BVP4C: Error: Unable to solve the collocation equations -- a singular Jacobian encountered

Hello, I am facing "Unable to solve the collocation equations -- a singular Jacobian encountered" error using BVP4C function.Below are the details:

Equations:

1--> f''' + ff'' - f'^2 + B(f''^2-ff'''')+Zg' - M^2f' + D^2 = 0

2--> Yg''+ fg' +f'g - I(2g +f'') = 0

3--> h'' +Prfh'+PrQh = 0

Boundary Conditions:

f(0)=0, f'(0)=1+Kf''(0), f'(infty)=D, f''(infty) = 0

h'(0)=-ga*(1-h(0)) , h(infty)=0

g(0)=-Mzf''

I have formulated equations & boundary conditions in m script as follows:

ff = [y(2)  
      y(3)
      y(4)
      y(5)
      (- y(1)*y(3) + y(2)^2 - B*y(3)^2 - Z* y(5)+ M^2*y(2)-D^2)
      (-y(1)*y(5)-y(2)*y(4)+I*(2*y(4)+y(3)))
      y(6)
      y(7)
      -pr * ((y(1)*y(7)) + Q*(y(6)))];

res = [ya(1)
       ya(2) - 1 - K*ya(3)
       yb(1)
       yb(2)-D
       ya(4)+Mz*ya(3) 
       yb(3)
       ya(5)
       ya(6) + ga * (1 - ya(5))
       yb(6)];

Where , y(1) = f, y(2) = f', y(3) = f'', y(4) = g, y(5) = g'',y(6) = h, y(7) = h'

All the parameters used for equation(M,I,B etc.) are known & defined as global variables.

Script is developed referring to the BVP4C samples.

Please help me resolve this error. Thanks.

[Torsten]

Please indicate which functions are meant by y(1),y(2),...,y(9).

Best wishes

Torsten.

[Sachin Ganjare]

Thanks for responding. Below is the information:

y(1) = f, y(2) = f', y(3) = f'', y(4) = g, y(5) = g'',y(6) = h, y(7) = h'

Answer 1

[Torsten]

y1'=y2
y2'=y3
y3'=y4
y4'=solve your first equation for f''''
y5'=y6
y6'=solve your second equation for g''
y7'=y8
y8'=solve your third equation for h''

(y1=f, y2=f', y3=f'', y4'=f''', y5=g, y6=g', y7=h, y8=h')

Now specify 8 boundary conditions (instead of 7 in the problem formulation or 9 in the code as above), and you are done.

Best wishes

Torsten.

[Sachin Ganjare]

Thank you so much for responding, but I got really confused with the syntax of equation & boundary condition formulation. Can you please help.

[Sachin Ganjare]

Now I formulated equation like below:

ff = [y(2)
      y(3)
      (- y(1)*y(3) + y(2)^2 - B*y(3)^2 - Z* y(6)+ M^2*y(2)-D^2)
      y(5)
      (-y(1)*y(5)-y(2)*y(4)+I*(2*y(4)+y(3)))/Y
      y(7)
      -pr * ((y(1)*y(7)) + Q*(y(6)))];

And boundary conditions like below:

res = [ya(1)
       ya(2) - 1 - K*ya(3)
       yb(3)
       yb(2)-D
       ya(4)+Mz*ya(3) 
       yb(4)
       ya(6) + ga * (1 - ya(5))];

Still facing same problem. Can you please help.

Thanks.

[Torsten]

ff=[y(2)
    y(3)
    y(4)
    (y(4)+y(1)*y(3)-y(2)^2+B*y(3)^2+Z*y(6)-M^2*y(2)+D^2)/(B*y(1))
    y(6)
    (I*(2*y(5)+y(3))-y(2)*y(5)-y(1)*y(6))/Y
    y(8)
    -(pr*y(1)*y(8)+pr*Q*y(7))];

res = [ya(1)
       ya(2) - 1 - K*ya(3)
       yb(3)
       yb(2)-D
       ya(5)+Mz*ya(3) 
       yb(7)
       ya(8) + ga * (1 - ya(7))];

By I, you really mean the imaginary unit ?

Best wishes

Torsten.

[Torsten]

You will need to add one more boundary condition since you supplied 8 ODEs, but only 7 boundary conditions.

Best wishes

Torsten.

[Sachin Ganjare]

Thank you so much for your help, Torsten. Really appreciate your help.

Everything is working fine now.

Note:I is not imaginary unit but just another parameter in equation.

Answer 2

[fatima malik]

initialguess=bvpinit(linspace(0,10,5000),@mat4init);
sol=bvp4c(@rhs_ode1,@bcs1,initialguess);
eta=linspace(0,10,5000);
f=deval(sol,eta);
plot(eta,f(11,:),'LineWidth',1.2);
hold on
end
axis([0 1.4 0 1.05]);
xlabel('\eta','FontSize',14,'FontAngle', 'italic','FontName','Arial','Color','k');
ylabel('f(\theta)',

how to solve for singular jacobian ???