# Items tagged with solutionsolution Tagged Items Feed

### Warning, solutions may have been lost...

November 22 2014
0 1

Hi

I need your help .. I solve a system and I get the error Warning, solutions may have been lost.

The maple code is attached.

testerror.mw

### Sytem of equation solution ...

November 21 2014
0 0

Dear all:

hello everybody;

I need your help to solve the system f(x,y)=0, and g(x,y)=0, such that there some parameter in the system, also all the parameter are positive and also our unkowns  x and y are also positive.

I try to write this code. I feel that under some condition we can have four solution or three or two. I need your help. Many thinks.

Systemsolve.mw

### Solving equation...

November 17 2014
0 2

Hello all,

I have the following equation:

N*exp(-(1/2)*eta*epsilon*(N*alpha*epsilon*w+2*N*w*C[max]-alpha*epsilon*z-2*Q1*alpha)/(w*N))*S1*upsilon*w-N*S1*upsilon*w+K1^2*alpha*eta*z*epsilon+K1*alpha*eta*z*epsilon*S1 = 0

in which I need to find solution for epsilon (analytical solution) when epsilon>0.

Thanks,

Dmitry

### Solving a system of non-linear equations for diffe...

November 14 2014
0 2

I have the following system of non-linear equations and want to find their solutions experimenting with my parameters. I also want to restrict the solutions to be non-negative. I have done the following, but i am sure it exist a more efficient way. Can somone help on this?

eqns := [A = (gr_c+delta)*kh^(1-alpha)/sav_rate, theta = Rk*(Rh-Rk)/(gamma*((Rh-Rk)^2+sigma^2)), theta = 1*1+kh, Rk = 1+rk-delta, Rh = 1+rh-delta, rk = A*alpha*kh^(alpha-1), rh = A*(1-alpha)*kh^alpha, sigma = sigmay/theta, varrho = Rap^((eps-1)*eps/(1-gamma)), Rap = Rk^(1-gamma)+(1-gamma)*Rk^(-gamma)*theta*(Rh-Rk)-.5*Rk^(-gamma-1)*gamma*(1-gamma)*theta^2*((Rh-Rk)^2+sigma^2), R = Rk+theta*(Rh-Rk), beta = ((1+gr_c)/R)^(1/eps)/varrho];
print(`output redirected...`); # input placeholder
[
[
[
[
[ (1 - alpha)
[ (gr_c + delta) kh
[A = ----------------------------,
[ sav_rate
[

Rk (Rh - Rk)
theta = ---------------------------, theta = 1 + kh,
/ 2 2\
gamma \(Rh - Rk) + sigma /

Rk = 1 + rk - delta, Rh = 1 + rh - delta,

(alpha - 1) alpha
rk = A alpha kh , rh = A (1 - alpha) kh ,

/(eps - 1) eps\
|-------------|
sigmay \ 1 - gamma /
sigma = ------, varrho = Rap , Rap =
theta

(1 - gamma) (-gamma)
Rk + (1 - gamma) Rk theta (Rh - Rk) - 0.5

(-gamma - 1) 2 / 2 2\
Rk gamma (1 - gamma) theta \(Rh - Rk) + sigma /,

/ 1 \]
|---|]
\eps/]
/1 + gr_c\ ]
|--------| ]
\ R / ]
R = Rk + theta (Rh - Rk), beta = ---------------]
varrho ]
]
vals := [alpha = .36, delta = 0.6e-1, sigmay = sqrt(0.313e-1), gamma = 3, eps = .5, gr_c = 0.2e-1, sav_rate = .23];
eval(eqns, vals);
print(`output redirected...`); # input placeholder
[
[
[
[ 0.64 Rk (Rh - Rk)
[A = 0.3478260870 kh , theta = -----------------------,
[ / 2 2\
[ 3 \(Rh - Rk) + sigma /

0.36 A
theta = 1 + kh, Rk = 0.94 + rk, Rh = 0.94 + rh, rk = ------,
0.64
kh

0.36 0.1769180601
rh = 0.64 A kh , sigma = ------------,
theta

0.1250000000
varrho = Rap ,

1 2 theta (Rh - Rk)
Rap = --- - -----------------
2 3
Rk Rk

2 / 2 2\
3.0 theta \(Rh - Rk) + sigma /
+ --------------------------------, R = Rk + theta (Rh - Rk),
4
Rk

2.000000000]
/1\ ]
1.0404 |-| ]
\R/ ]
beta = ---------------------]
varrho ]
]
eqns := [A = .3478260870*kh^.64, theta = (1/3)*Rk*(Rh-Rk)/((Rh-Rk)^2+sigma^2), theta = 1+kh, Rk = .94+rk, Rh = .94+rh, rk = .36*A/kh^.64, rh = .64*A*kh^.36, sigma = .1769180601/theta, varrho = Rap^.1250000000, Rap = 1/Rk^2-2*theta*(Rh-Rk)/Rk^3+3.0*theta^2*((Rh-Rk)^2+sigma^2)/Rk^4, R = Rk+theta*(Rh-Rk), beta = 1.0404*(1/R)^2.000000000/varrho];
print(`output redirected...`); # input placeholder
[
[
[
[ 0.64 Rk (Rh - Rk)
[A = 0.3478260870 kh , theta = -----------------------,
[ / 2 2\
[ 3 \(Rh - Rk) + sigma /

