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I want to solve system of non linear odes numerically.

I encounter following error

Error, (in dsolve/numeric/bvp) cannot determine a suitable initial profile, please specify an approximate initial solution

how to correct it

regards

i want to solve these two coupled eqaut with finite boundary conditionsions. Can some one help me

eq1:=diff(f(eta),eta,eta,eta,eta)+2*(epsilon/(1+epsilon*theta(eta)))^2*diff(f(eta),eta,eta)*((diff(theta(eta),eta))^2)-(epsilon/(1+epsilon*theta(eta)))*(diff(theta(eta),eta,eta))*(diff(f(eta),eta,eta))=0;

eq2:=diff(theta(eta),eta,eta)+Pr*Re*f(eta)*diff(theta(eta),eta)=0;

Re:=1:Pr:=1:epsilon:=0.25:

bc:=f(1)=0,D(f)(-1)=0,D(f)(1)=1,D(f)(-1)=1,theta(-1)=0,theta(1)=0;

 

Hi,

 

I want to solve the Falkner-Skan equation numbercally using maple. The Falkner-Skan equation is 

f′′′ + ff′′ + (1 − (f′)^2)=0 ,

with subject to the boundary conditions,

f(0) =0 ; f′(0) = 0,

f′(∞) = 1. 

could you  please help me to have loop to solve this problem from t=0..30.

And then save the data in DATfile in order to plot using Gnuplot?

 

Regards

 

Hai everyone

may i ask why solution have an error?

hope i have an answer

r

 

NULL

restart

with(plots):

Pr := 6.8:

Eq1 := (101-100*lambda)*(diff(f(eta), `$`(eta, 3)))+f(eta)*(diff(f(eta), `$`(eta, 2)))+2*delta*theta(eta)+2*delta*Nc*gamma(eta)-2*delta*Nr*phi(eta);

(101-100*lambda)*(diff(diff(diff(f(eta), eta), eta), eta))+f(eta)*(diff(diff(f(eta), eta), eta))+2*theta(eta)+2*gamma(eta)-2*phi(eta)

(1)

Eq2 := (101-100*lambda)*(diff(theta(eta), `$`(eta, 2)))+Pr*f(eta)*(diff(theta(eta), eta))+Pr*Nb*(diff(theta(eta), eta))*(diff(phi(eta), eta))+Pr*Nt*(diff(theta(eta), eta))^2;

(101-100*lambda)*(diff(diff(theta(eta), eta), eta))+6.8*f(eta)*(diff(theta(eta), eta))+3.40*(diff(theta(eta), eta))*(diff(phi(eta), eta))+3.40*(diff(theta(eta), eta))^2

(2)

Eq3 := (101-100*lambda)*(diff(phi(eta), `$`(eta, 2)))+Le*f(eta)*(diff(phi(eta), eta))+Nt*(diff(theta(eta), `$`(eta, 2)))/Nb;

(101-100*lambda)*(diff(diff(phi(eta), eta), eta))+.1*f(eta)*(diff(phi(eta), eta))+1.000000000*(diff(diff(theta(eta), eta), eta))

(3)

Eq4 := (101-100*lambda)*(diff(gamma(eta), `$`(eta, 2)))+Sc*s*(diff(theta(eta), `$`(eta, 2)))+Sc*f(eta)*(diff(gamma(eta), eta));

(101-100*lambda)*(diff(diff(gamma(eta), eta), eta))+.30*(diff(diff(theta(eta), eta), eta))+.6*f(eta)*(diff(gamma(eta), eta))

(4)

VBi := [10, 20, 30]:

etainf := 5:

bcs := f(0) = 0, (D(f))(0) = 0, (D(theta))(0) = -Bi*(1-theta(0)), phi(0) = 1, gamma(0) = 1, (D(f))(etainf) = 1, theta(etainf) = 0, phi(etainf) = 0, gamma(etainf) = 0;

f(0) = 0, (D(f))(0) = 0, (D(theta))(0) = -Bi*(1-theta(0)), phi(0) = 1, gamma(0) = 1, (D(f))(5) = 1, theta(5) = 0, phi(5) = 0, gamma(5) = 0

