IJRRAS 6 (2) ● February 2011
www.arpapress.com/Volumes/Vol6Issue2/IJRRAS_6_2_07.pdf
ON AVERAGING METHOD FOR SOLUTION OF DIFFERENTIAL
EQUATIONS
E. M. Roshdy
Department of Mathematics, Military Technical College, Cairo, Egypt.
ABSTRACT
This method is used to solve non linear differential equations with linear and non linear small dissipative terms and
or with time dependent parameters examples also are given.
1. INTRODUCTION
In these days non linear differential equations are usually solved numerically by using digital computers this is an
efficient and well- paved to obtain a particular solutions however, if the system of differential equations is Lagrange
subsidiary system of characteristic equations for first order partial differential equation such as vlasou or if the
influence of the parameters of the equation and /or of the initial or boundary conditions on the solution are of
interest, the computer solution can be very expensive or even impossible if the computing time exceeds a reasonable
time interval. Actually some authors have recently proposed analytic methods for the solution of the non linear but
nearly linear, differential equations. Krylov and Bogolyubov [3] investigated equations of the type
𝑥 ∙∙ + 𝜔2 𝑥 = 𝜖 𝑓(𝑥, 𝑥 ∙∙ )
(1)
Where 𝜖 is a small parameter the method starts from the so called generating solution
𝑥 = 𝐴 sin 𝜔𝑡 + 𝜑 , 𝑥 ∙ = 𝐴𝜔 cos(𝜔𝑡 + 𝜑)
(2)
Which satisfies (1) at 𝜖=0 in order to solve (1) it is assumed that the constants integration A & 𝜑 depend on time in
(2) 𝐴 → 𝐴 𝑡 , 𝜑 → 𝜑 𝑡 .
Expressing 𝑓(𝑥, 𝑥 ∙ ) in a Fourier series in the total phase ψ = ωt + φ
And assuming that 𝜖 is small, so that the amplitude A and the phase φ change very slowly during one period of
oscillation i.e.
A.
φ.
≪ω ,
≪ψ
(3)
A
φ
One obtains the first order of 𝜖 by averaging over one period
𝑑𝐴
𝜖 1
<
>=
𝑑𝑡
𝜔 2𝜋
2𝜋
𝐹( 𝐴 sin 𝜓 , 𝐴 𝜔 cos Ψ) cos Ψ 𝑑Ψ
0
𝑑𝜑
−𝜀 1
<
>=
𝑑𝑡
𝐴𝜔 2𝜋
(4)
2𝜋
𝐹( 𝐴 sin 𝜓 , 𝐴 𝜔 cos Ψ) sin Ψ 𝑑Ψ
0
(5)
Where A and 𝜑 are assumed to be time independent under the integrals. higher order solutions can be obtained .this
method though it Is restricted to equations of type (1) (i.e. to nearly linear equations) has been used extensively in
plasma physics, theory of octillions, control theory etc[2]. The method was also used to solve partial differential
equation [4].
Kruskal [2] extended the Krylov - Bogolyubov method to solve
x ∙∙ = f (x, x . , ε)
The solutions of these fully non linear equations are based on recurrent relations and are given in the form of power
series of the small parameter 𝜀.
In this paper we are investigating an averaging method for the solution of the equations of the type
𝑥 . + 𝜔𝑓 𝑋 = 𝜀𝑓 𝑥, 𝑥 .
(7)
Using elliptic functions we begin with the exact solution of the fully non linear equations
𝑥 .. + 𝜔2 𝑓 𝑥 = 0
Without developing f(x) into a power series with respect to the small parameter 𝜀.
