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I am trying to solve a 4th order differential equation using shooting method by disintegrating the ODE into four coupled first order ODEs. I only have initial conditions. So I am trying to perturb y(1) and y(4) and use them as a parameter to achieve a target value for y(3). But all I get is NaNs on my output matrices for y(1), y(2), y(3) and y(4). Any inputs will be appreciated.

Here is my code:

%%Constants

A = 1e-20; % Hamaker Constant

R_FC = 24.597; % Gas Constant of FC-72 (J/kg-K)

c = 1; % Accomodation Coefficient

sigma = 0.01; % Surface Tension Coefficient (N/m)

rho_l = 1593.84; % Liquid Density (kg/m3)

k_l = 0.0544; % Liquid Conductivity (W/m-K)

mu_l = 0.0004683; % Liquid Viscosity (kg/m2-s)

rho_v = 11.61; % Vapor Density (kg/m3)

h_lv = 88000; % Enthalpy of Vaporization (J/kg)

Twall = 334; % Wall Temperature (K), 10 degrees superheat

Tsat = 329; % Saturation Temperature of FC-72 at 1 atm (K)

R_int = ((2 - c)/(2*c))*(Tsat^(1.5))*((2*pi*R_FC)^(0.5))/(rho_v*(h_lv^2));

delta_ad = (A/(rho_l*h_lv*((Twall/Tsat) - 1)))^(1/3);

e1 = delta_ad/1000;

deltaP_ad = A/((delta_ad)^3);

deltaP_ad_corr = A/((delta_ad+e1)^3);

eQ = 1;

%%Function

tpcl = @(xi,y) [y(2); (1/sigma)*((1+(y(2)*y(2)))^(1.5))*(y(3)-(A/(y(1)^3))); (3*mu_l/(rho_l*h_lv))*(-y(4)/(y(1)^3)); (Twall-Tsat-(Tsat*y(3)/(rho_l*h_lv)))/((y(1)/k_l)+R_int)];

xi_end = 5e-7;

N = 1000;

[xi,y1] = rk4(tpcl,[0 xi_end],[delta_ad 0 deltaP_ad 0],N);

[xi,y2] = rk4(tpcl,[0 xi_end],[delta_ad+e1 0 deltaP_ad_corr eQ],N);

delta_a = delta_ad;

delta_b = delta_ad + e1;

Q_a = 0;

Q_b = eQ;

slope_a = y1(2,end);

slope_b = y2(2,end);

%%Shooting Method

tol = 1e-5;

iter = 10;

slope_target = tan(pi/18);

for i = 1:iter

Q_new = Q_a + (Q_b-Q_a)/(slope_b-slope_a)*(slope_target-slope_a);

delta_new = delta_a + (delta_b-delta_a)/(slope_b-slope_a)*(slope_target-slope_a);

deltaP_new = A/((delta_new)^3);

[xi,y] = rk4(tpcl,[0 xi_end],[delta_new 0 deltaP_new 0],N);

fprintf('iter:%2d, delta(0)=%17.15f, slope(xi_end)=%17.15f\n',i,delta_new,y(2,end));

if (abs(y(2,end)-slope_target) <= tol)

break;

end

Q_a = Q_b;

Q_b = Q_new;

delta_a = delta_b;

delta_b = delta_new;

slope_a = slope_b;

slope_b = y(2,end);

end

Jan
on 4 Jul 2013

Edited: Jan
on 4 Jul 2013

Anonymous functions are hard to debug. Better define the ODE as a function and use http://www.mathworks.de/matlabcentral/answers/1971 to define the constants on demand.

Then you can use the debugger to step through the program line by line until you find the source of the NaN's by your own. This is more efficient to let all forum users debug the code for you, especially when the function rk4() is not known.

Jan
on 7 Jul 2013

This could have different reasons: A bug in the code, a division by zero or another mathematical effect. It is your turn to find the exact cause for the creation of the NaNs. E.g. this might be useful:

dbstop if naninf

Gentian Zavalani
on 7 Jul 2013

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