(5)

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

for i to 3 do Bi := VBi[i]; dsol[i] := dsolve(dsys, numeric, continuation = lambda); print(Bi); print(dsol[i](0)) end do

Error, (in dsolve/numeric/bvp) cannot determine a suitable initial profile, please specify an approximate initial solution

 

NULL

NULL

 

Download soret.mw

 

 

hi.please remove error in attached file

thanks...gfhf.mw

using FDM or FEM rather than dsolve?

ode.docxode.docx

Hi, Im now trying to run my code. But it took like years to even getting the results. may I know any solutions on how to get faster results? Because I have run this code for almost 4 hours yet there is still 'Evaluating...' at the corner left. And when I tried to stop the program, it will stop at 'R1...'.

 

Digits := 18;
with(plots):n:=1.4: mu(eta):=(diff(U(eta),eta)^(2)+diff(V(eta),eta)^(2))^((n-1)/(2)):
Eqn1 := 2*U(eta)+(1-n)*eta*(diff(U(eta), eta))/(n+1)+diff(W(eta), eta) = 0;
Eqn2 := U(eta)^2-(V(eta)+1)^2+(W(eta)+(1-n)*eta*U(eta)/(n+1))*(diff(U(eta), eta))-mu(eta)*(diff(U(eta), eta, eta))-(diff(U(eta), eta))*(diff(mu(eta), eta)) = 0;
Eqn3 := 2*U(eta)*(V(eta)+1)+(W(eta)+(1-n)*eta*U(eta)/(n+1))*(diff(V(eta), eta))-mu(eta)*(diff(V(eta), eta, eta))-(diff(V(eta), eta))*(diff(mu(eta), eta)) = 0;
bcs1 := U(0) = 0, V(0) = 0, W(0) = 0;
bcs2 := U(4) = 0, V(4) = -1;
R1 := dsolve({Eqn1, Eqn2, Eqn3, bcs1, bcs2}, {U(eta), V(eta), W(eta)}, initmesh = 30000, output = listprocedure, numeric);
Warning, computation interrupted
for l from 0 by 2 to 4 do R1(l) end do;
plot1 := odeplot(R1, [eta, U(eta)], 0 .. 4, numpoints = 2000, color = red);

 

Thankyou in advance :)

how can i improve doing the graph with various parameter

do anyone have abother numerical method in maple rather than rk45 felhberg

like keller box/homotopy/ or anything

i have attchassignment.mwsassignment.pdf

restart

with(plots)

%?

Eq1 := diff(f(eta), `$`(eta, 3))+(diff(f(eta), `$`(eta, 2)))*f(eta)-(diff(f(eta), eta))^2+4 = 0

diff(diff(diff(f(eta), eta), eta), eta)+(diff(diff(f(eta), eta), eta))*f(eta)-(diff(f(eta), eta))^2+4 = 0

(1)

%?

Eq2 := diff(theta(eta), `$`(eta, 2))+Pr*(diff(theta(eta), eta))*f(eta) = 0

diff(diff(theta(eta), eta), eta)+Pr*(diff(theta(eta), eta))*f(eta) = 0

(2)

%?

VPr := [0.1e-1, 0.2e-1, 0.3e-1]

etainf := 27

bcs := (D(f))(0) = 0, f(0) = 0, (D(theta))(0) = -1, (D(f))(etainf) = 2, theta(etainf) = 0

(D(f))(0) = 0, f(0) = 0, (D(theta))(0) = -1, (D(f))(27) = 2, theta(27) = 0

(3)

dsys := {Eq1, Eq2, bcs}

for i to 3 do Pr := VPr[i]; dsol[i] := dsolve(dsys, numeric); print(Pr); print(dsol[i](0)) end do

0.1e-1

[eta = 0., f(eta) = HFloat(0.0), diff(f(eta), eta) = HFloat(0.0), diff(diff(f(eta), eta), eta) = HFloat(3.4862842650940435), theta(eta) = HFloat(9.305856096452466), diff(theta(eta), eta) = HFloat(-0.9999999999999998)]

