ode45 Error differential equation system

I'm trying to solve a system of differential equations with ode45 but an appears. Could someone help me spot the mistake(s)?

 %%Parameters
  R_la= 0.4; 
  R_sa_b= 5.03;
  R_sv= 1.32;
  R_lv= 0.56;
  P_a_b= 100;
  P_v= 6;
  V_la=1;
  V_sa_b= 12;
  P_ic= 10;
  Ca= 0.205; 
  k_ven= 0.186; 
  P_v1= -2.25; 
  V_vn= 28;
  G_q= 3; 
  tau_q= 20; 
  Pa_co2_b= 40; 
  tau_co2= 40;

%%State parameters

F=@(t,V_sa,P1,P2) [ Ca.*(P1-P_ic);
                              ((P_a_b-P1)./(R_la + 0.5 .*R_sa_b) - (P1-P2)./(0.5 .*R_sa_b+R_sv))./Ca;
                                       ((P1-P2)./(0.5 .*R_sa +R_sv)-(P2-P_v)./R_lv)./ (1./(k_ven.*(P2-P_ic-P_v1))) ]

[t,V_sa,P1,P2]= ode45(F,[0 10],[0 0 0]);

q= (P1-P2)./0.5 .*R_sa + R_sv ;

F1=@(t,xq,xc) [ (-xq+G_q .*(q-q_b)./q_b)./tau_q ;               
                               (-xc +0.3+3.*tanh(Pa_co2./Pa_co2_b -1.1))./tau_co2 ]

[t,xq,xc]= ode45(F1,[0 10],[0 0 0]);

Error message:

Not enough input arguments.

Error in
CBF_v2>@(t,V_sa,P1,P2)[Ca.*(P1-P_ic);((P_a_b-P1)./(R_la+0.5.*R_sa_b)-(P1-P2)./(0.5.*R_sa_b+R_sv))./Ca;((P1-P2)./(0.5.*R_sa+R_sv)-(P2-P_v)./R_lv)./(1./(k_ven.*(P2-P_ic-P_v1)))]

Error in odearguments (line 87)
f0 = feval(ode,t0,y0,args{:});   % ODE15I sets args{1} to yp0.

Error in ode45 (line 115)
    odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);

Error in CBF_v2 (line 37)
[t,V_sa,P1,P2]= ode45(F,[0 10],[0 0 0]);

Answer 1

[Walter Roberson]

Change

F=@(t,V_sa,P1,P2) [ Ca.*(P1-P_ic);
                              ((P_a_b-P1)./(R_la + 0.5 .*R_sa_b) - (P1-P2)./(0.5 .*R_sa_b+R_sv))./Ca;
                                       ((P1-P2)./(0.5 .*R_sa +R_sv)-(P2-P_v)./R_lv)./ (1./(k_ven.*(P2-P_ic-P_v1))) ]

[t,V_sa,P1,P2]= ode45(F,[0 10],[0 0 0]);

to

F = @(t, V_saP1P2) [ Ca.*(V_saP1P2(2)-P_ic);
                            ((P_a_b-V_saP1P2(2))./(R_la + 0.5 .*R_sa_b) - (V_saP1P2(2)-V_saP1P2(3))./(0.5 .*R_sa_b+R_sv))./Ca;
                                     ((V_saP1P2(2)-V_saP1P2(3))./(0.5 .*R_sa +R_sv)-(V_saP1P2(3)-P_v)./R_lv)./ (1./(k_ven.*(V_saP1P2(3)-P_ic-P_v1))) ];

 [t, V_saP1P2] = ode45(F, [0 10], [0 0 0]);
 V_sa = V_saP1P2(:,1);
 P1 = V_saP1P2(:,2);
 P2 = V_saP1P2(:,3);

and similar changes for your F1

[gorilla3]

Thank you for your help but I still get the error:

Not enough input arguments.

Error in
CBF_v2>@(t,V_sa,P1,P2)[Ca.*(P1-P_ic);((P_a_b-P1)./(R_la+0.5.*R_sa_b)-(P1-P2)./(0.5.*R_sa_b+R_sv))./Ca;((P1-P2)./(0.5.*R_sa+R_sv)-(P2-P_v)./R_lv)./(1./(k_ven.*(P2-P_ic-P_v1)))]

Error in odearguments (line 87)
f0 = feval(ode,t0,y0,args{:});   % ODE15I sets args{1} to yp0.

Error in ode45 (line 115)
    odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);

Error in CBF_v2 (line 36)
[t, V_saP1P2] = ode45(F, [0 10], [0 0 0]);

[Torsten]

The error message clearly shows that you didn't incorporate Walters's changes.

Best wishes

Torsten.

[gorilla3]

I'm sorry I just replaced the 2nd part of the code at frist. It works now, thank you both!

[gorilla3]

Hi,

I'm sorry for troubling you again but I realized that I didn't express the 1st eq correctly. It should stand for:

• d/dt (V_sa)= d/dt(Ca*(P1-P_ic))

However, I'm unsure on how to express d/dt P1 in the equation, since that itself is an equation of the system.

Could you help me reformulate it please? It would be great!

[Torsten]

This means that

V_sa(t) = V_sa(t=0) + Ca*(P1(t)-P1(t=0))

So you don't need to include V_sa in your system of equations.

Best wishes

Torsten.

[gorilla3]

But if I elaborate: d/dt V_sa= d/dtCa * (P1-P_ic) + Ca* d/dt (P1- P_ic).

where Ca and P_ic are numbers so their derivative is 0, but P1 is not a number. Indeed, one of the equations in the system is:

• d/dt P1 = ( (P_a_b-P1)/(R_la+0.5*R_sa_b) - (P1-P2)/(0.5*R_sa_b+R_sv) )/Ca_b

So I can't just remove V_sa.

My question is how to express:

• d/dtCa * (P1-P_ic) + Ca* d/dt (P1- P_ic)

as the 1st equation of the system, since I can't use the d/dt symbol in it.

[gorilla3]

Hi Torsten,

in case you find my last comment confusing, here is a better overview: https://uk.mathworks.com/matlabcentral/answers/362979-ode-45-formulation-of-equations-in-system

Thanks again!

[Torsten]

Integrating your first equation from t'=0 to t'=t gives

 V_sa(t)-V_sa(0)= Ca*(P1(t)-P_ic) - Ca*(P1(0)-P_ic)

This means that V_sa(t) can be expressed by P1(t) as

 V_sa(t)=
 V_sa(0)+ Ca*(P1(t)-P_ic) - Ca*(P1(0)-P_ic)=
 V_sa(0)+Ca*(P1(t)-P1(0))

Consequently, you don't need to include a differential equation for V_sa in your system. The solution can be derived from the solution for P1 by the formula from above.

If you insist on solving a differential equation for V_sa:

d/dt(V_sa) = d/dt(Ca*(P1-P_ic)) = Ca*d/dt(P1)

Now for d/dt(P1), insert the expression from the right-hand side of the differential equation for P1.

Best wishes

Torsten.