MATLAB: Second order ODE – BVP

bvpMATLABode

Hi,
I am trying to obtain the solution of the following second order ODE but I struggle.
and on the interval [0, 0.5]. It is my understanding that 1) transoformation to the system of first order ODE and a Matlab bvp4c solver should be used. I wrote therefore the code:
function bvp5
xlow=0; xhigh=0.5;
solinit = bvpinit(linspace(xlow,xhigh,10),[0 1]); 
sol = bvp4c(@bvp5ode,@bvp5bc,solinit);
xint = linspace(xlow,xhigh);
Sxint = deval(sol,xint);
plot(xint,Sxint(1,:))
% -----------------------------------------------

function dydx = bvp5ode(x,y)
dydx = [ y(2) 0.64*y(1)-(2/x)*y(2) ];
% -----------------------------------------------
function res = bvp5bc(ya,yb)
res = [ ya(1)-0.2 yb(2) ];
I obtain the folelowing error: Unable to solve the collocation equations — a
singular Jacobian encountered.
1) How to form a guess ? I dont have any idea …
2) What is the problem with my equation or my code?
Best wishes,

Best Answer

I used an alternative method that makes use of Matlab's fzero function. You can also see one way of avoiding the problem at x = 0:
xlow=0; xhigh=0.5;
xspan = [xlow xhigh];
dydx0 = 0;
y0 = 0;  % Initial guess for y(x=0)
% Use fzero to get value of y0 that makes y(x=0.5) = 0.1;
y0 = fzero(@(y0) y0val(y0, xspan), y0);
disp(['y(x=0) = ' num2str(y0)])
% Now use ode45 to integrate from x=0 to x=0.5 with good starting value
% for y(x=0)
Y0 = [y0 0]; 
[x, Y] = ode45(@odefn, xspan, Y0);
plot(x,Y(:,1)),grid
xlabel('x'),ylabel('y')
function Z = y0val(y0, xspan)
         Y0 = [y0, 0];
         [~, Y] = ode45(@odefn, xspan, Y0);
         yend = Y(end,1);
         Z = yend - 0.1;
         
end
function dYdx = odefn(x, Y)
         y = Y(1); dydx = Y(2);
         d2ydx2 = 0.64*y;
         if dydx>0
             d2ydx2 = d2ydx2 - (2/x)*dydx;
         end
         dYdx = [dydx;
                 d2ydx2];
         
 end
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