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teliko.mwMapl_doc.docxHello,

After I have substitute all the variables by hand ,maple still continious  processing(27 hours and still evaluating)for a simple

summarisation by the time that live math2 needed only 2 mins.Have i done something wrong?I want to solve the following equation which is shown in the attached file with the experimental data

I actually had the same problem on Maple 16. When i go to plot an equation (usually on implicit plots) it just repeats the equation in blue text rather than showing any sort of plot. Im sure i am just missing a simple option somewhere, but for the life of me I cant find it. Any and all help would be appreciated. 

I'm trying to solve a differential equation in Maple given by:

 when y(3) = -7.

 

I should get the result

but for some reason I get 

 

I solved it by writing:

 

I have Maple 18 if it makes a difference.

Dear Maple enthusiasts,

I am unable to find a working method to solve a system of 8 equations, of which 4 are differential equations. The system contains 8 unknown variables and the goal is to find an expression for each of these variables as a function of the time t. I have attached the code of my project at the bottom of this message.

I have tried the following:

  1. Using solve/dsolve to solve all 8 equations at once. This results in Maple eating up all of my memory and never finishing its calculations.
  2. First using solve to solve the 4 non-differential equations so that I get 4 out of 8 variables as a function of the 4 remaining variables. This results in an expression containing RootOf() for each of the 4 veriables I'm solving for, which prevents me from using these expressions in the 4 remaining differential equations.
  3. First using dsolve to solve the differential equations, which gives once again an expression for 4 variables as a function of the 4 remaining variables. I then use solve to solve the 4 remaining equations with the new found expressions. This results in an extremely long solution for each of the variables.

The code below contains the 3rd option I tried.

Any help or suggestions would be greatly appreciated. I have been scratching my head so much that I'm getting bald and whatever I search for on google or in the Maple help, I can't find a good reference to a system of differential equations together with other equations.

 

 

restart:

PARK - Mixed control

 

 

Input parameters

 

 

Projected interface area (m²)

A_int:=0.025^2*Pi:

 

Temperature of the process (K)

T_proc:=1873:

 

Densities (kg/m³)

Rho_m:=7000: metal

Rho_s:=2850: slag

 

Masses (kg)

W_m:=0.5: metal

W_s:=0.075: slag

 

Mass transfer coefficients (m/s)

m_Al:=3*10^(-4):

m_Si:=3*10^(-4):

m_SiO2:=3*10^(-5):

m_Al2O3:=3*10^(-5):

 

Weight percentages in bulk at t=0 (%)

Pct_Al_b0:=0.3:

Pct_Si_b0:=0:

Pct_SiO2_b0:=5:

Pct_Al2O3_b0:=50:

 

Weight percentages in bulk at equilibrium (%)

Pct_Al_beq:=0.132:

Pct_Si_beq:=0.131:

Pct_SiO2_beq:=3.13:

Pct_Al2O3_beq:=52.12:

 

Weight percentages at the interface (%)

Constants

 

 

Atomic weights (g/mol)

AW_Al:=26.9815385:

AW_Si:=28.085:

AW_O:=15.999:

AW_Mg:=24.305:

AW_Ca:=40.078:

 

Molecular weights (g/mol)

MW_SiO2:=AW_Si+2*AW_O:

MW_Al2O3:=2*AW_Al+3*AW_O:

MW_MgO:=AW_Mg+AW_O:

MW_CaO:=AW_Ca+AW_O:

 

Gas constant (m³*Pa/[K*mol])

R_cst:=8.3144621:

 

Variables

 

 

with(PDEtools):
declare((Pct_Al_b(t),Pct_Al_i(t),Pct_Si_b(t),Pct_Si_i(t),Pct_SiO2_b(t),Pct_SiO2_i(t),Pct_Al2O3_b(t),Pct_Al2O3_i(t))(t),prime=t):

Equations

 

4 rate equations

 

 

Rate_eq1:=diff(Pct_Al_b(t),t)=-A_int*Rho_m*m_Al/W_m*(Pct_Al_b(t)-Pct_Al_i(t));

 

Rate_eq2:=diff(Pct_Si_b(t),t)=-A_int*Rho_m*m_Si/W_m*(Pct_Si_b(t)-Pct_Si_i(t));

 

Rate_eq3:=diff(Pct_SiO2_b(t),t)=-A_int*Rho_s*m_SiO2/W_s*(Pct_SiO2_b(t)-Pct_SiO2_i(t));

 

Rate_eq4:=diff(Pct_Al2O3_b(t),t)=-A_int*Rho_s*m_Al2O3/W_s*(Pct_Al2O3_b(t)-Pct_Al2O3_i(t));

 

3 mass balance equations

 

 

