Two-Point Boundary Value Problems - Numerical Methods in Engineering with MATLAB - page 305

function shoot4
% Shooting method for 4th-order boundary value
% problem in Example 8.4.

global XSTART XSTOP H % Make these params. global.

XSTART = 0; XSTOP = 1; % Range of integration.

H = 0.5; % Step size.

freq = 1; % Frequency of printout.

u = [0 1]; % Trial values of u(1).

% and u(2).

x = XSTART;
u = newtonRaphson2(@residual,u);
[xSol,ySol] = bulStoer(@dEqs,x,inCond(u),XSTOP,H);
printSol(xSol,ySol,freq)

functionF=dEqs(x,y) %Differentialequations.

F = [y(2) y(3) y(4) x;];

functiony=inCond(u) %Initialconditions;u(1)

y = [0 u(1) 0 u(2)]; % and u(2) are unknowns.

functionr=residual(u) %Boundaryresiduals.
global XSTART XSTOP H
r = zeros(length(u),1);
x = XSTART;
[xSol,ySol] = bulStoer(@dEqs,x,inCond(u),XSTOP,H);
lastRow = size(ySol,1);
r(1)= ySol(lastRow,1);
r(2) = ySol(lastRow,3);

Here is the output:

>> x y1 y2 y3 y4
0.0000e+000 0.0000e+000 1.9444e-002 0.0000e+000 -1.6667e-001
5.0000e-001 6.5104e-003 1.2150e-003 -6.2500e-002 -4.1667e-002
1.0000e+000 -4.8369e-017 -2.2222e-002 -5.8395e-018 3.3333e-001

Noting that

w 0 L 4
E I

1
L =

w 0 L 3
E I

dv
dx =

dv
d

d ξ

dx =

dy
d

dy
d ξ ξ ξ