Solving a nonlinear system differential equation using symbols

Hello,

I'm trying to solve a systems of non linear equations at equillbrium.

First I solved each equation individually and then made a system of equations to solve. I realized that I was getting 3 instead of 5 solutions

used the return conditions function and got the following using this code

syms a b c d e k_off_1p Pd1_0 k_on_1p Ab_initial_conc k_on_2p k_off_1p k_off_1c Cd137_0 k_on_1c k_on_2c k_off_1c
Eqns = [0 == -k_on_1p.*Ab_initial_conc*a + k_off_1p.*c - k_on_2p.*a*d + k_off_1p.*e;
        0 == -k_on_1c.*Ab_initial_conc*b+k_off_1c.*d-k_on_2c.*b*c+k_off_1c.*e;
        0 == k_on_1p.*Ab_initial_conc*a-k_off_1p.*c-k_on_2c.*b*c+k_off_1c.*e;
        0 == k_on_1c.*Ab_initial_conc*b-k_off_1c.*d-k_on_2p.*a*d+k_off_1p.*e;
        0 ==  k_on_2c.*b*c+k_on_2p.*a*d-(k_off_1c+k_off_1p)*e;        
        ];
[sola,solb,solc,sold,sole] = solve(Eqns, [a,b,c,d,e],'ReturnConditions',true)

I got the error "Inconsistent output with 3 variables for input argument with 5 variables"

Trying to solve by hand, I also got three variables as an out put. using 2 other equations , I could redefine the 3rd and 4th eqns. My intention is to have matlab use the 3rd and 4th equation to substitute the values of c and d into the remaining eqns.

syms a b c d e k_off_1p Pd1_0 k_on_1p Ab_initial_conc k_on_2p k_off_1p k_off_1c Cd137_0 k_on_1c k_on_2c k_off_1c
Eqns = [0 == -k_on_1p.*Ab_initial_conc*a + k_off_1p.*c - k_on_2p.*a*d + k_off_1p.*e;
        0 == -k_on_1c.*Ab_initial_conc*b+k_off_1c.*d-k_on_2c.*b*c+k_off_1c.*e;
        c==Pd1_0 - a -e;
        d==Cd137_0 - b -e;
        0 ==  k_on_2c.*b*c+k_on_2p.*a*d-(k_off_1c+k_off_1p)*e;        
        ];
[sola,solb,solc,sold,sole] = solve(Eqns, [a,b,c,d,e],'ReturnConditions',true)

I recieved the following error

Error using sym/cat>checkDimensions (line 68)

CAT arguments dimensions not consistent.

Error in sym/cat>catMany (line 33)

[resz, ranges] = checkDimensions(sz,dim);

Error in sym/cat (line 25)

ySym = catMany(dim, args);

Error in sym/vertcat (line 19)

ySym = cat(1,args{:});

Error in tewst (line 65)

Eqns = [0 == -k_on_1p.*Ab_initial_conc*a + k_off_1p.*c - k_on_2p.*a*d + k_off_1p.*e;

I am unsure how to proceed from here, because the dimensions are consistent in my mind, and I'm unsure how to introduce the 2 new equations from another perspective.

Thanks

Answer 1

[Paul]

When using ReturnConditions, the outputs need to include Parameters and Conditions, or use only a single output

syms a b c d e k_off_1p Pd1_0 k_on_1p Ab_initial_conc k_on_2p k_off_1p k_off_1c Cd137_0 k_on_1c k_on_2c k_off_1c
Eqns = [0 == -k_on_1p.*Ab_initial_conc*a + k_off_1p.*c - k_on_2p.*a*d + k_off_1p.*e;
        0 == -k_on_1c.*Ab_initial_conc*b+k_off_1c.*d-k_on_2c.*b*c+k_off_1c.*e;
        0 == k_on_1p.*Ab_initial_conc*a-k_off_1p.*c-k_on_2c.*b*c+k_off_1c.*e;
        0 == k_on_1c.*Ab_initial_conc*b-k_off_1c.*d-k_on_2p.*a*d+k_off_1p.*e;
        0 ==  k_on_2c.*b*c+k_on_2p.*a*d-(k_off_1c+k_off_1p)*e;        
        ];
[sola,solb,solc,sold,sole,parameters,conditions] = solve(Eqns, [a,b,c,d,e],'ReturnConditions',true)

