BVP Error, help with explaination

Hello all,

I am struggeling a bit with understanding why I get an error message here. The code works fine with bigger numbers for A or with smaller numbers for B but is crashing for example with the given parameters. I already figured that y(1) is becoming NaN at some point which causes the code to crash. Could somebody help me understand why this is happening?

bvp()

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

Error in solution>bvp (line 8)
sol = bvp4c(@ode, @bcfun, sol_init);

function bvp
A = 50e-3;
B = 1;

xmesh = linspace(0, 1, 100);
sol_init = bvpinit(xmesh, @init);
sol = bvp4c(@ode, @bcfun, sol_init);

    function dydx = ode(x, y)
        dydx(1) = y(2);
        dydx(2) = exp(y(1)*log(10)/A);
        %disp(y(1))        
    end

% Define the boundary conditions
    function res = bcfun(ya, yb)
        res(1) = ya(1) - B;
        res(2) = yb(1);
    end

% Define the initial guess for the solution
    function guess = init(x)
        guess(1) = B;
        guess(2) = 0;
    end
end

Answer 1

[Ayush Anand]

The magnitude of the exponential function exp(y(1)*log(10)/A) blows up very rapidly as A is . This is most probably causing the overflow, leading to an Inf (infinity) value in MATLAB. Further calculations involving Inf can lead to NaN (Not a Number), as these are operations that MATLAB cannot numerically evaluate to a real or infinite number.

Also, the solver bvp4c iteratively adjusts y(1) and y(2) to satisfy both the differential equation and the boundary conditions. If during its iterations, y(1) becomes large enough, exp(y(1)*log(10)/A) will overflow. Once y(1) or any subsequent value becomes NaN, the solver cannot proceed with numeric computations and thus the error you see.

[Thanh Hoang]

Thanks a lot Ayush!

Is there a way to overcome this? Or is this system not solvable numerically?

[Ayush Anand]

I think as mentioned by Torsten in the other answer, the solver reaches the limits of computation/precision possible on the machine and hence can't proceed with any further feasible steps. It seems to be associated with the system of equations itself and you may want to change the formulation of the system.

Answer 2

[Torsten]

The solver is no longer able to capture the steep gradient of your solution at x=0.

Look at sol.x in the last step of the continuation method from below and how fine the mesh has to be chosen around x = 0.

bvp()

ans = 1x64

1.0e+00 * 0 0.000000000125289 0.000000000250578 0.000000000375868 0.000000000751735 0.000000001127603 0.000000002255205 0.000000006201814 0.000000010148424 0.000000025371059 0.000000040593694 0.000000060890541 0.000000213116894 0.000000365343247 0.000000548014871 0.000000730686495 0.000001803882283 0.000002877078072 0.000005822658004 0.000008768237935 0.000017536475870 0.000026304713805 0.000048225308642 0.000092546011329 0.000136866714015 0.000199340409301 0.000329630944865 0.000459921480429 0.000717008956755 0.000974096433081

<mw-icon class=""></mw-icon>

<mw-icon class=""></mw-icon>

function bvp
  A_array = [5e-1,3e-1,2e-1,1.5e-1,1e-1,0.9e-1,0.8e-1,0.7e-1,0.6e-1,0.55e-1,0.5e-1];
  B = 1;
  xmesh = linspace(0, 1, 100);
  sol_init = bvpinit(xmesh, [0 B]);
  hold on
  for i = 1:numel(A_array)
    A = A_array(i);
    sol = bvp4c(@ode, @bcfun, sol_init);
    plot(sol.x,sol.y(1,:))
    init = @(x)interp1(sol.x.',sol.y.',x);
    sol_init = bvpinit(sol.x,init);
  end
  format long
  sol.x
  hold off
  grid on

  function dydx = ode(x, y)
    dydx(1) = y(2);
    dydx(2) = exp(y(1)*log(10)/A);
     %disp(y(1))        
  end

% Define the boundary conditions
  function res = bcfun(ya, yb)
    res(1) = ya(1) - B;
    res(2) = yb(1);
  end
end