0.2e-1

[eta = 0., f(eta) = HFloat(0.0), diff(f(eta), eta) = HFloat(0.0), diff(diff(f(eta), eta), eta) = HFloat(3.4862842653392216), theta(eta) = HFloat(6.7064688361073745), diff(theta(eta), eta) = HFloat(-1.0)]

0.3e-1

[eta = 0., f(eta) = HFloat(0.0), diff(f(eta), eta) = HFloat(0.0), diff(diff(f(eta), eta), eta) = HFloat(3.4862842648459234), theta(eta) = HFloat(5.552583608770695), diff(theta(eta), eta) = HFloat(-0.9999999999999998)]

(4)

SDf1 := odeplot(dsol[1], [eta, f(eta)], 0 .. etainf, color = green, axes = box); SDf2 := odeplot(dsol[2], [eta, f(eta)], 0 .. etainf, color = red); SDf3 := odeplot(dsol[3], [eta, f(eta)], 0 .. etainf, color = blue)

display([SDf1, SDf2, SDf3], labels = ["η", "f (η)"], labeldirections = [horizontal, vertical], labelfont = [italic, 16, bold], axes = boxed, axesfont = [times, 14], thickness = 3)
%?

 

%?

SDfd1 := odeplot(dsol[1], [eta, diff(f(eta), eta)], 0 .. etainf, color = green, axes = box); SDfd2 := odeplot(dsol[2], [eta, diff(f(eta), eta)], 0 .. etainf, color = red); SDfd3 := odeplot(dsol[3], [eta, diff(f(eta), eta)], 0 .. etainf, color = blue)

%?

display([SDfd1, SDfd2, SDfd3], labels = ["η", "f  ' (η)"], labeldirections = [horizontal, vertical], labelfont = [italic, 16, bold], axes = boxed, axesfont = [times, 14], thickness = 3)

 

%?

`Sθ1` := odeplot(dsol[1], [eta, theta(eta)], 0 .. etainf, color = green, axes = box); `Sθ2` := odeplot(dsol[2], [eta, theta(eta)], 0 .. etainf, color = red); `Sθ3` := odeplot(dsol[3], [eta, theta(eta)], 0 .. etainf, color = black)

display([`Sθ1`, `Sθ2`, `Sθ3`], labels = ["η", "θ (η)"], labeldirections = [horizontal, vertical], labelfont = [italic, 16, bold], axes = boxed, axesfont = [times, 14], thickness = 3)

 

%?

`Sθd1` := odeplot(dsol[1], [eta, diff(theta(eta), eta)], 0 .. etainf, color = green, axes = box); `Sθd2` := odeplot(dsol[2], [eta, diff(theta(eta), eta)], 0 .. etainf, color = red); `Sθd3` := odeplot(dsol[3], [eta, diff(theta(eta), eta)], 0 .. etainf, color = black)

display([`Sθd1`, `Sθd2`, `Sθd3`], labels = ["η", "θ '(η)"], labeldirections = [horizontal, vertical], labelfont = [italic, 16, bold], axes = boxed, axesfont = [times, 14], thickness = 3)

 

%?

 

 

Download assignment.mws

 

Hi. I want to solve this system of equations by varying the value of n. I managed to aolve and plot for n=1, 1.1 and 1.2 but it happens to be a problem when I let n=1.3.

>restart;

>Digits := 15;

>with(plots):n:=1.3: mu(eta):=(diff(U(eta),eta)^(2)+diff(V(eta),eta)^(2))^((n-1)/(2)):

>Eqn1 := 2*U(eta)+(1-n)*eta*(diff(U(eta), eta))/(n+1)+diff(W(eta), eta) = 0;

>Eqn2 := U(eta)^2-(V(eta)+1)^2+(W(eta)+(1-n)*eta*U(eta)/(n+1))*(diff(U(eta), eta))-mu(eta)*(diff(U(eta), eta, eta))-(diff(U(eta), eta))*(diff(mu(eta), eta)) = 0;