Warning: Unable to find explicit solution. For options, see help.

 
sola =
 
Empty sym: 0-by-1
 
 
solb =
 
Empty sym: 0-by-1
 
 
solc =
 
Empty sym: 0-by-1
 
 
sold =
 
Empty sym: 0-by-1
 
 
sole =
 
Empty sym: 0-by-1
 
 
parameters =
 
Empty sym: 1-by-0
 
 
conditions =
 
Empty sym: 0-by-1
 

% or
sol = solve(Eqns, [a,b,c,d,e],'ReturnConditions',true)

Warning: Unable to find explicit solution. For options, see help.

sol = struct with fields:
             a: [0×1 sym]
             b: [0×1 sym]
             c: [0×1 sym]
             d: [0×1 sym]
             e: [0×1 sym]
    parameters: [1×0 sym]
    conditions: [0×1 sym]

solve() can't find the explicit soution, but that's a different problem.

[Reginald]

Thank you. Would this mean an explicit solution is not possible, or is my attempt flawed to obtain one?

[Torsten]

From Octave, I get a parametrized solution in d and e, i.e.

a = a(d,e), b = b(d,e), c= c(d,e)

meaning that your equations define a 2d-manifold.

[Paul]

@Reginald

Try using IgnoreAnalyticConstraints. Also, consider including other assumptions on the variables, e.g. "real", "postive", etc., if approropriate. Unfortunately, I've never been able to figure out how to determine which analytic constraints are being ignored.

syms a b c d e k_off_1p Pd1_0 k_on_1p Ab_initial_conc k_on_2p k_off_1p k_off_1c Cd137_0 k_on_1c k_on_2c k_off_1c
Eqns = [0 == -k_on_1p.*Ab_initial_conc*a + k_off_1p.*c - k_on_2p.*a*d + k_off_1p.*e;
        0 == -k_on_1c.*Ab_initial_conc*b+k_off_1c.*d-k_on_2c.*b*c+k_off_1c.*e;
        0 == k_on_1p.*Ab_initial_conc*a-k_off_1p.*c-k_on_2c.*b*c+k_off_1c.*e;
        0 == k_on_1c.*Ab_initial_conc*b-k_off_1c.*d-k_on_2p.*a*d+k_off_1p.*e;
        0 ==  k_on_2c.*b*c+k_on_2p.*a*d-(k_off_1c+k_off_1p)*e;        
        ];
sol = solve(Eqns, [a,b,c,d,e],'ReturnConditions',true,'IgnoreAnalyticConstraints',true)

sol = struct with fields:
             a: [2×1 sym]
             b: [2×1 sym]
             c: [2×1 sym]
             d: [2×1 sym]
             e: [2×1 sym]
    parameters: [z    z1]
    conditions: [2×1 sym]

sol.a % etc.

ans = 

@Torsten

Does this solution match Octave? How does Octave handle "analytic constraints"?

[Torsten]

As I said, a,b and c depend on d and e in the octave solution.

I don't see this in the solution given. Or do z and z1 parametrize here ?