Mass_eq1:=0=(Pct_Al_b(t)-Pct_Al_i(t))+4*AW_Al/(3*AW_Si)*(Pct_Si_b(t)-Pct_Si_i(t));

 

Mass_eq2:=0=(Pct_Al_b(t)-Pct_Al_i(t))+4*Rho_s*m_SiO2*W_m*AW_Al/(3*Rho_m*m_Al*W_s*MW_SiO2)*(Pct_SiO2_b(t)-Pct_SiO2_i(t));

 

Mass_eq3:=0=(Pct_Al_b(t)-Pct_Al_i(t))+2*Rho_s*m_Al2O3*W_m*AW_Al/(Rho_m*m_Al*W_s*MW_Al2O3)*(Pct_Al2O3_b(t)-Pct_Al2O3_i(t));

 

1 local equilibrium equation

 

 

Gibbs free energy of the reaction when all of the reactants and products are in their standard states (J/mol). Al and Si activities are in 1 wt pct standard state in liquid Fe. SiO2 and Al2O3 activities are in respect to pure solid state.

 

delta_G0:=-720680+133*T_proc:

 

Expression of mole fractions as a function of weight percentages (whereby MgO is not taken into account, but instead replaced by CaO ?)

x_Al2O3_i(t):=(Pct_Al2O3_i(t)/MW_Al2O3)/(Pct_Al2O3_i(t)/MW_Al2O3 + Pct_SiO2_i(t)/MW_SiO2 + (100-Pct_SiO2_i(t)-Pct_Al2O3_i(t))/MW_CaO);
x_SiO2_i(t):=(Pct_SiO2_i(t)/MW_SiO2)/(Pct_Al2O3_i(t)/MW_Al2O3 + Pct_SiO2_i(t)/MW_SiO2 + (100-Pct_SiO2_i(t)-Pct_Al2O3_i(t))/MW_CaO);

 

Activity coefficients

Gamma_Al_Hry:=1: because very low percentage present  during the process (~Henry's law)

Gamma_Si_Hry:=1: because very low percentage present  during the process (~Henry's law)

Gamma_Al2O3_Ra:=1: temporary value!

Gamma_SiO2_Ra:=10^(-4.85279678314968+0.457486603678622*Pct_SiO2_b(t)); very small activity coefficient?
plot(10^(-4.85279678314968+0.457486603678622*Pct_SiO2_b),Pct_SiO2_b=3..7);

 

Activities of components

a_Al_Hry:=Gamma_Al_Hry*Pct_Al_i(t);
a_Si_Hry:=Gamma_Si_Hry*Pct_Si_i(t);
a_Al2O3_Ra:=Gamma_Al2O3_Ra*x_Al2O3_i(t);
a_SiO2_Ra:=Gamma_SiO2_Ra*x_SiO2_i(t);

 

Expressions for the equilibrium constant K

K_cst:=exp(-delta_G0/(R_cst*T_proc));

Equil_eq:=0=K_cst*a_Al_Hry^4*a_SiO2_Ra^3-a_Si_Hry^3*a_Al2O3_Ra^2;

 

Output

 

 

with(ListTools):
dsys:=Rate_eq1,Rate_eq2,Rate_eq3,Rate_eq4:
dvars:={Pct_Al2O3_b(t),Pct_SiO2_b(t),Pct_Al_b(t),Pct_Si_b(t)}:
dconds:=Pct_Al2O3_b(0)=Pct_Al2O3_b0,Pct_SiO2_b(0)=Pct_SiO2_b0,Pct_Si_b(0)=Pct_Si_b0,Pct_Al_b(0)=Pct_Al_b0:
dsol:=dsolve({dsys,dconds},dvars):

Pct_Al2O3_b(t):=rhs(select(has,dsol,Pct_Al2O3_b)[1]);
Pct_Al_b(t):=rhs(select(has,dsol,Pct_Al_b)[1]);
Pct_SiO2_b(t):=rhs(select(has,dsol,Pct_SiO2_b)[1]);
Pct_Si_b(t):=rhs(select(has,dsol,Pct_Si_b)[1]);

sys:={Equil_eq,Mass_eq1,Mass_eq2,Mass_eq3}:
vars:={Pct_Al2O3_i(t),Pct_SiO2_i(t),Pct_Al_i(t),Pct_Si_i(t)}:
sol:=solve(sys,vars);

,


Download Park_-_mixed_control_model.mw

Hi:

i will solve the three equations below with numerical method,how?

