system of second order implicit ode and implicit functions Matlab
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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
回答（1 个）
Steven Lord
2016-9-20
0 个投票
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.
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2016-9-20
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