TABLE 12.3 Perturbation methods of nonlinear mechanics and theoretical physics the third column gives n leading asymptotic terms with respect to the small parameter ε.. Method name Examp
Trang 1TABLE 12.3 Perturbation methods of nonlinear mechanics and theoretical physics
(the third column gives n leading asymptotic terms with respect to the small parameter ε).
Method
name
Examples of problems solved by the method
Form of the solution sought
Additional conditions and remarks Method
of scaled
parameters
(0≤t<∞)
One looks for periodic solutions of the equation
y tt +ω2y = εf (y, y t);
see also Paragraph 12.3.5-3
y (t) = n–1
k=0 ε y k (z),
t = z
1+n–1
k=1
ε ω k
Unknowns: y k and ω k;
y k+1 /y k = O(1); secular terms are eliminated through selection
of the constants ω k
Method
of strained
coordinates
(0≤t<∞)
Cauchy problem:
y t = f (t, y, ε); y(t0) = y0 (f is of a special form);
see also the problem in the
method of scaled parameters
y (t) = n–1
k=0 ε y k (z),
t = z + n–1
k=1 ε ϕ k (z)
Unknowns: y k and ϕ k;
y k+1 /y k = O(1),
ϕ k+1 /ϕ k = O(1)
Averaging
method
(0≤t<∞)
Cauchy problem:
y tt +ω2y = εf (y, y t ),
y(0) = y0, y t (0) = y1; for more general problems,
see Paragraph 12.3.5-4, Item2◦
y = a(t) cos ϕ(t), the amplitude a and phase ϕ
are governed by the equations
da
dt= –ω ε
0f s (a), dϕ
dt = ω0–aω ε
0f c (a)
Unknowns: a and ϕ;
fs=2π1 72π
0 sin ϕF dϕ,
fc=2π1 72π
0 cos ϕF dϕ,
F = f (a cos ϕ,–aω0 sin ϕ)
Krylov–
Bogolyubov–
Mitropolskii
method
(0≤t<∞)
One looks for periodic solutions of the equation
y tt +ω2y = εf (y, y t);
Cauchy problem for this
and other equations
y = a cos ϕ+ n–1
k=1 ε y k (a, ϕ),
a and ϕ are determined
by the equations
da
dt=n
k=1 ε A k (a), dϕ
dt = ω0+n
k=1 ε Φk (a)
Unknowns: y k , A k,Φk;
y kare2π-periodic
functions of ϕ; the y kare assumed
not to contain cos ϕ
Method
of two-scale
expansions
(0≤t<∞)
Cauchy problem:
y tt +ω2y = εf (y, y t ),
y(0) = y0, y t (0) = y1; for boundary value problems,
see Paragraph 12.3.5-5, Item2◦
y=n–1
k=0 ε y k (ξ, η), where
ξ = εt, η = t
1+n–1
k=2 ε ω k
,
d
dt = ε ∂ξ ∂ + 1+ε2ω2+· · ·) ∂
∂η
Unknowns: y k and ω k;
y k+1 /y k = O(1); secular terms are eliminated through
selection of ω k
Multiple
scales
method
(0≤t<∞)
One looks for periodic solutions of the equation
y tt +ω2y = εf (y, y t);
Cauchy problem for this
and other equations
y=n–1
k=0 ε y k, where
y k = y k (T0, T1, , T n ), T k = ε k t
d
dt=∂T ∂
0+ε ∂T ∂
1+· · ·+ε n ∂
∂Tn
Unknowns: y k;
y k+1 /y k = O(1);
for n =1, this method
is equivalent to the averaging method
Method of
matched
asymptotic
expansions
(0≤x≤b)
Boundary value problem:
εy xx +f (x, y)y x = g(x, y),
y(0) = y0, y(b) = y b (f assumed positive);
for other problems, see Paragraph 12.3.5-6, Item2◦
Outer expansion:
y=n–1
k=0 σ k (ε)y k (x), O(ε)≤x≤ b;
inner expansion (z = x/ε):
2y= n–1 k=0 2σ k (ε) 2y k (z), 0≤x≤O (ε)
Unknowns: y k,2y k , σ k,2σ k;
y k+1 /y k = O(1),
2y k+1 / 2y k = O(1); the procedure of matching expansions is used:
y (x →0) =2y(z →∞)
Method of
composite
expansions
(0≤x≤b)
Boundary value problem:
εy xx +f (x, y)y x = g(x, y),
y(0) = y0, y(b) = y b (f assumed positive);
boundary value problems
for other equations
y = Y (x, ε)+ 2 Y (z, ε),
Y=n–1
k=0 σ k (ε)Y k (x),
2
Y=n–1
k=0 2σ k (ε) 2 Y k (z), z = x
ε; here, 2Y k →0 as z →∞
Unknowns: Y k, 2Y k , σ k,2σ k;
Y (b, ε) = y b,
Y(0, ε)+ 2 Y(0, ε) = y0; two forms of representation
of the equation
(in terms of x and z)
are used to obtain solutions
Trang 212.3.5-2 Method of regular (direct) expansion in powers of the small parameter.