eq1 := -2.517407096*10^12*q[1](t)^2-5.292771429*10^12*q[1](t)-1.888055322*10^12*q[2](t) = 0
eq2 := 2.246321962*10^12*q[1](t)^2+1.684741471*10^12*q[2](t)+8.110113889*10^12*q[1](t)-7.480938859*10^10*q[3](t) = 0
eq3 := int((-3.826000000*10^11*q[2](t)*cos(Pi*x)*Pi^2-3.826000000*10^11*q[1](t)^2*cos(Pi*x)*Pi^3*sin(Pi*x)+3.414000000*10^11*q[1](t)^2*sin(Pi*x)^2*Pi^4-3.414000000*10^11*q[1](t)^2*cos(Pi*x)^2*Pi^4+7*(int(exp(10*tau), tau = -infinity .. t))+q(x, t))*sin(Pi*x), x = 0 .. 1) = 0

hi,

   here are  equations like this

 sol := [abs(r)^2+abs(t)^2 = 1, r*conjugate(t)+t*conjugate(r), abs(r) = abs(t)]

when i solve this equations using command solve,the result  is none. and i used r=x+I*y,t=u+I*v in the equations,

sol:=[u^2+v^2+x^2+y^2 = 1, 2*u*x+2*v*y, sqrt(x^2+y^2) = sqrt(u^2+v^2)]

i still can't get a result.why,can you help me.

thanks.

 

I am a problem with solve differential equation, please help me: THANKS 

g := (y^2-1)^2; I4 := int(g^4, y = -1 .. 1); I5 := 2*(int(g^3*(diff(g, y, y)), y = -1 .. 1)); I6 := int(g^3*(diff(g, y, y, y, y)), y = -1 .. 1); with(Student[Calculus1]); I10 := ApproximateInt(6/(1-f(x)*g)^2, y = -1 .. 1, method = simpson);

dsys3 := {I4*f(x)^2*(diff(f(x), x, x, x, x))+I5*f(x)^2*(diff(f(x), x, x))+I6*f(x)^3 = I10, f(-1) = 0, f(1) = 0, ((D@@1)(f))(-1) = 0, ((D@@1)(f))(1) = 0};

dsol5 := dsolve(dsys3, numeric, output = array([0.]));

              Error, (in dsolve/numeric/bvp) system is singular at left endpoint, use midpoint method instead

****************FORMAT TWO ********************************************************

g := (y^2-1)^2; I4 := int(g^4, y = -1 .. 1); I5 := 2*(int(g^3*(diff(g, y, y)), y = -1 .. 1)); I6 := int(g^3*(diff(g, y, y, y, y)), y = -1 .. 1); with(Student[Calculus1]); I10 := ApproximateInt(6/(1-f(x)*g)^2, y = -1 .. 1, method = simpson);
dsys3 := {I4*f(x)^2*(diff(f(x), x, x, x, x))+I5*f(x)^2*(diff(f(x), x, x))+I6*f(x)^3 = I10, f(-1) = 0, f(1) = 0, ((D@@1)(f))(-1) = 0, ((D@@1)(f))(1) = 0};

dsol5 := dsolve(dsys3, method = bvp[midrich], output = array([0.]));
%;
                                   Error, (in dsolve) too many levels of recursion

I DONT KNOW ABOUT THIS ERROR

PLEASE HELP ME

THANKS A LOT

 

Hy all.

I want to solve this equation, with„dd” as numerical result. What do I do wrong? Thanks. Nico

restart;
TTot := 70;
TC := 17;
GM := .26;
QMax := 870;
V := 3600*GM*QMax*TTot;
eq := V = int(QMax*exp((-t+TC)/dd)*(1+(t-TC)/TC)^(TC/dd), t = 0 .. TTot);
fsolve(eq, dd);

Hi,

I have a system of diff equations (see below). I am trying to obtain analytical solution. when I assume that z=wN, I receive such solution. Do anybody have idea if I know that z>wN, does this system has an analytical solution?

diff(K(t), t) = -(1/2)*(Q(t)^2*alpha^2*eta*upsilon-2*eta*alpha*(N*upsilon*w*C[max]-z*alpha*K(t))*Q(t)+N*w*(-2*C[max]*z*eta*alpha*K(t)+upsilon*((-N*w+z)*alpha+N*C[max]^2*w*eta)))*K(t)/((C[max]*w*N-alpha*Q(t))*upsilon*N*w)

diff(Q(t), t) = (1/2)*(-z*(Q(t)^2*alpha^2*eta-2*N*Q(t)*alpha*eta*w*C[max]+w*(w*(eta*C[max]^2-alpha)*N+z*alpha)*N)*K(t)-2*N*upsilon*w*(N*w-z)*(C[max]*w*N-alpha*Q(t)))/((C[max]*w*N-alpha*Q(t))*upsilon*N*w)

