Unable to solve the collocation equations -- a singular Jacobian encountered.

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Farooq Aamir
Farooq Aamir 2024-1-27
编辑： Torsten 2024-1-27
function Sc_1
% Define constants
phi = 0.02;
R_s = 1738;
R_f = 1053;
S_s = 230000000;
S_f = 0.18;
Cp_s = 1046.7;
Cp_f = 3594;
K_s = 156;
K_f = 0.492;
We = 0.3;
Ha = 0.3;
A = 0.1;
Pr = 4;
Q_star = 0.1;
Ec = 0.1;
s_1 = 0.1;
Sc = 0.3;
s_2 = 0.1;
Kr = 0.2;
s_3 =0.1;
Lb = 0.3;
Pe = 0.1;
delta_1 = 0.1;
k_prime = 0.2;
k_2prime = 0.1;
M_0 = 0.6;
alpha = pi/2;
n = -0.803;
B_2 = (1 - phi)^-2.5;
B_1 = (1-phi)+phi*(R_s/R_f);
B_4 = ((1-phi)+phi*((R_s*Cp_s)/(R_f*Cp_f)));
B_3 = ((S_s+2*S_f)-2*phi*(S_f-S_s))/((S_s+2*S_f)+phi*(S_f-S_s));
B_5 = ((K_s+2*K_f)-2*phi*(K_f-K_s))/((K_s+2*K_f)+phi*(K_f-K_s));
% Solve the BVP
% Create an options structure with specified tolerances and a Jacobian function
x = linspace(0, 1, 10);
options = bvpset('RelTol',1e-6,'AbsTol',1e-6);
solinit = bvpinit(x, [1 1 1 0 0 0 0 0 0]);
sol = bvp4c(@bvpexam2, @bcfun, solinit, options);
x_vals = sol.x;
y_vals = sol.y;
% Plot the solutions
figure(1);
plot(x_vals, y_vals(2,:), 'LineWidth', 1.3);
hold on; % Keep the plot for next iterations
figure(2);
plot(x_vals, y_vals(4,:), 'LineWidth', 1.3);
hold on; % Keep the plot for next iterations
figure(3);
plot(x_vals, y_vals(6,:), 'LineWidth', 1.3);
hold on; % Keep the plot for next iterations
figure(4);
plot(x_vals, y_vals(8,:), 'LineWidth', 1.3);
hold on; % Keep the plot for next iterations
% Boundary and ODE functions
function res = bcfun(ya, yb)
res = [Pr*ya(1) + (B_5/B_1)*M_0*ya(5)-0;
ya(2) - 1;
ya(4)+k_prime*ya(5)-0;
ya(6)+k_2prime*ya(7)-0;
ya(8)-0;
yb(2) - A;
yb(4) - (1 - s_1);
yb(6) - (1 - s_2);
yb(8) - (1 - s_3)];
end
function ysol = bvpexam2(x, y)
yy1 = (B_1*(y(2)^2 - y(1) * y(2)) + B_3*sin(alpha)^2*Ha*(y(2)-A) - A^2)/(B_2 + (3*(n- 1)/2)*B_2*We*y(3)*y(3));
yy2 = ((Pr*B_4)*(s_1*y(2) + y(2)*y(4) - y(1)*y(5)) ...
- B_2*Pr*Ec*((y(3))^2)*((1+(3*(n-1)/2))*We*(y(3))^2) + Q_star*(y(4)+s_1) ...
- B_3*Ha*Ec*Pr*((sin(alpha))^2)*((y(2) - A)^2))/B_5;
yy3 = Sc*(y(2)*y(6) + s_2*y(2) - y(1)*y(7)) - Kr*(y(6)+ 1/s_2);
yy4 = Lb*(y(8)+s_3) - Lb*y(9) + Pe*(y(9)*y(7) + (y(8)+delta_1+s_3));
ysol= [y(2); y(3); yy1; y(5); yy2; y(7); yy3; y(8); yy4];
end
end
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Torsten
Torsten 2024-1-27
编辑：Torsten 2024-1-27
There can be many reasons that your code fails with this error message.
Mostly, it's due to programming errors, wrong parameter values or bad initial guesses for the functions.

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