172
IJRRAS 6 (2) ● February 2011
Roshdy ● Averaging Method for Solution of Differential Equations
2. ELLIPTIC FUNCTIONS
The Krylov - Bogolyubov technique is based on the harmonic solutions if e.g. one considers Duffing's equation
𝑥 .. + 𝜔2 𝑥 = 𝜀𝑥 3
(9)
Or Einstein's equation for the perihelion shift
𝑥 .. + 𝜔2 𝑥 = 𝜀𝑥 2 + 𝑎
(10)
Then the Krylov - Bogolybov technique starts from (2) and is valid for 𝜀 ≪ 1
One obtain from (4) the result A=constant and the frequency modification (Amplitude dispersion) is contained in (5)
.it is possible to solve (9) exactly and any other equation of type (8) for any 𝜀 by multiplying (8) by 𝑥 . and
integrating twice one obtains
𝑥
𝑑𝑥
𝑡 − 𝑡0 =
(−2𝜔 2
0
𝑥
0
(11)
𝑓 𝑥 𝑑𝑥 + 2𝐸 )
Where 𝑡0 , 𝐸 are integration constant. if f(X) is a polynomial of degree three or a simple harmonic function such as
sin (Lx) then (11) is an elliptic integral and its inverse function may be expressed by a jacobi elliptic function [5]
we are now going to consider the Langevin equation of motion of electrons in a periodic space dependent electric
field. this equation is of importance not only in plasma physics. in kinetic theory, but also a form of Froude
equations for rolling ships or damped pendulum. the equation is ( 𝜔, 𝑙, 𝜀 𝑎𝑟𝑒 𝑔𝑖𝑣𝑒𝑛 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 )
𝑥 .. + 𝜔2 sin 𝑙𝑥 = 𝜀𝑓 𝑥, 𝑥 .
(12)
Where 𝜀 is a collision frequency (e.g. electrons in neutralizing ion background) the solution for 𝜀 = 0 (the un
perturbed equation a generating solution in the sense of Krylov- Bogolyubov) is
2
𝑥 𝑡 = 𝑎𝑟𝑐𝑠𝑖𝑛 𝑘𝑠𝑛 ( 𝑙, 𝜓, 𝑘)
(13)
𝑙
Where Sn is the Jacobi elliptic sine function, 𝜓 = 𝜔𝑡 + 𝜙 , 𝑘, the modulus of the elliptic function (amplitude of the
oscillation) and 𝜙 are integration constants.
3. AVERAGING METHOD WITH ELLIPTIC FUNCTIONS:
In order to solve (12) w now replaces 𝑘 → 𝑘(𝑡) 𝜑 → 𝜑 𝑡
(14)
And set up a generating solution (13) and
2𝑘𝑤
𝑥. =
𝑐𝑛 𝑙 , 𝜓, 𝑘
(15)
𝑙
Which may be obtained by differentiating (13) with k and𝜙 kept constant and then applying (14), also the relations
1 − 𝑘 2 𝑠𝑛2 𝑙, 𝜓, 𝜅 = 𝑑 2 𝑛 ( 𝑙, 𝜓, 𝜅)
𝜕𝑠𝑛
= 𝑙, 𝑐𝑛 𝑙, 𝜓, 𝜅 𝑑𝑛 𝑙, 𝜓, 𝜅
𝜕𝜓
Were used .differentiating (13) and observing, (14) then equating it to (15) gives
𝑑𝑠𝑛
−𝜅 𝑠𝑛 + 𝜅 𝑑𝜅
𝜑=
(17)
𝜅
𝜄. 𝑐𝑛 𝑑𝑛
𝑑
𝑑
𝑑
𝑑𝑘
When applying to (13) we use = ∗ .
𝑑𝑡
𝑑𝑡
𝑑𝑘
𝑑𝑡
Since the jacabi function depend on k directly and since its argument 𝜓
𝜕𝑠
𝜕𝑠
𝜕𝜓
𝑠
Depends on k we have to use the total derivative with respect to k, i.e. 𝑑 𝑛 𝑑𝑘 = 𝑛 + 𝑛 ∗
equation (17) is
𝜕𝑘
𝜕𝜓
𝜕𝜓
independent of
𝑑𝑐 .
The form f(x,𝑥 ′ ). differentiating (15) and substituting into (12), we obtain
𝑘 ′ 𝑐𝑛 − 𝑘𝜑 𝑙 𝑠𝑛 𝑑𝑛 + 𝑘𝑘′ 𝑛 =
𝑑𝑘
∈ 𝑙
2𝑤
.
𝐹(𝑥, 𝑥 )
Here we used
(18)
𝜕𝑐𝑛
𝜕𝜓
= − 𝑙 𝑠𝑛 𝑑𝑛 .
