I want to solve the following Cauchy problem \begin{equation} y' = y^2 + \frac{t^4 - 6t^3 + 12t^2 - 14t + 9}{(1+t)^2} \end{equation}

with initial condition: $y(0) = 2$ for $t \in [0,1.6]$ using a 3 steps Adam-Moulton method implemented in Matlab as follows:

function [t,u] = AM3(fun,t0,T,y0,N)
  h = (T-t0)/N; % integration step
  t = t0:h:T; %time mesh
  [t(1:3),u(1:3)] = RK4(fun,t0,t0+2*h,y0,2); % first 3 steps

  for n=1:N-2
    fn = feval(fun,t(n),u(n));
    fn1 = feval(fun,t(n+1),u(n+1));
    fn2 = feval(fun,t(n+2),u(n+2));
    F = @(z) z - u(n+2) - (h/24)*(9*feval(fun,t(n+3),z) + 19*fn2 - 5*fn1 + fn);
    z0 = u(n+1);
    z1 = u(n+2);
    dF = @(z) 1 - (h/24)*9*(2*z);
    u(n+3) = newton(F,dF,z0);

N is the number of integration nodes used (i.e. the size of the time mesh). The function returns a discretized solution of the problem in the array $u$. Since the method is implicit at every iteration I use Newton's method written as

function y = newton(f,df,x)
 fx = f(x);
 tol = 1e-10;
 itermax = 1e3;
 iter = 0;
 while abs(fx) > tol && iter < itermax
    x = x - f(x)/df(x);
    fx = f(x);

My problem is that it seems that the equation is very ill-conditioned since even with a tiny integration step $h$ what I obtain is something like this: enter image description here the problem persisting with much bigger $N$.

Despite the equation being pretty ugly it has en exact solution: $y(t) = \frac{(t-1)(t-2)}{t+1}$ which is relatively simple.

Can you help me please getting around this? I double checked the algorithms and they seems fine; I also tried to change Newton's method with some other (secant, fixed point, etc) but no novelty occured.

I also tried changing altoghether the method used with an Adam-Bashforth or Runge-Kutta one but to no avail.

  • 1
    $\begingroup$ Did you try to run it for the simple test problem $y'= -5y, y(0)=1$ ? Does it have the right order for that equation? If not, your implementation is wrong $\endgroup$ – VoB Jan 7 at 21:11
  • 2
    $\begingroup$ If you claim that $y(t) = \frac{(t-1)(t-2)}{t+1}$ has to be a solution, then the equation should have a minus, not a plus: $$\frac{dy}{dt} \, = \, y^2 \, - \, \frac{t^4 - 6t^3 +12t^2 - 14t + 9}{(t+1)^2}$$ $\endgroup$ – Futurologist Jan 8 at 5:10

You can de-singularize Riccati equations $y'=y^2+a$ by setting $y=-\frac{u'}{u}$ to get $u''+au=0$. If $a$ is continuous on the integration interval, the solution will be well-behaved on that interval. Solving with $u(0)=1$ and $u'(0)=-y(0)=-2$ gives the plot

enter image description here

with a root of $u$ and thus a singularity of $y$ at about $t=0.35230$. In any numerical integration, all that follows after that is to be considered random noise.

In the task as you obviously intended it, you have an equation $y'=y^2+p'-p^2$, that is, $a=p'-p^2$, for some given solution function $p$. The 4th degree term in the nominator comes from the second term, so needs to have a negative coefficient. With this corrected sign in the equation the solution for the transformed equation looks like

enter image description here

so the denominator $u$ is a solid distance away from zero. Solving the equation directly also confirms this. There is no divergence even under heavy distortion of the initial value

enter image description here


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