MATLAB: Trying to write an ODE solver using Backward Euler with Newton-Raphson method

backward eulernewton raphsonode

Hi, I'm trying to write a function to solve ODEs using the backward euler method, but after the first y value all of the next ones are the same, so I assume something is wrong with the loop where I use NewtonRoot, a root finding function I wrote previously. Do I need to include a separate loop over the Newton-Raphson method? If so, I'm not sure what index I would use.
function [t,y]=BackwardEuler(F,a,b,y0,N,err,imax)
h=(b-a)/N;
y(1)=y0;
t=a:h:b;
syms f(u)
f(u)=u-y0-h*F(u);
df=diff(f,u);
for ii=1:N
  y(ii+1)=NewtonRoot(f,df,y(ii),err,imax);
end
end

Best Answer

Alternatively, you can write a function.
function [t,yb,h] = backwardEuler(a,b,N,M,y0,f,fx)
h   =  (b-a)/N;
t     =  zeros(1,N);
yb    =  zeros(1,N);
t(1)  =  a;
yb(1) =  y0;
for j=1:N
x = yb(j);
t(j+1)=t(j)+h;
for i=1:M
x = x-(x-yb(j)-h*f(t(j+1),x))/(1-h*fx(t(j+1),x));
end
yb(j+1)= x;
end
end
f =  @(t,x) -0.800*x.^(3/2)+10.*2000 *(1 - exp(-3*t));
fx = @(t,x) -0.800*1.5*x.^(1/2);
a = 0;
b = 2;
N = 250;
M = 20;
err = 0.001;
y0 = 2000;
[t,yb,h] = backwardEuler(a,b,N,M,y0,f,fx);
figure()
plot(t,yb,'linewidth',1.5,'color','b')
grid;
a = title('Backward Euler Method');
set(a,'fontsize',14);
a = ylabel('n');
set(a,'Fontsize',14);
a = xlabel('t(s)');
set(a,'Fontsize',14);
axis([0 0.5 0 2000]),
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