Error solving bvp4c - Singular jacobian

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Jesús Parejo
Jesús Parejo el 18 de Mayo de 2022
Comentada: Jesús Parejo el 18 de Mayo de 2022
Hello everyone.
I'm trying to solve in Matlab2017b an ODE with the boundary conditions:
,
For this purpose, I have used the solver bvp4c. I think that this equation must be solvable for all values of [z1 z2 z3] because it has the form of a generic forced oscillator. However, there are many for which appears the error: singular jacobian (like the values I have written down) and I cannot guess which is the problem. Any idea?
Thank you in advance!
%Constants
hb = 6.626e-34/(2*pi);
m = 9*1.660538921e-27;
w0 = 2e6*2*pi;
Cc = (1.6e-19)^2/(4*pi*8.854e-12);
R = 10;
gammam = 1;
gammap = (3*R^3/(R^3+2))^(1/4);
NT = 50;
n = 100;
tf = 3.2e-6;
%Initial seed
z=[-104.2545 628.8529 33.2914];
z1=z(1); z2 =z(2); z3=z(3);
New1 = bvp4c(@(t,y) new_qubic(t, y, z1, z2, z3, gammam, gammap, tf, w0, Cc, m),@bvp4bc,solinit,options);
function dydx=new_qubic(t, y, z1, z2, z3, gammam, gammap, tf, w0, Cc, m)
z4=0;
rhop=1+(-126-z1-5*z2-15*z3-35*z4+126*gammap)*t^5/tf^5+...
(420+5*z1+24*z2+70*z3+160*z4-420*gammap)*t^6/tf^6+...
(-540-10*z1-45*z2-126*z3-280*z4+540*gammap)*t^7/tf^7+...
(315+10*z1+40*z2+105*z3+224*z4-315*gammap)*t^8/tf^8+...
(-70-5*z1-15*z2-35*z3-70*z4+70*gammap)*t^9/tf^9+...
z1*t^10/tf^10+z2*t^11/tf^11+z3*t^12/tf^12+z4*t^13/tf^13;
rho1p=5*(-126-z1-5*z2-15*z3-35*z4+126*gammap)*t^4/tf^5+...
6*(420+5*z1+24*z2+70*z3+160*z4-420*gammap)*t^5/tf^6+...
7*(-540-10*z1-45*z2-126*z3-280*z4+540*gammap)*t^6/tf^7+...
8*(315+10*z1+40*z2+105*z3+224*z4-315*gammap)*t^7/tf^8+...
9*(-70-5*z1-15*z2-35*z3-70*z4+70*gammap)*t^8/tf^9+...
10*z1*t^9/tf^10+11*z2*t^10/tf^11+12*z3*t^11/tf^12+13*z4*t^12/tf^13;
rho2p=20*(-126-z1-5*z2-15*z3-35*z4+126*gammap)*t^3/tf^5+...
30*(420+5*z1+24*z2+70*z3+160*z4-420*gammap)*t^4/tf^6+...
42*(-540-10*z1-45*z2-126*z3-280*z4+540*gammap)*t^5/tf^7+...
56*(315+10*z1+40*z2+105*z3+224*z4-315*gammap)*t^6/tf^8+...
72*(-70-5*z1-15*z2-35*z3-70*z4+70*gammap)*t^7/tf^9+...
90*z1*t^8/tf^10+110*z2*t^9/tf^11+132*z3*t^10/tf^12+156*z4*t^11/tf^13;
rho3p=60*(-126-z1-5*z2-15*z3-35*z4+126*gammap)*t^2/tf^5+...
120*(420+5*z1+24*z2+70*z3+160*z4-420*gammap)*t^3/tf^6+...
210*(-540-10*z1-45*z2-126*z3-280*z4+540*gammap)*t^4/tf^7+...
336*(315+10*z1+40*z2+105*z3+224*z4-315*gammap)*t^5/tf^8+...
504*(-70-5*z1-15*z2-35*z3-70*z4+70*gammap)*t^6/tf^9+...
720*z1*t^7/tf^10+990*z2*t^8/tf^11+1320*z3*t^9/tf^12+1716*z4*t^10/tf^13;
rho4p=120*(-126-z1-5*z2-15*z3-35*z4+126*gammap)*t^1/tf^5+...
360*(420+5*z1+24*z2+70*z3+160*z4-420*gammap)*t^2/tf^6+...