K(0) = K0, Q(0) = Q0

Thanks,

Dmitry

Hello,
I have a system of first order diff. equations which I would like to solve symbolically. Unfortunately, Maple does not solve the system. Do anybody have suggestions how can I solve this system (please see below):

diff(S(t), t) = -eta*(C[max]*w*N-alpha*Q(t))*K(t)*S(t)/(w*N*(S(t)+K(t))),

diff(K(t), t) = S(t)*((z*eta*alpha*(C[max]*w*N-alpha*Q(t))*S(t)-upsilon*(eta*alpha^2*Q(t)^2-2*C[max]*w*N*eta*alpha*Q(t)+((-N*w+z)*alpha+N*C[max]^2*w*eta)*N*w))*K(t)^2+(2*((1/2)*z*eta*(C[max]*w*N-alpha*Q(t))*S(t)+N*w*upsilon*(N*w-z)))*S(t)*alpha*K(t)+N*S(t)^2*w*alpha*upsilon*(N*w-z))/((K(t)^2*alpha*z+3*S(t)*K(t)*alpha*z+S(t)*(2*S(t)*z*alpha+upsilon*(C[max]*w*N-alpha*Q(t))))*(S(t)+K(t))*N*w),

diff(Q(t), t) = (-alpha*z*(z*eta*(C[max]*w*N-alpha*Q(t))*K(t)+N*w*upsilon*(N*w-z))*S(t)^2+(-z^2*eta*alpha*(C[max]*w*N-alpha*Q(t))*K(t)^2-(eta*alpha^2*Q(t)^2-2*C[max]*w*N*eta*alpha*Q(t)+N*w*((2*N*w-2*z)*alpha+N*C[max]^2*w*eta))*z*upsilon*K(t)-N*w*upsilon^2*(N*w-z)*(C[max]*w*N-alpha*Q(t)))*S(t)-N*w*z*alpha*upsilon*K(t)^2*(N*w-z))/((2*S(t)^2*alpha*z+(3*z*alpha*K(t)+upsilon*(C[max]*w*N-alpha*Q(t)))*S(t)+K(t)^2*alpha*z)*N*w*upsilon)

where initials conditions are:

S(0) = S0, K(0) = K0, Q(0) = Q0

Thanks,

Dmitry

 

 

 

Hello guys ...

I used a numerically method to solve couple differential equation that it has some boundary conditions. My problem is that some range of answers has 50% error . Do you know things for improving our answers in maple ?

my problem is :

a*Φ''''(x)+b*Φ''(x)+c*Φ(x)+d*Ψ''(x)+e*Ψ(x):=0

d*Φ''(x)+e*Φ(x)+j*Ψ''(x)+h*Ψ(x):=0

suggestion method by preben Alsholm:

a,b,c,d,e,j,h are constants.suppose some numbers for these constants . I used this code:


VR22:=0.1178*diff(phi(x),x,x,x,x)-0.2167*diff(phi(x),x,x)+0.0156*diff(psi(x),x,x)+0.2852*phi(x)+0.0804*psi(x);
VS22:=0.3668*diff(psi(x),x,x)-0.0156*diff(phi(x),x,x)-0.8043*psi(x)-0.80400*phi(x);
bok:=evalf(dsolve({VR22=0,VS22=0}));

PHI,PSI:=op(subs(bok,[phi(x),psi(x)]));
Eqs:={eval(PHI,x=1.366)=1,eval(diff(PHI,x),x=1.366)=0,eval(PHI,x=-1.366)=1,eval(diff(PHI,x),x=-1.366)=0,
eval(PSI,x=1.366)=1,eval(PSI,x=1.366)=1};
C:=fsolve(Eqs,indets(%,name));
eval(bok,C);
SOL:=fnormal(evalc(%));


I used digits for my code at the first of writting.

please help me ... what should i do?

Hi dear users:

i will plot the equation below abs(y) in terms of x,(note:abs(y) and x is real values),can every body help me?

eq:

-32.46753247/(Pi*x^2)+1.053598444*10^8*Pi^2*y/x^2-5.342210338*10^14*Pi^2*y*(2.574000000*10^8*Pi^2-.7700000000*x^2)/((-3.904240733*10^6*x^2+1.305131902*10^15*Pi^2-159.8797200*Pi^2*x^2+2.672275320*10^10*Pi^4+2.391363333*10^(-7)*x^4)*x^2)+1.504285714*10^9*Pi^4*y^3/x^2 = y

i have a non linear equation that depends on three variables e, theta and z.

i have done calculations to calculate e while varying theta and z. theta varied among the vector [0, Pi/4, Pi/3, Pi/2] and z was varying between 1 and 20

when plotting my data it gives the following plot where z is represented on the x-axis and each curve correspond to one theta

 

i am currently able of fitting one plot to one equation i would like to fit the data points using the nonlinearfit function and to only get one equation for all the plots. is that possible in maple or not

 

How can I solve a differential equation set of the type,

dy(x)/dx +y^2 =P(x); dP(x)/dx = R(P) numerically

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