[Walter Roberson]

syms a b c d e k_off_1p Pd1_0 k_on_1p Ab_initial_conc k_on_2p k_off_1p k_off_1c Cd137_0 k_on_1c k_on_2c k_off_1c
Eqns = [0 == -k_on_1p.*Ab_initial_conc*a + k_off_1p.*c - k_on_2p.*a*d + k_off_1p.*e;
        0 == -k_on_1c.*Ab_initial_conc*b+k_off_1c.*d-k_on_2c.*b*c+k_off_1c.*e;
        0 == k_on_1p.*Ab_initial_conc*a-k_off_1p.*c-k_on_2c.*b*c+k_off_1c.*e;
        0 == k_on_1c.*Ab_initial_conc*b-k_off_1c.*d-k_on_2p.*a*d+k_off_1p.*e;
        0 ==  k_on_2c.*b*c+k_on_2p.*a*d-(k_off_1c+k_off_1p)*e;        
        ];
sol = solve(Eqns, [a,b,c,d,e],'ReturnConditions',true,'IgnoreAnalyticConstraints',true)

sol = struct with fields:
             a: [2×1 sym]
             b: [2×1 sym]
             c: [2×1 sym]
             d: [2×1 sym]
             e: [2×1 sym]
    parameters: [z    z1]
    conditions: [2×1 sym]

sol.conditions

ans = 

Notice the conditions are symtrue . So the solutions apply for any real or complex-valued z and z1

[Paul]

Yse, z and z1 are free parameters such that d = z and e = z1, and then a,b, and c each have two solutions for given those same parameters.

Answer 2

[Torsten]

Here is the sequence of commands to obtain the solution in Octave.

One can use the "solve" directly for all variables, but I deduced appropriate solution steps from the equations.

syms A B C D E f g h k l m n
% Rename Variables
% A = a
% B = b
% C = c
% D = d
% E = e
% f = k_on_1p
% g = Ab_initial_conc
% h = k_off_1p
% k = k_on_2p
% l = k_on_1c
% m = k_off_1c
% n = k_on_2c
%Eqns = [0 == -f*g*A        + h*C     + h*E     - k*A*D;
%        0 ==        -l*g*B      +m*D + m*E              - n*B*C;
%        0 ==  f*g*A        - h*C     + m*E              - n*B*C;
%        0 ==         l*g*B      -m*D + h*E     - k*A*D;
%        0 ==                         -(m+h)*E  + k*A*D  + n*B*C;
%        ];
Eqns = [0 == -f*g*A + h*C + h*E - k*A*D;...
        0 == -l*g*B + m*D + m*E - n*B*C;...
        0 == f*g*A - h*C + m*E - n*B*C;...
        0 == l*g*B - m*D + h*E - k*A*D;...
        0 == -(m+h)*E + k*A*D + n*B*C;...
       ];        
%[solA solB solC solD solE] = solve(Eqns,[A B C D E])
Bsol = solve(Eqns(4),B)
Csol = solve(Eqns(1),C)
eqnA = subs(Eqns(3),[B,C],[Bsol,Csol])
Asol = solve(eqnA,A)
Bsol = subs(Bsol,A,Asol)
Csol = subs(Csol,A,Asol)
Check11 = simplify(subs(Eqns(1),[A,C],[Asol(1),Csol(1)]))
Check12 = simplify(subs(Eqns(1),[A,C],[Asol(2),Csol(2)]))
Check21 = simplify(subs(Eqns(2),[B,C],[Bsol(1),Csol(1)]))
Check22 = simplify(subs(Eqns(2),[B,C],[Bsol(2),Csol(2)]))
Check31 = simplify(subs(Eqns(3),[A,B,C],[Asol(1),Bsol(1),Csol(1)]))
Check32 = simplify(subs(Eqns(3),[A,B,C],[Asol(2),Bsol(2),Csol(2)]))
Check41 = simplify(subs(Eqns(4),[A,B],[Asol(1),Bsol(1)]))
Check42 = simplify(subs(Eqns(4),[A,B],[Asol(2),Bsol(2)]))
Check51 = simplify(subs(Eqns(5),[A,B,C],[Asol(1),Bsol(1),Csol(1)]))
Check52 = simplify(subs(Eqns(5),[A,B,C],[Asol(2),Bsol(2),Csol(2)]))