840*(-540-10*z1-45*z2-126*z3-280*z4+540*gammap)*t^3/tf^7+...
1680*(315+10*z1+40*z2+105*z3+224*z4-315*gammap)*t^4/tf^8+...
3024*(-70-5*z1-15*z2-35*z3-70*z4+70*gammap)*t^5/tf^9+...
5040*z1*t^6/tf^10+7920*z2*t^7/tf^11+11880*z3*t^8/tf^12+17160*z4*t^9/tf^13;
rhom=126*(gammam-1)*t^5/tf^5-420*(gammam-1)*t^6/tf^6+...
540*(gammam-1)*t^7/tf^7-315*(gammam-1)*t^8/tf^8+70*(gammam-1)*t^9/tf^9+1;
rho1m=630*(gammam-1)*t^4/tf^5-2520*(gammam-1)*t^5/tf^6+...
3780*(gammam-1)*t^6/tf^7-2520*(gammam-1)*t^7/tf^8+630*(gammam-1)*t^8/tf^9;
rho2m=2520*(gammam-1)*t^3/tf^5-12600*(gammam-1)*t^4/tf^6+...
22680*(gammam-1)*t^5/tf^7-17640*(gammam-1)*t^6/tf^8+5040*(gammam-1)*t^7/tf^9;
rho3m=7560*(gammam-1)*t^2/tf^5-50400*(gammam-1)*t^3/tf^6+...
113400*(gammam-1)*t^4/tf^7-105840*(gammam-1)*t^5/tf^8+35280*(gammam-1)*t^6/tf^9;
rho4m=15120*(gammam-1)*t^1/tf^5-151200*(gammam-1)*t^2/tf^6+...
453600*(gammam-1)*t^3/tf^7-529200*(gammam-1)*t^4/tf^8+211680*(gammam-1)*t^5/tf^9;
wp=sqrt((sqrt(3)*w0)^2/rhop^4-rho2p/rhop);
w1p=1/2/wp*(-4*(sqrt(3)*w0)^2*rho1p/rhop^5-(rho3p*rhop-rho2p*rho1p)/rhop^2);
w2p=-1/4/wp^3*(-4*(sqrt(3)*w0)^2*rho1p/rhop^5-(rho3p*rhop-rho2p*rho1p)/rhop^2)^2+...
1/(2*wp)*(-(4*(sqrt(3)*w0)^2*rho2p*rhop-20*(sqrt(3)*w0)^2*rho1p^2)/rhop^6-(rho4p*rhop^2-rho2p^2*rhop-2*rho3p*rho1p*rhop-2*rho2p*rho1p^2)/rhop^3);
wm=sqrt(w0^2/rhom^4-rho2m/rhom);
w1m=1/2/wm*(-4*w0^2*rho1m/rhom^5-(rho3m*rhom-rho2m*rho1m)/rhom^2);
w2m=-1/4/wm^3*(-4*w0^2*rho1m/rhom^5-(rho3m*rhom-rho2m*rho1m)/rhom^2)^2+...
1/(2*wm)*(-(4*w0^2*rho2m*rhom-20*w0^2*rho1m^2)/rhom^6-(rho4m*rhom^2-rho2m^2*rhom-2*rho3m*rho1m*rhom-2*rho2m*rho1m^2)/rhom^3);
ddd=(4*2^(2/3)*Cc^(1/3)*(-2*m*wm*w1m+2*m*wp*w1p)^2)/(9*(-m*wm^2+m*wp^2)^(7/3))-...
(2^(2/3)*Cc^(1/3)*(-2*m*w1m^2+2*m*w1p^2-2*m*wm*w2m+2*m*wp*w2p))/(3*(-m*wm^2+m*wp^2)^(4/3));
dydx=[y(2) -sqrt(m/2)*ddd-wp^2*y(1)];
end
function res = bvp4bc(ya,yb)
res = [ya(1) ya(2)];
end
  2 comentarios
Torsten
Torsten el 18 de Mayo de 2022
Neither "solinit" nor "options" is supplied.
