, x)),

   #Eq 2:
   A4*Pr*phi*(diff(Theta(x), x, x))/A3+f(x)*(diff(Theta(x), x))+Q*Theta(x)
   
   #All these ODEs are implicitly equated to 0.
]:

<ODEs[]>; #Display the ODEs.

Vector(2, {(1) = A1*(diff(diff(diff(f(x), x), x), x))/(A2*phi)-(diff(f(x), x))^2-M^2*f(x)+f(x)*(diff(diff(f(x), x), x)), (2) = A4*Pr*phi*(diff(diff(Theta(x), x), x))/A3+f(x)*(diff(Theta(x), x))+Q*Theta(x)})

 (2)
> 

Params := Record(fw = .2, M = .5, Q = .5, Pr = 6.2, phi = 0.5e-1, rf = 997.1, kf = .613, cpf = 4179, btf = 0.3e-4, p1 = 0.1e-1, p2 = 0.5e-1, p3 = 0.5e-1, rs1 = 5100, ks1 = 3007.4, cps1 = 410, bs1 = 0.2e-3, rs2 = 2200, ks2 = 5000, cps2 = 790, bs2 = 0.5e-3, rs3 = 3970, ks3 = 40, cps3 = 765, bs3 = 0.4e-3, A1 = B1*p1+B2*p2+B3*p3, B1 = 1+2.5*phi+6.2*phi^2, B2 = 1+13.5*phi+904.4*phi^2, B3 = 1+37.1*phi+612.6*phi^2, B4 = (ks1+2*kf-2*phi*(kf-ks1))/(ks1+2*kf+phi*(kf-ks1)), B5 = (ks2+3.9*kf-3.9*phi*(kf-ks2))/(ks2+3.9*kf+phi*(kf-ks2)), B6 = (ks3+4.7*kf-4.7*phi*(kf-ks3))/(ks3+4.7*kf+phi*(kf-ks3)), A2 = 1-p1-p2-p3+p1*rs1/rf+p2*rs2/rf+p3*rs3/rf, A3 = B4*p1+B5*p2+B6*p3, A4 = 1-p1-p2-p3+p1*rs1*cps1/(rf*cpf)+p2*rs2*cps2/(rf*cpf)+p3*rs3*cps3/(rf*cpf))

Record(fw = .2, M = .5, Q = .5, Pr = 6.2, phi = 0.5e-1, rf = 997.1, kf = .613, cpf = 4179, btf = 0.3e-4, p1 = 0.1e-1, p2 = 0.5e-1, p3 = 0.5e-1, rs1 = 5100, ks1 = 3007.4, cps1 = 410, bs1 = 0.2e-3, rs2 = 2200, ks2 = 5000, cps2 = 790, bs2 = 0.5e-3, rs3 = 3970, ks3 = 40, cps3 = 765, bs3 = 0.4e-3, A1 = B1*p1+B2*p2+B3*p3, B1 = 1+2.5*phi+6.2*phi^2, B2 = 1+13.5*phi+904.4*phi^2, B3 = 1+37.1*phi+612.6*phi^2, B4 = (ks1+2*kf-2*phi*(kf-ks1))/(ks1+2*kf+phi*(kf-ks1)), B5 = (ks2+3.9*kf-3.9*phi*(kf-ks2))/(ks2+3.9*kf+phi*(kf-ks2)), B6 = (ks3+4.7*kf-4.7*phi*(kf-ks3))/(ks3+4.7*kf+phi*(kf-ks3)), A2 = 1-p1-p2-p3+p1*rs1/rf+p2*rs2/rf+p3*rs3/rf, A3 = B4*p1+B5*p2+B6*p3, A4 = 1-p1-p2-p3+p1*rs1*cps1/(rf*cpf)+p2*rs2*cps2/(rf*cpf)+p3*rs3*cps3/(rf*cpf))

 (3)
> 

LB, UB := 0, 1; BCs := [`~`[`=`](([f(x), diff(f(x), x), Theta])(LB), [fw, 1, 1])[], `~`[`=`](([diff(f(x), x), Theta])(UB), [0, 0])[]]

[(f(x))(0) = fw, (diff(f(x), x))(0) = 1, Theta(0) = 1, (diff(f(x), x))(1) = 0, Theta(1) = 0]

