Dmitry

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These are questions asked by Dmitry

I have the following double integral:

In the above integral, r and sigma are the random variables: r is distributed normally with a mean and standard deviation equal to sigma, which is a random variable by itself (k near sigma in integral is a known parameter). Sigma is distributed lognormally, with a known mean and standard deviation. The probability density function of the sigma is defined by fs(sigma) in the above integral. How is it possible to solve this integral in Maple?

Thanks in advance 

Hello All,

I have an autonomous system of ordinary nonlinear equations. In order to investigate it near the fixed points, I would like to reduce it to the normal or hyper-normal form. I see in the literature that some authors developed unique algorithms for the very specific differential systems (e.g., the systems with two/three eigenvalues), which is not my case.  

Maybe you know, if Maple has any specific commands that may conduct such type of reduction?

Thanks in advance,

Dmitry

 

 

Dear All,

I have the following differential equation:

(diff(lambda(t), t) = lambda(t)*(rho - A*beta*h^(1 - beta + a)/(A*h^(1 - beta + a)*t*(1 - beta) + k^(1 - beta))))     (1)

where rho, A, beta, a, h and k, are known parameters. Note, that all of them are positive and parameters rho, beta and a are less than one.

When solving this equation by dsolve, I get:

lambda(t) = _C1*exp(rho*t)*(A*h^(1 - beta + a)*t*beta - A*h^(1 - beta + a)*t - k^(1 - beta))^(beta/(-1 + beta))    (2)

which when substituting, for instance, beta=0.3 into the obtained solution, transforms into:

lambda(t) = _C1*exp(rho*t)/(-0.7*A*h^(0.7 + a)*t - k^0.7)^0.4285714286                                                                (3)

However, if I already have put beta=0.3 into the original differntial equation (see (1)) and solve it, I receive:

lambda(t) = _C1*exp(rho*t)/(10*k^(7/10) + 7*A*h^(7/10 + a)*t)^(3/7)                                                                       (4)

Clearly, the resultant solutions in (3) and (4) are quite different since the solution of (3) is any complex number and in (4) it is a real number. Note, that when solving the differential function (1) numerically, there exists a solution for any value of beta.

I also have tried to rename beta (supposing that maybe beta is any protected name) but the result remains the same. Also note, that if such manipulation is done with any other parameters, the solutions (3) and (4) are the same.

I will be thankful for any clarification regarding this issue. 

Thanks in advance,

Dmitry

 

Hello all,

I have an ODE system (please see bellow) where my unknowns are S(t) and K(t), all the the other symbols are known parameters. This system, by given the initial values for S and K, that is, S(0)=100 and K(0)=20, I can solve numerically.

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

In addition, I have an algebraic equation:

eq1:= -c2 + (K(t2)*S(t2)+w*N*S(t2))*z=0,

where S(t2) and K(t2) are the solutions of my ODE sys in S and K at t=t2. The t2 is unknown time variable. 

My question is: how can I find t2 such that my algebraic equation (eq1) is satisfied.

Thanks in advance,

Dmitry

 

 

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.

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