The Optimal Parameters To Select The Method Of Numerical Solution Of Differential Equations Cycle

In this thesis, we consider the following periodic boundary value problem of differential equationwhere x(t) = (x₁(t),…,x₂(t))^T∈Rⁿ,t∈[0,T],T is the minimum positive periodictime , and f =[f₁,…,f_n]^T∈Rⁿ. It is well known that system (1) has at least oneperiodic solution which is stability under some conditions. We mainly discuss a kind of computation methods for problem (1).Since initial value is unknown, numerical methods for the initial value of ordinary differential equations, such as Euler method and the Runge-Kutta method, can not be used to solve it. In this thesis, we will introduce a new numerical method. The periodic problem can be transferred into an optimal parameter selection problem through considering the initial value problem as unknown parameter vectors. It is as follows.Consider following system:whereξ= (ξ₁,ξ₂,…,ξ_n)^T∈Rⁿ and define the cost functional:J(ξ)=1/2||x(T)-ξ||² (3)An optimal parameter selection problem(P) is: Given the system(2), find a systemparameterξ∈Rⁿ, such that the cost functional (3) is minimized overξ∈Rⁿ.Optimal parameter selection problems are basically mathematical programming problems, and hence all standard mathematical programming techniques can be applied to optimal parameter selection problems. When gradient formulas for the cost functional is derived, efficient gradient type algorithms for solving mathematical programming problems can be applied to solve the parameter selection problem. Here, two new algorithms are presented for known and unknown periods respectively.In addition, the numerical solutions of the periodic boundary value problem forimpulsive differential system are also presented. Since the statex(t)depends on theimpulsive times and the impulsive conditions, optimal parameter selection problems cannot be solved directly by available optimization techniques. By using a time scaling transform,y_i(S) = x(t_i-1+(t_i-t_i-1)s), 0≤S≤1,i=1,2,…,N.the impulsive problem is transferred to an equivalent optimal parameter selection problem. The gradient formulae of the cost function are obtained, and then a gradient-based computational method is established.At last, Four numerical examples are illustrated to our results by using the optimal control software, MISER 3.3. There are periodic boundary value problem of differential equation and equations with known periodic time, periodic boundary value problem of differential equation with unknown periodic time, and the periodic boundary value problem for impulsive differential system.