Polynomial Approximation Of A Nonlinear Ordinary Differential Equation

Many physical systems are modeled by nonlinear autonomous differential equations of the following formwhere f : R→R is a nonlinear function,α_i are real numbers , k is the order of the system . If we put system (1) is written equivalently asWe associate an initial state x1(t)=x10,...,xk(t0) = xk0tosystem (2). We assume that conditions on f for which the solution of the initial value problem exists and is unique are satisfied . In general , the nonlinear vector field is a function of all the state variables, system(2)is a particular case. Of course, the dynamics of the system depend crucially on the scalar nonlinear function .System(2) associated with an initial state has the following general formwhere x=(x1,x2,...,xk) is the state vector, f : R~k→R~k is the vector field.Using a simple function to approximate a complicated function plays an important role in almost field of mathematics. Linear systems are not only simpler than nonlinear systems but also more complete than nonlinear systems on the facet of theory. A classical example is the asymptotic stability, where the method the most used for the determination of the stability type of an equilibrium point is based on Frechet derivative at the equilibrium point. So at the equilibrium point. we can using linear systems of approximation, when we deal with nonlinear systems.In the second chapter of this paper, giving two type of approximation which is used frequently. One is the classical linear approximation.Unfortunately the classical linearization near an equilibrium point has the following drawbacks(1) When presents at least a zero eigenvalue or a pair of purely imaginary eigenvalues, there is no equivalence between the linear and the nonlinear systems. i.e., the behavior of the linearized system can be very different from the nonlinear system. This statement results from the Hartment-Grobman theorem.(2)The classical linearization is a first order approximation, the quality of approximation degenerates for highly nonlinear system.(3)In many cases, it happens that the Jacobian matrix at the equilibrium point dose not exist. Classical linearization theorems do not apply in this case.As a result to these drawbacks, various complementary methods have been suggested to be used for some specific problems, and especially to overcome the drawbacks of the classical linearization.A least squares linearization method is suggested to approximate nonlinear cricuits equations. Benouaz and Arino gave the method the mathematical validity and they show the applicability of the method to solve stability problems. At each step, it gives a linear map, starting from the Jacobian matrix Df(x) estimated at the initial value x0. The optimal approximation of the nonlinear equation is obtained as a limit of the sequence of linear maps determined by the procedure. This can be used to solve nonlinear state equations. This method proves that the order of the approximation is two, or higher. Furthermore, in the scalar case, we give the analytic expression of the approximation.Consider the following system of nonlinear ordinary differential equation:Which can be approximated near the origin by the following linear systemwhere a is a real number given bywhere the approximation is not defined for x0 = 0 , So, a dose not depend on t, which only depends on x0. The real number a was computed under the assumption that the nonlinear vector field presents negative spectrum around the origin. shi ye qiong first relax the assumption of the negative spectrum and introduce a family of linearizations which is a generalization of equation(3):where n is a positive integer. Approximation value of a_n is determined by the minimum value of the following function Because polynomial has good character, in the third section, we introduce polynomial approximation to nonlinear ordinary differential equation. By compute the minimum of the following functionAccording to a necessary and sufficient condition for maximum-minimum principle of function of several variables, foraj,bj,they are existence and uniqueness at each step.We know the element of sequence of aj,bj is convergence, by provingεj<εj-1 < ...<ε1. The limit of aj, bj is the polynomial approximation near the initial point and prove that the order of the approximation is three or higher, the error between approximation solution and numerical solution is small.