Shooting Method On Harmonic Equation

I'm really new to Matlab, so this may be ridicously easy what I'm about to ask, but bare with me please.

I'm trying to integrate the Harmonic equation y'' +(a^2)*y=0 with a=2.4, with BC y(0)=y'(pi)=0. I'm doing this to try and "shoot" for the actual value of y at y=pi which we obviously can find analytically but I need to get my head around the code so I can apply this to a more complicated problem.

Thanks.

Answer 1

[Torsten]

function main
  ydot0_start = 1.0;
  a = 2.4;
  iflag = 0;
  sol = fzero(@(x)fun_shooting(x,a,iflag),ydot0_start);
  iflag = 1;
  y_at_pi = fun_shooting(sol,a,iflag)
end

function res = fun_shooting(x,a,iflag)
  fun_ode = @(t,y)[y(2);-a^2*y(1)];
  tspan = [0,pi];
  y0 = [0;x];
  [t,y] = ode45(fun_ode,tspan,y0);
  if iflag == 0
    res = y(end,2);
  else
    res = y(end,1);
  end
end

[Alexander Kimbley]

Thank you so much! However I've tried using the code & i get the error message "function definition not supported in this context. Create functions in code file." I have MATLAB_R2018b, do I need to download another package or is it somehting else?

Thanks.

[Torsten]

If you save the above file as "main.m", load it in MATLAB and run it, there should be no problems.

[Alexander Kimbley]

Brilliant, thanks.

[Alexander Kimbley]

Hi, just to double check what are the conditions youve used on Y? Y(0)=0 and what other?

Thanks.

[Torsten]

I used y(0)=0 and y'(0)=x in "fun_shooting" and adjust x such that y'(pi)=0.

[Alexander Kimbley]

ahh okay, brilliant thanks, you've been a great help!!

[Alexander Kimbley]

So how would I go about changing the conditions to y(0)=0, y'(0)=1, where we alter 'a' to get y(pi)=0 to 10^-8 accuaracy say, assuming we dont actually know the true value of a.

Thanks.

[Torsten]

function main
  a0 = 4.5;
  a = fzero(@fun_shooting,a0)
end

function res = fun_shooting(x)
  fun_ode = @(t,y)[y(2);-x^2*y(1)];
  tspan = [0,pi];
  y0 = [0;1];
  [t,y] = ode45(fun_ode,tspan,y0);
  res = y(end,1);
end