Jesús Parejo
Jesús Parejo el 18 de Mayo de 2022
ups, sorry, these are solinit and options
nt= 2000;
tf=3.2e-6;
solinit = bvpinit(linspace(0,tf,nt),[0 0]);
options = bvpset('RelTol',10^(-6));

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Respuesta aceptada

Torsten
Torsten el 18 de Mayo de 2022
Editada: Torsten el 18 de Mayo de 2022
Try this code following John's suggestion:
%Constants
hb = 6.626e-34/(2*pi);
m = 9*1.660538921e-27;
w0 = 2e6*2*pi;
Cc = (1.6e-19)^2/(4*pi*8.854e-12);
R = 10;
gammam = 1;
gammap = (3*R^3/(R^3+2))^(1/4);
NT = 50;
n = 100;
tf = 3.2e-6;
%Initial seed
z=[-104.2545 628.8529 33.2914];
z1=z(1); z2 =z(2); z3=z(3);
[T,Y]=ode15s(@(t,y) new_qubic(t, y, z1, z2, z3, gammam, gammap, tf, w0, Cc, m),[0 tf],[0 0]);
plot(T,Y(:,1))
function dydx=new_qubic(t, y, z1, z2, z3, gammam, gammap, tf, w0, Cc, m)
z4=0;
rhop=1+(-126-z1-5*z2-15*z3-35*z4+126*gammap)*t^5/tf^5+...
(420+5*z1+24*z2+70*z3+160*z4-420*gammap)*t^6/tf^6+...
(-540-10*z1-45*z2-126*z3-280*z4+540*gammap)*t^7/tf^7+...
(315+10*z1+40*z2+105*z3+224*z4-315*gammap)*t^8/tf^8+...
(-70-5*z1-15*z2-35*z3-70*z4+70*gammap)*t^9/tf^9+...
z1*t^10/tf^10+z2*t^11/tf^11+z3*t^12/tf^12+z4*t^13/tf^13;
rho1p=5*(-126-z1-5*z2-15*z3-35*z4+126*gammap)*t^4/tf^5+...
6*(420+5*z1+24*z2+70*z3+160*z4-420*gammap)*t^5/tf^6+...
7*(-540-10*z1-45*z2-126*z3-280*z4+540*gammap)*t^6/tf^7+...
8*(315+10*z1+40*z2+105*z3+224*z4-315*gammap)*t^7/tf^8+...
9*(-70-5*z1-15*z2-35*z3-70*z4+70*gammap)*t^8/tf^9+...
10*z1*t^9/tf^10+11*z2*t^10/tf^11+12*z3*t^11/tf^12+13*z4*t^12/tf^13;
rho2p=20*(-126-z1-5*z2-15*z3-35*z4+126*gammap)*t^3/tf^5+...
30*(420+5*z1+24*z2+70*z3+160*z4-420*gammap)*t^4/tf^6+...
42*(-540-10*z1-45*z2-126*z3-280*z4+540*gammap)*t^5/tf^7+...
56*(315+10*z1+40*z2+105*z3+224*z4-315*gammap)*t^6/tf^8+...
72*(-70-5*z1-15*z2-35*z3-70*z4+70*gammap)*t^7/tf^9+...
90*z1*t^8/tf^10+110*z2*t^9/tf^11+132*z3*t^10/tf^12+156*z4*t^11/tf^13;
rho3p=60*(-126-z1-5*z2-15*z3-35*z4+126*gammap)*t^2/tf^5+...
120*(420+5*z1+24*z2+70*z3+160*z4-420*gammap)*t^3/tf^6+...
210*(-540-10*z1-45*z2-126*z3-280*z4+540*gammap)*t^4/tf^7+...
336*(315+10*z1+40*z2+105*z3+224*z4-315*gammap)*t^5/tf^8+...
504*(-70-5*z1-15*z2-35*z3-70*z4+70*gammap)*t^6/tf^9+...
720*z1*t^7/tf^10+990*z2*t^8/tf^11+1320*z3*t^9/tf^12+1716*z4*t^10/tf^13;
rho4p=120*(-126-z1-5*z2-15*z3-35*z4+126*gammap)*t^1/tf^5+...
360*(420+5*z1+24*z2+70*z3+160*z4-420*gammap)*t^2/tf^6+...
840*(-540-10*z1-45*z2-126*z3-280*z4+540*gammap)*t^3/tf^7+...
1680*(315+10*z1+40*z2+105*z3+224*z4-315*gammap)*t^4/tf^8+...
3024*(-70-5*z1-15*z2-35*z3-70*z4+70*gammap)*t^5/tf^9+...