0.36 A
theta = 1 + kh, Rk = 0.94 + rk, Rh = 0.94 + rh, rk = ------,
0.64
kh

0.36 0.1769180601
rh = 0.64 A kh , sigma = ------------,
theta

0.1250000000
varrho = Rap ,

1 2 theta (Rh - Rk)
Rap = --- - -----------------
2 3
Rk Rk

2 / 2 2\
3.0 theta \(Rh - Rk) + sigma /
+ --------------------------------, R = Rk + theta (Rh - Rk),
4
Rk

2.000000000]
/1\ ]
1.0404 |-| ]
\R/ ]
beta = ---------------------]
varrho ]
]

solve(eqns, [Rk, Rh, varrho, Rap, beta, R, A, sigma, theta, rk, rh, kh]);

### Numerical solution of intersection of two curves o...

November 02 2014
0 5

Hello I am a Maple 15 user and I am using the command fsolve to solve for the intersection of two curves over a specified interval in x, namely from 0 to the lim defined in the Maple document. The specified interval contains asymptotes and when I specify the full interval only one of the three solutions is returned even if I can see that there are three distinct solutions by looking at the plot of RHS and LHS. Should I use another technique to find the solution or is my implementation of fsolve command wrong?

 (1)

 (2)

 (3)

### Analytical solution of 2D diffusion equation in po...

October 23 2014
0 2

Hello,

I have been trying to compute the analytical solution of two dimensional diffusion equation with zero neumann boundary conditions (no-flux) in polar coordinates using the solution in Andrei Polyanin's book. When I use 2d Gaussian function as initial condition, i cannot get the result. If I use some nicer function like f(r,phi)=1-r; there is no problem.

Any idea why this happens? or any suggestion to compute the analytical solution?

Thanks!

HB

 >
 >
 >
 >
 >
 >

### inequations: how to find a partical numeric soluti...

October 21 2014
1 2

Having solution of an inequations system, is there a way/function/algorithm to find a particular numeric solution (as simplex[minimize] can do) ?

ex:

Q := {1 < x - y, x + y < 1};

R := solve(Q);

{ x < 1 - y, y < 0, y + 1 < x }

manually it's easy to find some numeric solutions:

y = -1, x = 1
y = -2, x = 0

but I need an automatic way.

s.py

### Solving a combined system of differential and part...

September 19 2014
1 4

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);

,

### solving a symbolic inequality...

September 04 2014
1 2

Hi, those who are in mapleprimes.

i have a problem in solving inequality with symbolic notated parameters.

I wrote the following code to solve for n(SPH), but couldn't obtain any result but an error message.

solve(-s*(-n(SPF)*tau+n(SPH))/(tau-1) <= n(SPH),n(SPH)) assuming (tau<1,s>0,s<1,tau>0);

The error was

Error, (in assuming) when calling 'unknown'. Received: 'invalid input: Utilities:-SetSolutions expects its 2nd argument, solutions, to be of type ({list, set})({piecewise, ({list, set})({name, relation})}), but received [s = -tau~+1, [SPF = SPF, s = s, tau~ <= 0]]'

Please tell me how I should do to solve the inequality.

taro

### Pde system with initial condition...

July 27 2014
1 8

Dear Maple Users,

I'm beginner in Maple.

I have this system of Pde:

with lambda experimental parameter and n,c,v dependent variables. I write this on Maple but I read on internet that the solution "float(undefined)" is an error.

I will insert this initial condition: c(x,0)=0,n(x,0)=0.4

Thanks everybody

### Nonlinear system of 14 equations/unknowns, 9 param...

July 17 2014
0 0

Hello,

this is the second time I'm writing.

I posted this question in June http://www.mapleprimes.com/questions/201781-System-Of-Parametric-Equations.

This time I have  a similar problem because I'm trying to find a solution for a parametric system of equations but the number of equations and parameters is much bigger and using the tips you gave me last time I couldn't reach any result.

