How can I solve the error "Unable to solve the collocation equations -- a singular Jacobian encountered" in bvp4c?

Question

J 2024-7-29

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2141026-how-can-i-solve-the-error-unable-to-solve-the-collocation-equations-a-singular-jacobian-encounte

评论： Torsten 2024-7-29

W2.m

I tried to solve the bvp using bvp4c procedure. But I have an error "Unable to solve the collocation equations -- a singular Jacobian encountered." How to resolve this issue?

proj()
Error using bvp4c (line 196)
Unable to solve the collocation equations -- a singular Jacobian encountered.

Error in solution>proj (line 31)
      sol= bvp4c(@projfun,@projbc,solinit,options);
function sol= proj
A = 0.5;
    Gr = 0.7;
    Gc = 0.5;
    Kp = 3.0;
    beta = 0.5;
    Pr = 0.3;
    Df = 0.2;
    Sc = 0.1;
    L0 = 0.5;
    Sr = 0.3;
    M = 1;
    Bi1 = 0.5;
    Bi2 = 0.5;
    K0 = 0.3;
myLegend1 = {};myLegend2 = {};
rr = [0.3 0.5 0.8];
for i =1:numel(rr)
    Re = rr(i);
   y0 = [1, 0, 1, 1, 0, 1, 0];
options =bvpset('stats','on','RelTol',1e-4);
    x = linspace(0,10,500);
   solinit = bvpinit(x,y0);
      sol= bvp4c(@projfun,@projbc,solinit,options);
      disp((sol.y(1,20)))
    figure(1)
    plot(sol.x,(sol.y(6,:)))
    grid on,hold on
    myLegend1{i}=['Pr = ',num2str(rr(i))];
xlabel('eta');
    ylabel('(thetas-thetaf)/thetas');
    i=i+1;
end
figure(1)
legend(myLegend1)
hold on
    function dy= projfun(~,y)
        dy= zeros(7,1);
          dy(1) = y(2);                                                       
          dy(2) = y(3);                                                       
          dy(3) = (0.5*A*y(3) - Gr*y(4) - Gc*y(6) + (M + Kp + A)*y(2)) / (1 + (1/beta));
          dy(4) = y(5);
          dy(5) = (Pr*(0.5*A*y(5)) - Pr*Df*Sc*(0.5*A*y(7) + 2*A*y(6) - L0*y(6))) / (1 - (Pr*Df*Sc*Sr));
          dy(6) = y(7);
          dy(7) = (Sc*(0.5*A*y(7) + 2*A*dy(6)) - Sr*Pr*(0.5*A*y(5) + 2*A*y(4))) / (1 - (Sr*Df*Pr));
    end
function res= projbc(ya,yb)
res= [
        ya(2) - (1 + K0*ya(3)); 
        ya(5) + Bi1*(1 - ya(4)); 
        ya(7) + Bi2*(1 - ya(6));  
        yb(2); 
        yb(4); 
        yb(6);
        yb(7)];
end
end

1 个评论
显示 -1更早的评论隐藏 -1更早的评论

Torsten 2024-7-29

How to resolve this issue?

It's not a technical issue. Usually, there is something wrong with the equations or the boundary conditions. So you should compare both of them with the mathematical description of the problem.

请先登录，再进行评论。

请先登录，再回答此问题。

Answer 1

Torsten 2024-7-29

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2141026-how-can-i-solve-the-error-unable-to-solve-the-collocation-equations-a-singular-jacobian-encounte#answer_1491931

编辑：Torsten 2024-7-29

在 MATLAB Online 中打开

Your ODE system is linear - thus you can determine its general solution having 7 free parameters. But incorporating of your boundary conditions leads to a linear system of equations where the coefficient matrix A is rank-deficient (rank 6 instead of rank 7). Consequently, your system is not solvable.