5040*z1*t^6/tf^10+7920*z2*t^7/tf^11+11880*z3*t^8/tf^12+17160*z4*t^9/tf^13;
rhom=126*(gammam-1)*t^5/tf^5-420*(gammam-1)*t^6/tf^6+...
540*(gammam-1)*t^7/tf^7-315*(gammam-1)*t^8/tf^8+70*(gammam-1)*t^9/tf^9+1;
rho1m=630*(gammam-1)*t^4/tf^5-2520*(gammam-1)*t^5/tf^6+...
3780*(gammam-1)*t^6/tf^7-2520*(gammam-1)*t^7/tf^8+630*(gammam-1)*t^8/tf^9;
rho2m=2520*(gammam-1)*t^3/tf^5-12600*(gammam-1)*t^4/tf^6+...
22680*(gammam-1)*t^5/tf^7-17640*(gammam-1)*t^6/tf^8+5040*(gammam-1)*t^7/tf^9;
rho3m=7560*(gammam-1)*t^2/tf^5-50400*(gammam-1)*t^3/tf^6+...
113400*(gammam-1)*t^4/tf^7-105840*(gammam-1)*t^5/tf^8+35280*(gammam-1)*t^6/tf^9;
rho4m=15120*(gammam-1)*t^1/tf^5-151200*(gammam-1)*t^2/tf^6+...
453600*(gammam-1)*t^3/tf^7-529200*(gammam-1)*t^4/tf^8+211680*(gammam-1)*t^5/tf^9;
wp=sqrt((sqrt(3)*w0)^2/rhop^4-rho2p/rhop);
w1p=1/2/wp*(-4*(sqrt(3)*w0)^2*rho1p/rhop^5-(rho3p*rhop-rho2p*rho1p)/rhop^2);
w2p=-1/4/wp^3*(-4*(sqrt(3)*w0)^2*rho1p/rhop^5-(rho3p*rhop-rho2p*rho1p)/rhop^2)^2+...
1/(2*wp)*(-(4*(sqrt(3)*w0)^2*rho2p*rhop-20*(sqrt(3)*w0)^2*rho1p^2)/rhop^6-(rho4p*rhop^2-rho2p^2*rhop-2*rho3p*rho1p*rhop-2*rho2p*rho1p^2)/rhop^3);
wm=sqrt(w0^2/rhom^4-rho2m/rhom);
w1m=1/2/wm*(-4*w0^2*rho1m/rhom^5-(rho3m*rhom-rho2m*rho1m)/rhom^2);
w2m=-1/4/wm^3*(-4*w0^2*rho1m/rhom^5-(rho3m*rhom-rho2m*rho1m)/rhom^2)^2+...
1/(2*wm)*(-(4*w0^2*rho2m*rhom-20*w0^2*rho1m^2)/rhom^6-(rho4m*rhom^2-rho2m^2*rhom-2*rho3m*rho1m*rhom-2*rho2m*rho1m^2)/rhom^3);
ddd=(4*2^(2/3)*Cc^(1/3)*(-2*m*wm*w1m+2*m*wp*w1p)^2)/(9*(-m*wm^2+m*wp^2)^(7/3))-...
(2^(2/3)*Cc^(1/3)*(-2*m*w1m^2+2*m*w1p^2-2*m*wm*w2m+2*m*wp*w2p))/(3*(-m*wm^2+m*wp^2)^(4/3));
dydx=[y(2); -sqrt(m/2)*ddd-wp^2*y(1)];
end
  3 comentarios
Torsten
Torsten el 18 de Mayo de 2022
From the description of your problem,
dydx=[y(2); -sqrt(m/2)*ddd-wp^2*y(1)];
should be
dydx=[y(2); -sqrt(m/2)*ddd+wp^2*y(1)];
shouldn't it ?
Jesús Parejo
Jesús Parejo el 18 de Mayo de 2022
No, no. Calculations are correct. I have mislead the interpretation.

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Más respuestas (1)

John D'Errico
John D'Errico el 18 de Mayo de 2022
Editada: John D'Errico el 18 de Mayo de 2022
You have a simple classical ODE, with two INITIAL conditions, not boundary conditions at different ends. So use a tool like ODE45, NOT a boundary value solver. This is exctly what the ODE solvers (ODE45, etc.) are designed to solve.

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