Here is the system:

1) alpha[1]=v*a*u*b ;
2) alpha[2]=v*a*u*(1-b);
3) alpha[3]= v*z*c*(1-a) ;
4) alpha[4]=v*z*(1-a)*(1-c) ;
5) alpha[11]=1/2*v*a* u* b* (-p*u*b+p*u*b*a+b*g-g);
6) alpha[22]=1/2*v*a*u*(1-b)* (p u b-p u b a-b g-p u+p u a);
7) alpha[33] =1/2*v*c*z*(1-a)* (c* (-z*p*a+q)-q);
8) alpha[44]=1/2*v*z*((1-a)*(1-c)* (c*z*p*a-z*p*a-q*c);
9) alpha[12]=v*a*u*b*(1- b)*(-p*u+p*u*a+g) ;
10) alpha[13]=v*a*u*b*z*c*p*(1-a) ;
11) alpha[14]=a*u*b*z*(1-a)*(1-c) ;
12) alpha[23]=a*u*z*c*(1-a)*(1-b);
13) alpha[24]=v*a*u*z*p*(1-a)*(1-b)*(1-c);
14) alpha[34]= v*c*z*(1-a)*(1-c)*(-z*p*a+q);

I have 14 equations/unknowns and 8 parameters (a, b, c, u, v, z, p, q).

I would like to write this system only in terms of alphas. In order to do so, I usually try to find the value for the parameters and the substitute them into the equations (and I have already found b,c,g,q using this technique) but I couldn't manage to find all of them.

Howveer, as you suggested me, with Maple there is the command "eliminate" that implement exactly what I'm looking for but I can't make it work.

This is my code:

> sys := {alpha[1] = v*a*u*(1-b), alpha[2] = v*a*u*b, alpha[3] = v*z*c*(1-a), alpha[4] = v*z*(1-a)*(1-c), alpha[11] = (1/2)*v*a*u*(1-b)*(p*u*b-p*u*b*a-b*g-p*u+p*u*a), alpha[12] = v*a*u*b*(1-b)*(-p*u+p*u*a+g), alpha[13] =      z*c*a*u*(1-a)*(1-b), alpha[14] = v*z*a*u*p*(1-a)*(1-b)*(1-c), alpha[22] = (1/2)*v*a*u*b*(-p*u*b+p*u*b*a+b*g-g), alpha[23] = v*z*c*a*u*b*p*(1-a), alpha[24] = z*a*u*b*(1-a)*(1-c), alpha[33] = (1/2)*v*c*z*(1-a)*(c*(-z*p*a+q)-q), alpha[34] = v*c*z*(1-a)*(1-c)*(-z*p*a+q), alpha[44] = (1/2)*v*z*(1-a)*(1-c)*(c*z*p*a-z*p*a-q*c)};

> eliminate(sys, {a,b,c, p, q, u, v, z});

> simplify(%, size);

I also tries to substitute in the system the four parameters I already found but still I can't find a solution.

What am I doing wrong? Or the problem is that it is too complicated?

Elena

### Warning, cannot evaluate the solution further righ...

July 09 2014
0 1

Pleaz i nees help i have probleme withe singularity

 >

Paramétres

 >
 (1.1)

Equation suivant x :

 >
 (2.1)

Equation suivant z :

 >
 (3.1)

Equation suivant y :

 >
 (4.1)

Equation suivant y

 >
 (5.1)

Résolution :

 > CI:= x(0)=0,z(0)=0,theta(0)=0,alpha(0)=0,D(x)(0)=0,D(alpha)(0)=0,D(z)(0)=0,D(theta)(0)=0;
 (6.1)
 > if theta(t) <> 0 then  solution:=dsolve([eq1,eq2,eq3,eq4,CI],numeric,maxfun=0):  odeplot(solution, [[t, x(t)]], t = 0 .. 100, thickness = 2);  odeplot(solution, [[t, z(t)]], t = 0 .. 100, thickness = 2);  odeplot(solution, [[t, theta(t)]], t = 0 .. 100, thickness = 2);  odeplot(solution, [[t, alpha(t)]], t = 0 .. 100, thickness = 2);  #odeplot(solution,[[t,x(t)],[t,alpha(t)],[t,z(t)],[t,theta(t)]], t=0..100, thickness=2);  end ;

thank you !

### Help using maple...

July 06 2014
1 6

Hey,

I wish to find the zero crossing of the following eq. (with respect to x)

a is typically 1.729, b is 1.139 and c is 0.0688. Ploting the eq. and using the solve command result in two values (zero crossings) when replacing the constants by their respective values, but when I wish to have the solution with respect to the constants a, b and c the result is only for one of them. Any commands that can be helpful in this situation?

### How do I compare solutions from numeric dsolve?...

July 04 2014
2 5

Hello,

This is probably a silly question, but I am trying to compare the difference between two variables in the numerical solution of a system of ODEs. Ideally, I would like a method to find the maximal difference that occurs between two variables.

The following is a highly simplified example of what I'm talking about. In this case I'd like some means to find the timepoint and magnitude of the maximal difference between y2(t) and y3(t) for t>0, which from the plot can be seen to occur at about 1.75 seconds. Note: I realise this particular case admits an analytic solution of y3(t) which could be exploited, but in the general case I'm interested in that won't be true.

 (1)

 (2)

### Problem on the solution of Trigonometric Equations...

June 25 2014
1 1

How can I know the solution is correct??

Here is the code:

s := solve(sin(x^2) = 1/2);
test := subs(x = s[1], sin(x^2) = 1/2);

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