>Eqn3 := 2*U(eta)*(V(eta)+1)+(W(eta)+(1-n)*eta*U(eta)/(n+1))*(diff(V(eta), eta))-mu(eta)*(diff(V(eta), eta, eta))-(diff(V(eta), eta))*(diff(mu(eta), eta)) = 0;

>bcs1 := U(0) = 0, V(0) = 0, W(0) = 0;

>bcs2 := U(20) = 0, V(20) = -1;

>R1 := dsolve({Eqn1, Eqn2, Eqn3, bcs1, bcs2}, {U(eta), V(eta), W(eta)}, initmesh = 30000, output = listprocedure, numeric);


Error, (in dsolve/numeric/bvp) precision is insufficient for required absolute error, suggest increasing Digits to approximately 23 for this problem

>for l from 0 by 2 to 20 do R1(l) end do;
>plot1 := odeplot(R1, [eta, U(eta)], 0 .. 20, numpoints = 2000, color = red);


Error, (in plots/odeplot) input is not a valid dsolve/numeric solution

 

I have tried increasing the Digits as suggested to 23, 25, 30, 31 up untill 500 yet still same error occur suggesting to increase the Digits. Is there any other way to solve this kind of error? Can someone help me? Thank you in advance.

 

Hi. I want to solve a system of equations. But I got this type of error. 

>restart;

>Digits := 15;
>with(plots):n:=0.7:Pr=1: mu(eta):=(diff(U(eta),eta)^(2)+diff(V(eta),eta)^(2))^((n-1)/(2)):
>Eqn1 := 2*U(eta)+(1-n)*eta*(diff(U(eta), eta))/(n+1)+diff(W(eta), eta) = 0:
>Eqn2 := U(eta)^2-(V(eta)+1)^2+(W(eta)+(1-n)*eta*U(eta)/(n+1))*(diff(U(eta), eta))-mu(eta)*(diff(U(eta), eta, eta))-(diff(U(eta), eta))*(diff(mu(eta), eta)) = 0:
>Eqn3 := 2*U(eta)*(V(eta)+1)+(W(eta)+(1-n)*eta*U(eta)/(n+1))*(diff(V(eta), eta))-mu(eta)*(diff(V(eta), eta, eta))-(diff(V(eta), eta))*(diff(mu(eta), eta)) = 0:
>Eqn4 := (W(eta)+(1-n)*eta*U(eta)/(n+1))*(diff(theta(eta), eta))-(mu(eta)*(diff(theta(eta), eta, eta))+(diff(mu(eta), eta))*(diff(theta(eta), eta)))/Pr = 0:
>bcs1 := U(0) = 0, V(0) = 0, W(0) = 0, theta(0) = 1:
>bcs2 := U(20) = 0, V(20) = -1, theta(20) = 0:
>R1 := dsolve({Eqn1, Eqn2, Eqn3, Eqn4, bcs1, bcs2}, {U(eta), V(eta), W(eta), theta(eta)}, initmesh = 20000, output = listprocedure, numeric);

Error, (in dsolve/numeric/bvp/convertsys) too few boundary conditions: expected 8, got 7

>for l from 0 by 2 to 20 do R1(l) end do;
>plot1 := odeplot(R1, [eta, theta(eta)], 0 .. 20, numpoints = 2000, color = red);

 

What is the problem actually because based on the paper that I refer to, there is only 7 bc. 

Can anyone help me?

Thankyou in advance.

Hello Everybody,

I was trying to apply the Newton-Raphon method("Newton" in maple) for the following problem to obtain the constants (c1...cM) without success.

eq[1] := -0.0139687 c[2] - 0.0132951 c[1] = 0
eq[2] := 24806.4 c[2] - 0.0139687 c[1] = 0

If anybody has experience with it or knows how to use it I would really appreciate your help.