A = 0.5;

Gr = 0.7;

Gc = 0.5;

Kp = 3.0;

beta = 0.5;

Pr = 0.3;

Df = 0.2;

Sc = 0.1;

L0 = 0.5;

Sr = 0.3;

M = 1;

Bi1 = 0.5;

Bi2 = 0.5;

K0 = 0.3;

Mat = zeros(7);

Mat(1,2) = 1;

Mat(2,3) = 1;

Mat(3,2) = (M + Kp + A) / (1 + 1/beta);

Mat(3,3) = 0.5*A/(1 + 1/beta);

Mat(3,4) = -Gr/(1 + 1/beta);

Mat(3,6) = -Gc/(1 + 1/beta);

Mat(4,5) = 1;

Mat(5,5) = Pr*0.5*A/(1 - Pr*Df*Sc*Sr);

Mat(5,6) = - Pr*Df*Sc*(2*A - L0) / (1 - Pr*Df*Sc*Sr);

Mat(5,7) = - Pr*Df*Sc*0.5*A/ (1 - Pr*Df*Sc*Sr);

Mat(6,7) = 1;

Mat(7,4) = - Sr*Pr*2*A / (1 - Sr*Df*Pr);

Mat(7,5) = - Sr*Pr*0.5*A / (1 - Sr*Df*Pr);

Mat(7,7) = Sc*(0.5*A+2*A) / (1 - Sr*Df*Pr);

syms x1(t) x2(t) x3(t) x4(t) x5(t) x6(t) x7(t)

eqns = [diff(x1);diff(x2);diff(x3);diff(x4);diff(x5);diff(x6);diff(x7)]-Mat*[x1;x2;x3;x4;x5;x6;x7]==[0;0;0;0;0;0;0];

sol = dsolve(eqns,'MaxDegree',4)

sol = struct with fields:

x2: C3*(exp((t*(2976375*((13875*3^(1/2)*342589119285075140699353^(1/2))/1883723499362492536448 - 2321348855894375/1076413428207138592256)^(1/6) - 9802324*((3364850596725*((... x1: C1 + C2*(exp((t*(2976375*((13875*3^(1/2)*342589119285075140699353^(1/2))/1883723499362492536448 - 2321348855894375/1076413428207138592256)^(1/6) - 9802324*((33648505967... x3: C6*exp(-(t*(865^(1/2) - 1))/24) + C7*exp((t*(865^(1/2) + 1))/24) - C2*(exp((t*(2976375*((13875*3^(1/2)*342589119285075140699353^(1/2))/1883723499362492536448 - 23213488... x4: C2*(exp((t*(2976375*((13875*3^(1/2)*342589119285075140699353^(1/2))/1883723499362492536448 - 2321348855894375/1076413428207138592256)^(1/6) - 9802324*((3364850596725*((... x5: C4*exp((t*(9802324*((3364850596725*((13875*3^(1/2)*342589119285075140699353^(1/2))/1883723499362492536448 - 2321348855894375/1076413428207138592256)^(1/3))/960855558009... x6: C3*(exp((t*(2976375*((13875*3^(1/2)*342589119285075140699353^(1/2))/1883723499362492536448 - 2321348855894375/1076413428207138592256)^(1/6) - 9802324*((3364850596725*((... x7: C4*exp((t*(9802324*((3364850596725*((13875*3^(1/2)*342589119285075140699353^(1/2))/1883723499362492536448 - 2321348855894375/1076413428207138592256)^(1/3))/960855558009...

eqn1 = subs(sol.x2,t,0)-(1+K0*subs(sol.x3,t,0))==0;

eqn2 = subs(sol.x5,t,0)+Bi1*(1-subs(sol.x4,t,0))==0;

eqn3 = subs(sol.x7,t,0)+Bi2*(1-subs(sol.x6,t,0))==0;

eqn4 = subs(sol.x2,t,10)==0;

eqn5 = subs(sol.x4,t,10)==0;

eqn6 = subs(sol.x6,t,10)==0;

eqn7 = subs(sol.x7,t,10)==0;

[A,b] = equationsToMatrix([eqn1,eqn2,eqn3,eqn4,eqn5,eqn6,eqn7])

A =