The problem with ‘bvp_RHS’ was actually not with it, but with the extra argument to bvpinit. Eliminating that then produced the error that ‘bvp_BC’ needs to return 11 boundary conditions, not 12 as it currently does. So, I commented-out the last one, and the code ran without error.
Check ‘bvp_BC’ and eliminate the extraneous boundary condition, since I might not have eliminated the correct one (I just wanted to see if the code otherwise worked).
phif = 0.1; % total length
phit = 0.05;
alpha = 0.3;
C_init = 1e-2;
solinit = bvpinit(linspace(-phif/2, phif/2, 100), @(x)mat4init(x, phif));
options = bvpset('Stats','on','RelTol',1e-5);
sol = bvp4c(@bvp_RHS, @(yL,yR)bvp_BC(yL, yR, phit, alpha), solinit, options)
figure
plot(sol.x, sol.y)
grid
legend(compose('$y_{%2d}$',1:11), 'Location','bestoutside', 'Interpreter','latex')
%-------------------------------------------------------------------------
function yinit = mat4init(x, phif)
yinit = [cos(pi/phif*x)
-pi/phif * sin(pi/phif*x)
0.1
0.1
0.1
-0.1
0.1
0
0
0
0
];
end
function dyds = bvp_RHS(C,y)
% Total 2+9=11 variables.
dyds = [ y(2)
-0.5*y(1)^3 - C*y(1) % -0.5*kappa^3 - C*kappa
y(6) % =tx
y(7) % =ty
y(8) % =tz
-y(3) + y(1)*y(9) % =-rx + kappa*nx
-y(4) + y(1)*y(10) % =-ry + kappa*ny
-y(5) + y(1)*y(11) % =-rz + kappa*nz
-y(1)*y(6) % =-kappa*tx
-y(1)*y(7) % =-kappa*ty
-y(1)*y(8) % =-kappa*tz
];
end
%-------------------------------------------------------------------------
function res = bvp_BC(yL, yR, phit, alpha)
% BCs at L and R boundaries, with constraint for eigenvalue C.
% Symmetrised BCs for r and t, so that L and R are at \pm phit/2.
res = [ yL(1)
yR(1)
yL(3)-cos(phit/2) % fix position: rx
yL(4)+sin(phit/2) % fix position: ry
yL(5) % fix position: rz
yR(3)-cos(phit/2) % fix position: rx
yR(4)-sin(phit/2) % fix position: ry
yR(5) % fix position: rz
yL(6)-sin(phit/2)*cos(alpha) % fix slope: tx
yL(7)-cos(phit/2)*cos(alpha) % fix slope: ty
yL(8)-sin(alpha) % fix slope: tz
% yR(6)-sin(phit/2)*cos(alpha) % fix slope: tx
];
end
The output doesn’t look very interesting, however if it does what you want, that may not be an issue.
.