 (4)
> 

NBVs := [A1*(diff(f(x), x, x))(0) = C*`*f`, -A4*(diff(Theta(x), x))(0) = `Nu*`]; Nu := `Nu*`; Cf := `C*__f`; x0 := Array([LB])
> 

NULL
> 

Solve := module () local nbvs_rhs, Sol, Dsolve, ModuleApply, AccumData, ModuleLoad; export SavedData, Pos, Init; nbvs_rhs := `~`[rhs](:-NBVs); Dsolve := proc (Sys, Params::(set(name = realcons))) option remember; Sol := dsolve(Sys, _rest, 'numeric'); AccumData(Params); eval(Sol) end proc; ModuleApply := subs(_Sys = {:-BCs[], :-NBVs[], :-ODEs[]}, proc ({ fw::realcons := Params:-fw, Pr::realcons := Params:-Pr, M::realcons := Params:-M, Q::realcons := Params:-Q, phi::realcons := Params:-phi }) Dsolve(_Sys, {_options}, {_rest}[]) end proc); AccumData := proc (params::(set(name = realcons))) local n, nbvs; if Sol::rtable then nbvs := seq(n = Sol[2, 1][1, Pos(n)], n = nbvs_rhs) else nbvs := `~`[`=`](nbvs_rhs, eval(nbvs_rhs, Sol(:-LB)))[] end if; SavedData[params] := Record[packed](params[], nbvs); return end proc; ModuleLoad := eval(Init); Init := proc () Pos := proc (n::name) local p; option remember; member(n, Sol[1, 1], 'p'); p end proc; SavedData := table(); return end proc; ModuleLoad() end module

NULL

> 

colseq := [red, green, blue, brown]
> 

#parameter values that remain fixed for the entire set of plots:
Pc:= phi=0.05:
 
> 

#parameter values that remain fixed with each of the four plots::
Ps:= [
   [fw=0.2, Pr=6.2, M=0.5],
   [fw=0.2, Q=0.3, M=0.5],
   [fw=0.2, Pr=6.2, Q=0.3],
   [Q=0.3, Pr=6.2, M=0.5]
]:

#parameter value for each curve
Pv:= [
   Q=[0.2, 0.4, 0.6, 0.8],
   Pr=[0.7, 1.4, 2.1, 2.8],
   M=[0.6, 1.2, 1.8, 2.4],
   fw=[1, 2, 3, 4]
]:
      
> 

for i to nops(Ps) do
   plots:-display(
      [seq(
         plots:-odeplot(
            Solve(lhs(Pv[i])= rhs(Pv[i])[j], Ps[i][], Pc),
            [x, Theta(x)], 'color'= colseq[j], 'legend'= [lhs(Pv[i])= rhs(Pv[i])[j]]
         ),
         j= 1..nops(rhs(Pv[i]))
      )],
      'axes'= 'boxed', 'gridlines'= false,
      'labelfont'= ['TIMES', 'BOLDOBLIQUE', 16],
      'caption'= nprintf(
         cat("\n%a = %4.2f, "$nops(Ps[i])-1, "%a = %4.2f\n\n"), (lhs,rhs)~(Ps[i])[]
      ),
      'captionfont'= ['TIMES', 16]
   )
od;

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

  
> 

for i to nops(Ps) do plots:-display([seq(plots:-odeplot(Solve(lhs(Pv[i]) = rhs(Pv[i])[j], Ps[i][], Pc), [x, D(f(x))], 'color' = colseq[j], 'legend' = [lhs(Pv[i]) = rhs(Pv[i])[j]]), j = 1 .. nops(rhs(Pv[i])))], 'axes' = 'boxed', 'gridlines' = false, 'labelfont' = ['TIMES', 'BOLDOBLIQUE', 16], 'caption' = nprintf(cat(`$`("\n%a = %4.2f, ", nops(Ps[i])-1), "%a = %4.2f\n\n"), `~`[lhs, rhs](Ps[i])[]), 'captionfont' = ['TIMES', 16]) end do

