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

Hello, I am receiving this error attempting to solve a 2nd order ODE with BVP4c.

I am solving for the internal temperature distribution of a nuclear fuel rod using the ode:

with Boundary Conditions:

where R = 0.2m and is the outer diameter

Let me know if you need anything else.

q and k are constants

r is a radius variable

%These are my setup functions. 
    function dydx = rod_BV(r,y) % Details ODE to be solved
        dydx = zeros(2,1);
        dydx(1) = y(2)
        dydx(2) = -(q/k)-((1/r)*y(1))  % This equation is invalid at r = 0
    end
    
    function res = rod_BC(ya,yb) % Details boundary conditions
        res = zeros(2,1);
        res(1) = ya(2);
        res(2) = yb(1) - T_s;
    end
   
% here is my implementation of BVP4C
solinit = bvpinit(linspace(0,R,4),[0,0]); % I have tried setting these 0 values to realmin 
options = bvpset('RelTol', 10e-4,'AbsTol',10e-7);
sol = bvp4c(@rod_BV,@rod_BC,solinit,options)

Answer 1

[Divija Aleti]

Hi Luke,

A singular Jacobian indicates that the initial guess causes the solution to diverge. The BVP4C function finds the solution by solving a system of nonlinear algebraic equations. Nonlinear solvers are only as effective as the initial guess they start with, so changing your starting guess may help. Also, BVP4C is responsible for providing a guess for the solution from one iteration to another.

A detailed explanation and possible solutions for this error are given in the link below:

MATLAB Answers

Also, based on the picture of the equation you are solving, I believe the rod_BV function should be written as follows:

function dydx = rod_BV(r,y) % Details ODE to be solved
    dydx = zeros(2,1);
    dydx(1) = y(2)
    dydx(2) = -(q/r)-((1/r)*y(2))  % This equation is invalid at r = 0
end

This is because T = y(1), T' = y(2), which implies, T'' = -(q/r)-(1/r)*y(2).

Hope this helps!

Regards,

Divija

Answer 2

[MOSLI KARIM]

Hi, here is the solution that I propose to you

%These are my setup functions. 
R=0.2;
q=1;
k=1;
eps=1.0000e-06;
r=linspace(0,R,4);
solinit = bvpinit(linspace(0,R,4),[0,0]); % I have tried setting these 0 values to realmin 
options = bvpset('RelTol', 10e-4,'AbsTol',10e-7);
sol = bvp4c(@(r,y)rod_BV(r,y,q,k),@rod_BC,solinit,options);
y=deval(sol,r);
plot(r,y(1,:))

function dydx = rod_BV(r,y,q,k) % Details ODE to be solved
        dydx = zeros(2,1);
        dydx(1) = y(2);
        dydx(2) = -(q/k)-((1/(r+eps))*y(1)) ; % This equation is invalid at r = 0
end
    
function res = rod_BC(ya,yb) % Details boundary conditions
        %res = zeros(2,1);
%         res(1) = ya(2);
%         res(2) = yb(1) - 200;
        res=[ya(2);yb(1)-200];
end
   
% here is my implementation of BVP4C
% solinit = bvpinit(linspace(0,R,4),[0,0]); % I have tried setting these 0 values to realmin 
% options = bvpset('RelTol', 10e-4,'AbsTol',10e-7);
% sol = bvp4c(@rod_BV,@rod_BC,solinit,options)

[Torsten]

I think it must be

dydx(2) = -(q/k)-((1/(r+eps))*y(2)) ; % This equation is invalid at r = 0

instead of

dydx(2) = -(q/k)-((1/(r+eps))*y(1)) ; % This equation is invalid at r = 0

[MOSLI KARIM]

yes sir torsten I forget to pute y(2)