Hello,

I have two regimes. Each regime is characterized by system of state differential equations (diff(S(t), t), diff(K(t), t)) and co-state differential equations (diff(psi[S](t), t), diff(psi[Iota](t), t)) as follows: 

(1)

diff(S(t), t) = -eta*K(t)*S(t)/(w*N*(S(t)+K(t))), diff(K(t), t) = eta*K(t)*S(t)/(w*N*(S(t)+K(t)))-upsilon

diff(psi[S](t), t) = eta*K(t)^2*(psi[S](t)-psi[Iota](t))/(w*N*(S(t)+K(t))^2), diff(psi[Iota](t), t) = eta*S(t)^2*(psi[S](t)-psi[Iota](t))/(w*N*(S(t)+K(t))^2)

(2)

diff(S(t), t) = -eta*K(t)*S(t)/(w*N*(S(t)+K(t))), diff(K(t), t) = eta*K(t)*S(t)/(w*N*(S(t)+K(t)))

diff(psi[S](t), t) = eta*K(t)^2*(psi[S](t)-psi[Iota](t))/(w*N*(S(t)+K(t))^2), diff(psi[Iota](t), t) = eta*S(t)^2*(psi[S](t)-psi[Iota](t))/(w*N*(S(t)+K(t))^2)

The first regime is employed from 0 to t1 (where t1 is unknown) and then regime 2, from t1 to T. I know the initial values for the state variables of the first system at t=0, that is, S(0)=S0 and K(0)=K0, as well as boundary conditions for the co-state variables for regime 2 at t=T, that is, psi_S(T)=a and psi_I(T)=b. 

I also know that my unknown t1 should satisfy the algebraic equation: -c - psi_S(t1)=0, where psi_S(t1) is the solution of the co-state diff equation of the regime 2 at t=t1. 

My question is: If I assume that the systems have not analytical solution, how can I found unknown t1 numerically? Moreover, asssume that all other parameters, such as T, eta, upsilon and others are given.

Hello all,

I have the following equation:

N*exp(-(1/2)*eta*epsilon*(N*alpha*epsilon*w+2*N*w*C[max]-alpha*epsilon*z-2*Q1*alpha)/(w*N))*S1*upsilon*w-N*S1*upsilon*w+K1^2*alpha*eta*z*epsilon+K1*alpha*eta*z*epsilon*S1 = 0

in which I need to find solution for epsilon (analytical solution) when epsilon>0.  

Thanks,

Dmitry

Hello,

I have the following condition with two unknowns: t1 and t3:

-N*exp(Q1*alpha*eta*t1/(N*w))*exp(-(((N*w-z)*t1^2+((-N*w+z)*t3+2*Q1)*t1+(1/2)*t3*(t3*(N*w-z)-2*Q1))*alpha-2*N*w*C[max]*(t1-(1/2)*t3))*eta/(N*w))*S1*upsilon*w+N*exp(Q1*alpha*eta*t1/(N*w))*exp(-(((N*w-z)*t1^2+((-N*w+z)*t1+2*Q1)*t1+(1/2)*t1*((N*w-z)*t1-2*Q1))*alpha-2*N*w*C[max]*(t1-(1/2)*t1))*eta/(w*N))*S1*upsilon*w+K1^2*exp((1/2)*t1^2*alpha*eta*z/(N*w))*exp(-(1/2)*t1^2*alpha*eta)*exp(t1*eta*C[max])*alpha*eta*t1*z-K1^2*exp((1/2)*t1^2*alpha*eta*z/(N*w))*exp(-(1/2)*t1^2*alpha*eta)*exp(t1*eta*C[max])*alpha*eta*t3*z+K1*exp((1/2)*t1^2*alpha*eta*z/(N*w))*exp(-(1/2)*t1^2*alpha*eta)*exp(t1*eta*C[max])*S1*alpha*eta*t1*z-K1*exp((1/2)*t1^2*alpha*eta*z/(N*w))*exp(-(1/2)*t1^2*alpha*eta)*exp(t1*eta*C[max])*S1*alpha*eta*t3*z = 0

I know that this condition holds when t1=t3. Does there exist an additional solution for t1 and t3 which satisfies this condition?

Thanks,

Dmitry

Hi,

I have a system of diff equations (see below). I am trying to obtain analytical solution. when I assume that z=wN, I receive such solution. Do anybody have idea if I know that z>wN, does this system has an analytical solution?

diff(K(t), t) = -(1/2)*(Q(t)^2*alpha^2*eta*upsilon-2*eta*alpha*(N*upsilon*w*C[max]-z*alpha*K(t))*Q(t)+N*w*(-2*C[max]*z*eta*alpha*K(t)+upsilon*((-N*w+z)*alpha+N*C[max]^2*w*eta)))*K(t)/((C[max]*w*N-alpha*Q(t))*upsilon*N*w)

diff(Q(t), t) = (1/2)*(-z*(Q(t)^2*alpha^2*eta-2*N*Q(t)*alpha*eta*w*C[max]+w*(w*(eta*C[max]^2-alpha)*N+z*alpha)*N)*K(t)-2*N*upsilon*w*(N*w-z)*(C[max]*w*N-alpha*Q(t)))/((C[max]*w*N-alpha*Q(t))*upsilon*N*w)

K(0) = K0, Q(0) = Q0

Thanks,

Dmitry

Hello,
I have a system of first order diff. equations which I would like to solve symbolically. Unfortunately, Maple does not solve the system. Do anybody have suggestions how can I solve this system (please see below):

diff(S(t), t) = -eta*(C[max]*w*N-alpha*Q(t))*K(t)*S(t)/(w*N*(S(t)+K(t))),

diff(K(t), t) = S(t)*((z*eta*alpha*(C[max]*w*N-alpha*Q(t))*S(t)-upsilon*(eta*alpha^2*Q(t)^2-2*C[max]*w*N*eta*alpha*Q(t)+((-N*w+z)*alpha+N*C[max]^2*w*eta)*N*w))*K(t)^2+(2*((1/2)*z*eta*(C[max]*w*N-alpha*Q(t))*S(t)+N*w*upsilon*(N*w-z)))*S(t)*alpha*K(t)+N*S(t)^2*w*alpha*upsilon*(N*w-z))/((K(t)^2*alpha*z+3*S(t)*K(t)*alpha*z+S(t)*(2*S(t)*z*alpha+upsilon*(C[max]*w*N-alpha*Q(t))))*(S(t)+K(t))*N*w),

diff(Q(t), t) = (-alpha*z*(z*eta*(C[max]*w*N-alpha*Q(t))*K(t)+N*w*upsilon*(N*w-z))*S(t)^2+(-z^2*eta*alpha*(C[max]*w*N-alpha*Q(t))*K(t)^2-(eta*alpha^2*Q(t)^2-2*C[max]*w*N*eta*alpha*Q(t)+N*w*((2*N*w-2*z)*alpha+N*C[max]^2*w*eta))*z*upsilon*K(t)-N*w*upsilon^2*(N*w-z)*(C[max]*w*N-alpha*Q(t)))*S(t)-N*w*z*alpha*upsilon*K(t)^2*(N*w-z))/((2*S(t)^2*alpha*z+(3*z*alpha*K(t)+upsilon*(C[max]*w*N-alpha*Q(t)))*S(t)+K(t)^2*alpha*z)*N*w*upsilon)

where initials conditions are:

S(0) = S0, K(0) = K0, Q(0) = Q0

Thanks,

Dmitry