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

  
> 

ParamPlot2d := proc (Y::{`module`, procedure}, X::(name = range(realcons)), FP::(list(name = realcons)), { dsolveopts::(list({name, name = anything})) := [] }) plot(proc (x) options operator, arrow; Y(Solve(lhs(X) = x, FP[], 'abserr' = 0.5e-4, 'interpolant' = false, 'output' = x0, dsolveopts[])) end proc, rhs(X), 'numpoints' = 25, 'axes' = 'boxed', 'gridlines' = false, 'labelfont' = ['TIMES', 'BOLDOBLIQUE', 16], 'caption' = nprintf(cat(`$`("%a = %4.2f, ", nops(FP)-1), "%a = %4.2f"), `~`[lhs, rhs](FP)[]), 'captionfont' = ['TIMES', 16], _rest) end proc
> 

#procedure that extracts Nusselt number from dsolve solution:
GetNu:= (Sol::Matrix)-> Sol[2,1][1, Solve:-Pos(:-Nu)]:

> 

Q:= [0.2, 0.4, 0.6]:
plots:-display(
   [seq(
      ParamPlot2d(
         GetNu, fw= 1..4, [M= 0.5],
         'dsolveopts'= [Q= Q[k], Pr=6.2,  phi=0.05],
         'legend'= [Q= Q[k]], 'color'= colseq[k], 'labels'= [fw, Nu]
      ),
      k= 1..nops(Q)
   )]
);

Error, invalid input: ParamPlot2d expects value for keyword parameter dsolveopts to be of type list({name, name = anything}), but received [[.2, .4, .6] = .2, Pr = 6.2, phi = 0.5e-1]

  
> 

NULL

Download surface_dinesh_paper.mw  please help me to solve the problem

How do I solve a nonlinear coupled diffe...

KIRAN SAJJAN 35 Maple 2018
February 01 2023
1 6

restart:
PDEtools[declare](f(x), prime = x):
PDEtools[declare](Theta(x), prime = x):
PDEtools[declare](Phi(x), prime = x):
N := 4; M := .1; Kp := .1; Gr := 0.1e-1; Gc := 0.1e-1; Pr := 1; S := 0.1e-1; Sc := .78; Kc := 0.1e-1; La := 1
f (x):=  sum((p^(i))*f [i] (x), i = 0 .. N) ;
Theta(x):=  sum((p^(i))*Theta[i] (x), i = 0 .. N) ;
Phi(x):= sum((p^(i))*Phi [i] (x), i = 0 .. N);
HPMEq1 := (1-p)*(diff(f(x), x, x, x))+p*(diff(f(x), x, x, x)+(1/2)*(diff(f(x), x, x))*f(x)-(M^2+Kp)*(diff(f(x), x)-La)+Gr*Theta(x)+Gc*Phi(x))
HPMEq2 := (1-p)*(diff(Theta(x), x, x))/Pr+p*((diff(Theta(x), x, x))/Pr+(1/2)*(diff(Theta(x), x))*f(x)+S*Theta(x))
HPMEq3 := (1-p)*(diff(Phi(x), x, x))/Sc+p*((diff(Phi(x), x, x))/Sc+(1/2)*(diff(Phi(x), x))*f(x)+Kc*Phi(x))
for i from 0 to N do equ[1][i] := coeff(HPMEq1, p, i) = 0 end do
for i from 0 to N do equ[1][i] := coeff(HPMEq2, p, i) = 0 end do
for i from 0 to N do equ[1][i] := coeff(HPMEq3, p, i) = 0 end do
cond[1][0] := f[0](0) = 0, (D(f[0]))(0) = 0, Theta[0](0) = 1, Phi[0](0) = 1, Theta[0](5) = 0, Phi[0](5) = 0, (D(f[0]))(5) = 1; for j to N do cond[1][j] := f[j](0) = 0, (D(f[j]))(0) = 0, Theta[j](0) = 0, Phi[j](0) = 0, Theta[j](5) = 0, Phi[j](5) = 0, (D(f[j]))(5) = 0 end do
for i from 0 to N do pdsolve({cond[1][i], equ[1][i]}, f[i](x)); f[i](x) := rhs(%) end do
f(x) := evalf(simplify(sum(f[n](x), n = 0 .. N))); convert(f(x), 'rational'); subs(x = 1, diff(f(x), x))

Please rectify the error
Thank you

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