and
𝑠𝑖𝑛 𝑙𝑥 = 2𝑠𝑖𝑛
= 2𝑘𝑠𝑛 𝑐𝑜𝑠 𝑎𝑟𝑐𝑐𝑜𝑠
𝑙𝑥
𝑙𝑥
𝑐𝑜𝑠
= 2𝑘 𝑠𝑛 𝑐𝑜𝑠 (𝑎𝑟𝑐𝑠𝑖𝑛 𝑘𝑠𝑛( 𝑙 ∗ 𝜓, 𝑘 )
2
2
1 − 𝑘𝑠𝑛( 𝑙 ∗ 𝜓, 𝑘
2
= 2𝑘 𝑠𝑛 𝑑𝑛
173
(19)
IJRRAS 6 (2) ● February 2011
Roshdy ● Averaging Method for Solution of Differential Equations
Solving (17) & (18) for k and 𝜑 we obtain
𝜀 𝑙
𝑘′ =
𝑓 𝑥, 𝑥 ′ 𝑐𝑛 𝑙, 𝜓, 𝑘
2𝜔
Here use of the relations
𝑠𝑛 2 + 𝑐𝑛 2 = 1,
Furthermore we have
𝜑′ = −
𝜀𝑓 𝑥,𝑥 ′
𝑘 2𝑤 𝑑 𝑛
(𝑠𝑛 + 𝑘
𝑠𝑛
𝑑𝑠𝑛
𝑑𝑘
(20)
𝑑𝑠𝑛
𝑑𝑘
=
𝑑 𝑠𝑛 2
𝑑𝑘 2
,
𝑠𝑛
𝑑 𝑠𝑛
𝑑𝑘
+ 𝑐𝑛
𝑑𝑐𝑛
𝑑𝑘
=0
(21)
)
(22)
Since the Jacobi function Sn, cn and dn are periodic with period 4K where the quarter period
𝐾 𝑘 =
𝜋
2
𝑑𝜃
1 − 𝑘 2 𝑠𝑖𝑛2 𝜃
0
Is the complete elliptic integral of first kind, we may average (20) and (22) without any Fourier series expansion, but
′
𝜑′
assuming the validity of 𝑘 𝑘 ≪ 𝑤 , 𝜑 ≪ 𝑤 Which is true for small 𝜀. defining u= 𝑙 . 𝜓 and
1 4𝑘
<. . >=
. . 𝑑𝑢
(23)
4𝑘 0
′
We now calculate <k'> and <𝜑 > in analogous manner to (4) and (5). We then have form (20)
𝑑𝑘
𝜀 𝑙
4𝑘
< >=
𝐹 𝑠𝑛 𝑢, 𝑐𝑛 𝑢 𝑐𝑛 𝑢 𝑑𝑢
(24)
𝑑𝑡
𝜔8𝑘 0
Where again 𝜑 𝑎𝑛𝑑 𝑘 are considered to be constant under the integral .furthermore from (22)
𝑑𝑠
4𝑘
𝑠𝑛 𝑢 + 𝑘 𝑛
𝑑𝜑
𝜀
𝑑𝑘 𝑑𝑢 (25)
<
>=
𝑓(𝑠𝑛 𝑢, 𝑠𝑛 𝑢 )
𝑑𝑡
𝜔8𝑘 0
𝑑𝑛 𝑢
We consider now two cases:
𝑓 𝑥, 𝑥 ′ = 𝑓 𝑥 = 𝑓 𝑠𝑛 𝑢
This case is of interest since (12) is then of the form (8) and can be integrated exactly. by multiplying (12)
By 𝑥 ′ and integrating twice one obtain
𝑑𝑥
𝑡 − 𝑡0 =
2
(−2 𝜔 𝑓 𝑥 − 𝜀𝑓 𝑥 𝑑𝑥 + 2𝐸)
b) 𝑓 𝑥, 𝑥 ′ = 𝑓 𝑥 ′ = 𝑓 𝑐𝑛 𝑢
We have from (24)
𝑑𝑘
𝜀 𝑙 4𝑘
<
>=
𝐹 𝑘𝑐𝑛 𝑢 𝑑𝑢
(27)
𝑑𝑡
𝜔8𝑘 0
𝑎𝑛𝑑 𝑓𝑟𝑜𝑚 25 𝑤𝑒 𝑔𝑒𝑡
𝑑𝑠 𝑢
4𝑘
𝑠𝑛 𝑢 + 𝑘 𝑛
𝑑𝜑
−𝜀
𝑑𝑘 𝑑𝑢 (28)
<
>=
𝑓 𝑘𝑐𝑛 𝑢
𝑑𝑡
𝜔𝑘8𝑘 0
𝑑𝑛 𝑢
It is difficult to obtain general conclusion without a knowledge of F .
