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EXTENSION AND APPLICATION OF NEWTON'S METHOD IN NONLINEAR OSCILLATION THEORY
Mamatova Mahliyo Adhamovna
Teacher at the Department of Physics, Fergana State University, Mamirjonova Gulnoza Shokirjon qizi
Master, Fergana State University
We all know that because the nonlinear oscillation is very important in theory and application, it is a main subject studied by mathematicians and mechanicians. The analytic method in the study of nonlinear oscillation is most important.
Because derivative systems of quasi-linear systems are the simplest linear ordinary differential equations with constant coefficients and the small parameters appear in the equations, the homotopy invariants between the original systems and derivative system was quite easily set up. Therefore, the theory and the method for studying the periodic solution to quasi-linear systems have already been obtained, such as asymptotic method, small parameter method and method of multiple scales [2;-4]. However, the study of general strong nonlinear systems is very difficult. The main difficulty is that in the general case the concept of the derivative systems is not clear. Even if the small parameters exist in the control equations, the derivative systems also show strong nonlinearity. Therefore, it is very difficult to find out the integration of the derivative systems. In spite of this, scholars of many countries have done a lot in the strong nonlinear oscillation. So far, the methods in the study of quasi-linear systems have been extended to the study of strong nonlinear oscillation, references [5;-6] apply asymptotic method to investigate quasi-conservative system and in reference [7] Burton discussed the similar problem by using time transfomation method. But at present it is evident to lack the unanimous effective analytic method for studying strong nonlinear nonautonomous systems, especially the case of large amplitude excitation. Although in some cases we can obtain the approximate periodic solutions of strong nonlinear systems by using the. approximate methods as direct variational method and Galerkin method and so on, to raise the calculation accuracy makes unknown numbers increase, sothat solving the transcendental equations is very difficult, and at the same time it is not probable to obtain the asymptotic analytic solutions with explicit occurrence. Therefore, the application of these direct approximate methods is restricted. Besides, Liapnov system method developed by I.G. Malkin is only suitable to small amplitude oscillation.
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On the other hand, we note that Newton's method used long ago to solve the algebraic and transcendental equations has the characters of making the nonlinear problems transformed into the linear problems, alternating programs being simpler with fast speed of convergence.
The eminent former Soviet scholar L.V.Contolovich first successfully applied Newton's method to study abstract operators in the functional, space, and at the same time he suggested simplified Newton's method and studied the existence of the periodic solution for the second order nonlinear system about some class. But his proof can not give the analytic method to calculate the periodic solutions.
In this paper we give up the classcial nonlinear method, that is, we use the method to find out the periodic solution of the derivative system. We directly establish the approximate concept based upon Newton's method 'and prove that general second order nonlinear systems may have the periodic solution, whose result is wider than L. V. Contolovich's, and at the same time we'can give an iteration method to calculate asymptotic analytic solutions of strong or weak nonlinear systems. Finally we'll further prove by some examples that the method we use in this paper is feasible.
We consider
and assume that <p(t) and f(t) are all continuous periodic functions of periodic 2n/a) so, they can be expanded into Fourier series of the complex form as follows:
We let
Putting (2 -4) and (5) into (1) and equating the coefficients of the same harmonics on both sides yields
+- <5Cr) + ir(oC-^)cu- = —an
(6a) and (6b)
Where
= it
-1, r = 0 r ^ 0
(7)
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(8)
(9)
R = 0 KI2F (10)
In addition, we also assume i(/(t) is continuous, so on the strength ofdifferentiable theorem term by term to Fourier series in reference [9] we have
And let the integer Then
r*(ii) = 4 n (13)
It clearly visible that when V r > r * (n) we r2 < 2 (n — r)2 So considering (8
-9),
(11 - 12) and (13), we have
Therefore
h = Y?=-M-t + «00 + ircoÇ_r\2 +
w
So
Let
< +oo
(14)
(15)
(16)
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i|>m+l
It is clearly visible that lim t]1 = 0
m->oo
And let
It is clearly visible that lim )]2 = 0
(18)
m-»oo
(20)
We can change the form of equations (6) into still simpler one as follows
The equations
Are called the cut equations of (6), which can take the place of (6) approximately, where /' is (2m + l)(2m +1) order unit matrix, a = [a„]J_0 ±1 ±2 ±m ; H' is (2m + l)(2m + 1) order matrix located at the upper left side in H. The operator <f> represents the following transformation
Where = 0+1+2 ■ ^clearly visible that
And
Where Af(j = 0, ±1,..., ±m) are the eigenvalues of the matrix, H' ■ H'. H' is Hermite adjoint matrix of H'. So from Ref. [8] we know that if the following inequalite is satisfied
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Then when, m oo; the solution of (22) must converge to the solution of (21). From formula (24), it is clear that norms |A.W_1|| for all m, which are arbitrary integers, must have the upper boundary. By this and considering (23) and (28), we know when m is sufficiently large integer, formula (25) can always be established. Therefore if the cut equations (22) have the solution, then when, m co5 the solution of (22) must converge to the solution of (21).
Now we will prove that periodic solution of (1) obtained by means of the above method is absolute uniform convergence. For the sake of our purpose, substituting (5) into (1) and considering (8 - 9) and (10), we obtain
where
When |an | is sufficiently large, so that |a„ | > s, where the integer s satisfies the following inequality
R
So we have
|aril - ЛГ2-я(1 + |п|) Therefore, from (5) and (29) we obtain
{n= ±1,±2,...)
(29)
It is clearly visible that the right-hand side of (30) is a bounded positive value, from which we can arrive at the following conclusion.
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SJIF 2024 = 7.404 / ASI Factor = 1.7
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