Approximate Methods of the Decision Differential the Equations for Continuous Models of Economy

The review of some models of economy based on application of the ordinary differential equations is provided in article linear and nonlinear, and also the review of approximate methods of their decision. Expediency of use of this or that numerical methods of the solution of the differential equations is shown. For some equations of continuous models of economy in work the approximate method is offered and reasonable. Estimates of convergence of approximate methods are given in the corresponding functional spaces. Results of the numerical solution of these equations in the form of tables and schedules are also received. The comparative analysis of the received results is carried out.


Introduction
The review of some models of economy based on application of the ordinary differential equations is provided in article linear and nonlinear, and also the review of approximate methods of their decision.Expediency of use of this or that numerical metods of the solution of the differential equations is shown.For some equations of continuous models of economy in work the approximate method is offered and reasonable.Estimates of convergence of approximate methods are given in the corresponding functional spaces.Results of the numerical solution of these equations in the form of tables and schedules are also received.The comparative analysis of the received results is carried out.

Theory
It is known (Crassus & Chuprynov, 2002;Ross, 2006) that some continuous models of economy, for example, investigating economic dynamics, defining evolutions of economic systems are closely connected with the differential equations, with systems of the differential equations of the first order which, in turn, at the decision lead to the equations of the highest orders.So, for example in the study of the dynamics of capital-labor ratio (in the neoclassical growth model) it is presented as a function of time t and obtain the following nonlinear first order differential equation with separable variables: Productivity as the function of time determined through Y = F (K, L) -the national income, where F -uniform production function of the first order for which is fair F (TK,TL) = TF (K, L), To -the volume of capital investments (business assets), L -the volume of expenses of work.We will enter into consideration k capital-labor size = K/L, then labor productivity is expressed by a formula: The purpose of a task is the description of dynamics of a capital-labor or its representation as functions from t time.As any model is based on certain prerequisites, we need to make some assumptions and to enter a number of the defining parameters.
Suppose that the natural increase in temporary workforce: (3)  Consequently, at constant input parameters of the problem L, α and β function of the assets-in this case tends to stable steady-state value, regardless of nachalnyh conditions.Such a stationary point K = K St is a point of stable equilibrium, which is consistent with the known results of economic theory (Zhilyakov, Perlov, & Revtova, 2004).As an illustration, consider the following approximate methods for linear differential equation for the model with the projected market prices.

We
" 2 ′ 5 15. (7) Consider the case 1. Assume that at P (0) = 4, and P'(0) = 1, then the matrix Z w (8) obtained by solving the linear inhomogeneous second order differential equation for the function P(T) by the Runge-Kutta fourth-order accuracy has three columns: the first column contains the values of t, in which the solution is sought; the second column contains the values of the found solution P (t) at the corresponding points and the third value P'(t). (8) Solutions were prepared in 200 points in the package Mathcad.Analyzing the results, it was found that all prices tend to the steady price P St =3 with fluctuations around it, and the amplitude of these oscillations decays with time.
Consider the case 2. Take the initial moment of P (0) = 4, and P'(0) = -3, then the matrix Z w1 (9), the resulting solution is as follows: (9) The results of the solution of equation ( 7), have been found to prices from time to time in these two cases are shown in the graph (Figure 2).
Figure 2. The family of integral curves of equation ( 7) for the cases 1 and 2 This resulting family of integral curves of equation ( 7) for the cases 1 and 2 completely coincides with the family of integral curves for the exact solution.
Next, perform the computation of approximate solutions using computational scheme of the Bubnov-Galerkin method proposed for the equation ( 6), according to which the solution is reduced to a system of linear algebraic equations .The elements of the matrix A (10), the column vector of free terms F (11) explicitly written below and calculated in the package Mathcad.To solve the system used the method of inverse matrix, the elements of which are also represented (13).As a result of an approximate solution u (x), where the coefficients are the elements of the vector Xi (12), whose graph is shown in Figure 3.
and practice of use of methods.The concept of practice of computing work is quite uncertain.However, despite such uncertainty, criterion of practice often bears in itself certain positive information which often at this stage of development of science can't be formalized or proved.