 

Just to introduce the previous steps of calculations leading to the problem:To calculate the critical buckling force N and the shape of a rectangular uniformly loaded plate the governing diff. equation is the following

D11*w''''(x)+(2*D12+4*D66)*w''(x)''(y)+D22*w''''(y)+N*w''(x)=0

For the follwing solution of a boundary value problem(Boundary conditions:clamped-clamped: w(x=0,x=a)=0 & w'(x=0,x=a)=0) I applied the Ritz method:

w:=sum(c[i]*(cos(2*i*Pi*x/a)-1)),i=1..M)

Thus the Potential Energy P is:

P:=16537.6*c[2]^2-0.0139687*c[2]*c[1]+1033,60*c[1]^2-9.86960*N*c[2]^2-2.46740*N*c[1]^2

 deriving P to each constant c and setting them =0 leads to:

eq[1] := -0.0139687 c[2] + 2067.20 c[1] - 4.93480 N c[1] = 0
eq[2] := 33075.2 c[2] - 0.0139687 c[1] - 19.7392 N c[2] = 0

After calculating N=418,902  and feeding eq1 and eq 2,the follwing equations if two terms are considered:

eq[1] := -0.0139687 c[2] - 0.0132951 c[1] = 0
eq[2] := 24806.4 c[2] - 0.0139687 c[1] = 0

Everything I tried resulted in any c1 = c2 = 0 which is not  realistic. Maybe I made a mistake earlier.

Thanks a lot in advance.

Sam

 Plate_Buckling.mw

Hello,

I have two regimes. Each regime is characterized by system of state differential equations (diff(S(t), t), diff(K(t), t)) and co-state differential equations (diff(psi[S](t), t), diff(psi[Iota](t), t)) as follows: 

(1)

diff(S(t), t) = -eta*K(t)*S(t)/(w*N*(S(t)+K(t))), diff(K(t), t) = eta*K(t)*S(t)/(w*N*(S(t)+K(t)))-upsilon

diff(psi[S](t), t) = eta*K(t)^2*(psi[S](t)-psi[Iota](t))/(w*N*(S(t)+K(t))^2), diff(psi[Iota](t), t) = eta*S(t)^2*(psi[S](t)-psi[Iota](t))/(w*N*(S(t)+K(t))^2)

(2)

diff(S(t), t) = -eta*K(t)*S(t)/(w*N*(S(t)+K(t))), diff(K(t), t) = eta*K(t)*S(t)/(w*N*(S(t)+K(t)))

diff(psi[S](t), t) = eta*K(t)^2*(psi[S](t)-psi[Iota](t))/(w*N*(S(t)+K(t))^2), diff(psi[Iota](t), t) = eta*S(t)^2*(psi[S](t)-psi[Iota](t))/(w*N*(S(t)+K(t))^2)

The first regime is employed from 0 to t1 (where t1 is unknown) and then regime 2, from t1 to T. I know the initial values for the state variables of the first system at t=0, that is, S(0)=S0 and K(0)=K0, as well as boundary conditions for the co-state variables for regime 2 at t=T, that is, psi_S(T)=a and psi_I(T)=b. 

I also know that my unknown t1 should satisfy the algebraic equation: -c - psi_S(t1)=0, where psi_S(t1) is the solution of the co-state diff equation of the regime 2 at t=t1. 

My question is: If I assume that the systems have not analytical solution, how can I found unknown t1 numerically? Moreover, asssume that all other parameters, such as T, eta, upsilon and others are given.

i have this problem -> f'^2 -ff''=f'''-k1(2f'f'''-ff''''-f''^2)+Ha^2(E1-f') with boundary conditions f(0)=0, f'(0)=1, f'(∞)=0.

since it is a fourth order equation, but only three bcs, it does not produce unique solution. so the solution of the equation may be seek in form of f=f0(eta)+k1f1(eta).

thus the equation will become 

f0'^2-f0f0''=f0'''+Ha^2(E1-f0')

and

f1'''-Ha^2f1'-2f0'f1'+f0f1''+f1f0''=2f0'f0'''-f0f0''''-f0''^2.

boundary conditions are 

f0(0)=0,f0'(0)=1,f0'(∞)=0

f1(0)=0,f1'(0)=0,f1'(∞)=0.

i had been clueless in solving this problem. please somebody help me with this problem.

Hello maple experts,

please is there any other way of solving this problem Wht.mw

Some complained that the graphs do not satisfy the boundary conditions smoothly.

Thanks.

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