Analytical Solutions Of The Simplified Mathieu’s Equation

Analytical solutions of the simplified Mathieu’s equation

Nicolae Marcov*

*Corresponding author

University of Bucharest, Faculty of Mathematics and Computer Science

Str. Academiei nr. 14, sector 1, 0101014, Bucharest, Romania

[anonimizat]

DOI: 10.13111/2066-8201.2016.8.1.X

Received: 13 January 2016/ Accepted: 4 February 2016

Copyright©2015 Published by INCAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Abstract: Consider a second order differential linear periodic equation. The periodic coefficient is an approximation of the Mathieu’s coefficient. This equation is recast as a first-order homogeneous system. For this system we obtain analytical solutions in explicit form. The first solution is a periodic function. The second solution is a sum of two functions first is a continuous periodic function, but the second is an oscillating function with monotone linear increasing amplitude. We give a formula to directly compute the slope of this increase, without knowing the second numeric solution. The periodic term of second solution may be computed directly. The coefficients of fundamental matrix of the system are analytical functions.

Key Words: linear differential equation, parametric resonance.

1. PROBLEM FORMULATION

Consider the following second order non-linear differential equation with respect to real dimensionless time t.

In what follows we assume that the coefficient Q to be a real positive continuous periodic function with t real argument.

The following function is a reasonable approximation of the Mathieu’s coefficient [1], [2], [3].

The set of solutions is two dimensional real space [4], [5], [6], [7]. The function x is a periodic solution.

Let y be a second important solution.

For the characteristic coefficient (q), [8], the yp function is a periodic term of the solution y.

The problem is to give analytical formula for this coefficient.

2. EXPLICIT CHARACTERISTIC COEFFICIENT

We recast the equation (1) as a first-order system. The following system is obtain.

Let u be the derivative of periodic solution x. Except that q is zero, the derivative v of the oscillating solution y is not a periodic function.

The functions x, u, y, v have the following integral property

Conseqently it results the expression of the fundamental matrix [3] for the system (6).

The y function is also a solution of fist-order linear equation

The periodic term yp is the solution of the following first-order linear differential equation.

Indeed, substituting the function y in above equation we find this linear equation (11).

y = yp t x, v = vp t u + x, x (vp + t u + x) u (yp + t u) = 1

where

In order to compute the periodic term we can also consider the fourth-order system (13).

Let C be the variable constant of integration for the equation (11).

We shall consider constant

We introduce the equivalent expressions for the exact x solution.

If we make the change of the variable of integration, then the function h(s) is the solution of the following equation

The rational function r has the equivalent expression

Hence we have an appropriate formula for r function.

Consequently it results the equation of the function h.

In order to obtain the value of the coefficient it is necessary to impose the following integral condition.

The characteristic coefficient may be found with the use of the formula

3. SECOND EXPLICIT SOLUTION

Let be the integral periodic function

The restriction of on the interval [0, /2) is a known function. Integration of the given above equation (14), we obtain the expression of C constant variable.

From (15) we can use the equivalent expression of the x exact solution

Hence

y* C * x = 2 [1+ (2 1) cos2t + cos2 t ] sin t = ) cos2t ] sint

Consider the following identity

1 2 /

Therefore y* can be written as

From this it results the periodic term of the y solution.

Therefore

It easy to find the derivative v*

Consequently we deduce successively

The characteristic matrix is the sum of a periodic matrix and an oscillating matrix

The matrix and * are unitary matrix.

The solution of the system (6) have the following expresssion

4. REsultats

The coefficients of the two-order differential linear system are real continuous functions. The unique variable coefficient is a function Q (q, t) in which q is real small parameter. In this particulary case it is known an analytical explicit solution (x, u)T. In order to find the expression of the characteristic coefficient, [8], [9], it was imposed an explicit necessary improper integral condition. So it was found the explicit analytical formula for the characteristic coefficient of one particulary two-order differential system. Consequently it was found the second explicit solution (y,v)T. The vector (ytx,vtu) is a periodic vector. The components of the fundamental matrix have explicit expressions in which y* and v* are trigonometric polynomial functions and ( t) is a definite integral on the real interval (0, t). For directly caculation of the periodic term yp it is usefull the fourth-order system (13), but if and only if the parameter is equal with the characteristic coefficient , [10], [11].

REFERENCES

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