MATLAB solving BVP using bvp4c

I am trying to solve a BVP in matlab using the bvp4c function. The following equation is a 3rd order linear homogeneous ODE with constant coefficients. I have solved second order linear and non-linear but I can't seem to figure out how to do a third order. I cannot find and documentation on how to make this adjustment.

The equation: y''' = a*h'

with boundary conditions y(0) = 0.3; y(15) = 0.7; y'(0) = 0;

The main code I have tried is below:

init = bvpinit(linspace(1,15,10),[0,0]);
sol = bvp4c(@rhs_bvp, @bc_bvp, init);
x = linspace(1,15,100);
BS = deval(sol, x);
plot(x,BS(1,:));

using the bc_bvp funtion below:

function [ bc ] = bc_bvp( yl, yr )
hi = 0.3;
ho = 0.7;
bc = [yl(1) - hi; 
      yl(2);
      yr(1) - ho];
end

and the rhs_bvp function below:

function [ rhs ] = rhs_bvp( x, y )
a = 26;
rhs = [y(3); a*y(2)];
end

and I get this error:

    Index exceeds matrix dimensions.

    Error in rhs_bvp (line 8)
    rhs = [y(3); a*y(2)];

    Error in bvparguments (line 105)
        testODE = ode(x1,y1,odeExtras{:});

    Error in bvp4c (line 130)
        bvparguments(solver_name,ode,bc,solinit,options,varargin);

Any help would be greatly appreciated.

[Torsten]

What is h' ?

Answer 1

[Torsten]

If h' = y', try

function main
init = bvpinit(linspace(1,15,10),[0,0,0]);
sol = bvp4c(@rhs_bvp, @bc_bvp, init);
x = linspace(1,15,100);
BS = deval(sol, x);
plot(x,BS(1,:));
end

function [ rhs ] = rhs_bvp( x, y )
a = 26;
rhs = [y(2); y(3); a*y(2)];
end

function [ bc ] = bc_bvp( yl, yr )
hi = 0.3;
ho = 0.7;
bc = [yl(1) - hi; 
      yl(2);
      yr(1) - ho];
end

Best wishes

Torsten.

[Taylor Nichols]

Thanks for the help Torsten! I guess including a picture of how I broke down the function would've helped. So if we have a 3rd order function we need to have 3 initial guesses? I'm also confused on why we set up the rhs variable that way too. I have only seen examples on 2nd order. And no detailed documentation on using this method exist. If you have anymore quick pointers it would be greatly appreciated!

[Torsten]

https://en.wikipedia.org/wiki/Ordinary_differential_equation

Section "Reduction of order"