system of second order implicit ode and implicit functions Matlab

Hello all,

Is there a way to solve a system of implicitly defined 2nd order ode equations

 f1(x, y(x), y’(x),y’’(x),M(x), sigma_x(x), sigma_y(x), mu_x(x), mu_y(x), R(x), ksi(x))=0

. .

 f8(x, y(x), y’(x),y’’(x),M(x), sigma_x(x), sigma_y(x), mu_x(x), mu_y(x), R(x), ksi(x))=0

with initial conditions on y(0), y’(0)

where

 a,a_, r, rho, tau are parameters,

 M(x), sigma_x(x), sigma_y(x), mu_x(x), mu_y(x), R(x), ksi(x)

are implicitly defined by the system of equations and together with 2nd order ode implicit.

In particular, equations I am trying to solve are

 M =  1-exp(mu_y +sigma*sigma_y-1/2*(sigma+sigma_y)^2*tau+(sigma+sigma_y)*sqrt(tau));

 sigma_x = 1/x*(1/M-1)*(sigma+sigma_y) ; 

 sigma_y = sigma_x*x*y'(x)/y(x); 

 mu_x =(1/x*(1/M*a/y(x)-rho+(1/M-1)*(mu_x+sigma*sigma_y-R)+(1-1/M)*(sigma+sigma_y)^2);

 mu_y = y’(x)/y(x)*mu_x+1/2*y’’(x)/y(x)*sigma_x; 

 (r*(1-x)+rho*x)*y(x) =a*x/M+(1-x/M)*a_;

 a_ /y(x) +mu_y+sigma*sigma_y-R =( (1-x/M)/(1-x)*(sigma+sigma_y));

 a/ y(x) +mu_y+sigma*sigma_y-R =( 1/M(sigma+sigma_y)^2 + ksi*y(x)*M);

The equations come from stochastic calculus.I tried to use ode15i but the problem is I don’t see how to reduce the system of 8 equations to 1 equation in this case.

Thank you,

Branka Markovic

Answer 1

[Steven Lord]

See the "Solve Robertson Problem as Implicit Differential Algebraic Equations (DAEs)" example on the ode15i documentation page. It shows the technique you can use to solve your system. In the example, the vector y that robertsidae accepts is [y_1; y_2; y_3] in the mathematical formulation of the system.

[Branka Markovic]

@ Steven Lord Thank you very much for your answer. The index of system of DAEs here is 3, while the index of Robertson Problem is 1. I found the link how to proceed if the index is higher than 1, any help is appreciated.