We present some application of the method
a)
a)
Linear damping we have from (15)
𝑓 𝑥 ′ = −𝑥 ′ = −
2𝑘𝑤
𝑙
𝑠𝑛 𝑢, 𝑘
(29)
But (24) gives
<
1
𝑑𝑘
−𝜀 4𝑘 2
−𝜀
𝐾 𝑘
>=
𝑐𝑛 𝑢 𝑑𝑢 = 2 𝐸 𝑘 − 1 − 𝑘 2 𝐾 𝑘
𝑘
𝑑𝑡
4 0
𝑘
𝜀
= 2𝐻 𝑘
(30)
𝑘
Where
𝜋
𝐸 𝑘 = 02 1 − 𝑘 2 𝑠𝑖𝑛2 𝜃 𝑑𝜃 , 𝐻 𝑘 = 𝐸 𝑘 − 1 − 𝑘 2 𝐾(𝑘) (31)
E (k) is the complete elliptic integral of the second kind using the identity
𝑑
𝑑𝐻
𝐾 𝑘 =
𝐸 𝑘 − (1 − 𝑘 2 )𝐾(𝑘) =
(32)
𝑑𝑘
𝑑𝑘
We have
174
IJRRAS 6 (2) ● February 2011
Roshdy ● Averaging Method for Solution of Differential Equations
−𝜀 𝑡 − 𝑡0 =
𝐻′ (𝑘)
𝑑𝑘 = 𝑙𝑛 𝐸 𝑘 − 1 − 𝑘 2 𝐾(𝑘)
𝐻 𝑘
(33)
Which gives k=k (t)
b) Quadratic damping:
−4𝑘 2 𝜔 2
We have 𝑓 𝑥 ′ = −𝑥′2 =
𝑐𝑛2
(34)
𝑙
And from (24) we get k'=0 this is understandable because, from physical consideration the damping function should
be an odd function of x, e.g. f(x') = 𝑥 . 𝑥 . , 𝑜𝑟 𝑓 = 𝑥 . + 𝑥 .2 however, variable damping is of more interest
c) Van der pol der pol damping :
We generalize the van der pol equation 𝑥 ′′ + 𝑥 − 𝜀 1 − 𝑥 2 𝑥 ′ = 0
To the form (12), i.e.
𝑥 ′′ + 𝜔2 𝑠𝑖𝑛 𝑙𝑥 = −𝜀 𝑥 2 − 1 𝑥 ′ = 𝜀𝑓
(36)
This would represent ,for example a bounded in a homogeneous plasma model. from (24) we again get 𝑘(𝑡) we
𝑑𝑘
−𝐻(𝑘)
𝑘𝐺 (𝐾)
obtain−𝜀(𝑡 − 𝑡𝑜 ) =
, 𝑤𝑕𝑒𝑟𝑒 𝑝 𝑘 =
+ 2
𝑝 𝑘
𝑘𝐻 (𝑘)
𝑙 𝑘(𝑘)
And where
4𝑘
G (k) = 0 𝑐𝑛2 𝑢 𝑎𝑟𝑐 𝑠𝑖𝑛2 𝑘 𝑠𝑛 𝑢 𝑑𝑢
A warning might be useful:
Before applying the method described here, one has to determining if (8) has a periodic at all .there are cases in
which (8) has a periodic solution but (7) does not .an example is the equation
𝑥 ′′ − 𝑎𝑥 + 𝑐𝑥 3 = −𝜀𝑥 ′
For 𝜀 = 0 , an elliptic function is the solution for 𝜀 2 < 8𝑎, we have a stabe focus and a damped oscillation and the
method of sec. 3 can be applied. For 𝜀 2 > 8𝑎, we have a stable node and no oscillation. The solution for 𝜀 = 0
a = c= 𝜔2 is given by 𝑥 =
𝑘 2
2𝑘 2 −1
𝑐𝑛(
𝜔𝜏 +𝜑
)
(2𝑘 2 −1)
And for 𝜀 ≠ 0 , 𝑑𝑘′ 𝑑𝑡 is determined in the oscillatory case by a function of k which is composed of K (k), E (k) and
powers of k
4. REFERENCES
[1]. J. hole oscillation in nonlinear systems McGraw-hill New York 1968.
[2]. H.Kruskal, Asymptotic theory of Hamiltonian and other system with all solution periodic. J.Math .phy3,
806(1962).
[3]. K. Krylov and Bogolyguv introduction to non linear mechanics Princeton University press (1947).
[4]. N. Minorsky : nonlinear mechanics Edwards.Annarbor (1947)
[5]. E.Neville, Jacobian elliptic functions Clarendon press oxford (1968).
[6]. T.O'Neil collision less damping of non linear oscillation, physics fluids.
175