We consider an equation of general form with a parameter ε:
y
tt+ f (t, y, yt , ε) = 0 (12.3.5.1)
We assume that the function f can be represented as a series in powers of ε:
f (t, y, yt , ε) =
∞
n=0
εnf
n(t, y, yt ). (12.3.5.2)
Solutions of the Cauchy problem and various boundary value problems for
equa-tion (12.3.5.1) with ε → 0 are sought in the form of a power series expansion:
y =
∞
n=0
εny
One should substitute expression (12.3.5.3) into equation (12.3.5.1) taking into account
(12.3.5.2) Then the functions fn are expanded into a power series in the small parameter
and the coefficients of like powers of ε are collected and equated to zero to obtain a system
of equations for yn:
y
y
1 + F (t, y0, y0)y1 + G(t, y0, y0)y1+ f1(t, y0, y0) = 0, F = ∂f0
∂y, G =
∂f0
∂y (12.3.5.5) Here, only the first two equations are written out The prime denotes differentiation with
respect to t To obtain the initial (or boundary) conditions for yn, the expansion (12.3.5.3)
is taken into account.
The success in the application of this method is primarily determined by the possibility
of constructing a solution of equation (12.3.5.4) for the leading term y0 It is significant to
note that the other terms yn with n ≥ 1 are governed by linear equations with homogeneous initial conditions.
Example 1 The Duffing equation
y tt + y + εy3=0 (12.3.5.6) with initial conditions
y(0) = a, y t (0) =0
describes the motion of a cubic oscillator, i.e., oscillations of a point mass on a nonlinear spring Here, y is the deviation of the point mass from the equilibrium and t is dimensionless time.
For ε →0, an approximate solution of the problem is sought in the form of the asymptotic expan-sion (12.3.5.3) We substitute (12.3.5.3) into equation (12.3.5.6) and initial conditions and expand in powers
of ε On equating the coefficients of like powers of the small parameter to zero, we obtain the following problems for y0 and y1:
y0 + y0=0, y0= a, y0=0;
y1 + y1 = –y3, y1=0, y1=0
The solution of the problem for y0 is given by
y0= a cos t.
Substituting this expression into the equation for y1and taking into account the identity cos3t= 14cos3t+ 3
4 cos t, we obtain
y1 + y1= –14a3(cos3t+3cos t), y1=0, y1=0
Trang 3Integrating yields
y1= –38a3t sin t +321 a3(cos3t–3cos t).
Thus the two-term solution of the original problem is given by
y = a cos t + εa3
–38t sin t +321(cos3t–3cos t)
+ O(ε2)
Remark 1 The term t sin t causes y1 /y0 → ∞ as t → ∞ For this reason, the solution obtained is
unsuitable at large times It can only be used for εt 1; this results from the condition of applicability of the
expansion, y0 εy1.
This circumstance is typical of the method of regular series expansions with respect to the small parameter;
in other words, the expansion becomes unsuitable at large values of the independent variable This method is
also inapplicable if the expansion (12.3.5.3) begins with negative powers of ε Methods that allow avoiding
the above difficulties are discussed below in Paragraphs 12.3.5-3 through 12.3.5-5
Remark 2 Growing terms as t → ∞, like t sin t, that narrow down the domain of applicability of
asymptotic expansions are called secular.
12.3.5-3 Method of scaled parameters (Lindstedt–Poincar´e method).
We illustrate the characteristic features of the method of scaled parameters with a specific example (the transformation of the independent variable we use here as well as the form of the expansion are specified in the first row of Table 12.3).
Example 2 Consider the Duffing equation (12.3.5.6) again On performing the change of variable
t = z(1+ εω1+· · ·),
we have
y zz+ (1+ εω1+· · ·)2(y + εy3) =0 (12.3.5.7)
The solution is sought in the series form y = y0(z) + εy1(z) + · · · Substituting it into equation (12.3.5.7) and
matching the coefficients of like powers of ε, we arrive at the following system of equations for two leading
terms of the series:
y1 + y1 = –y3–2ω1y0, (12.3.5.9)
where the prime denotes differentiation with respect to z.
The general solution of equation (12.3.5.8) is given by
y0= a cos(z + b), (12.3.5.10)
where a and b are constants of integration Taking into account (12.3.5.10) and rearranging terms, we reduce
equation (12.3.5.9) to
y1 + y1 = –14a3cos
3(z + b)
–2a 3
8a2+ ω1
cos(z + b). (12.3.5.11)
For ω1≠–38a2, the particular solution of equation (12.3.5.11) contains a secular term proportional to z cos(z+b).
In this case, the condition of applicability of the expansion y1 /y0= O(1) (see the first row and the last column
of Table 12.3) cannot be satisfied at sufficiently large z For this condition to be met, one should set
In this case, the solution of equation (12.3.5.11) is given by
y1= 321a3cos
3(z + b)
Subsequent terms of the expansion can be found likewise
With (12.3.5.10), (12.3.5.12), and (12.3.5.13), we obtain a solution of the Duffing equation in the form
y = a cos(ωt + b) + 321εa3cos
3(ωt + b)
+ O(ε2),
ω=
1–38εa2+ O(ε2)– 1
=1+38εa2+ O(ε2)
Trang 412.3.5-4 Averaging method (Van der Pol–Krylov–Bogolyubov scheme).
1◦ The averaging method involved two stages First, the second-order nonlinear equation
y
tt+ ω02y = εf (y, yt (12.3.5.14)
is reduced with the transformation
y = a cos ϕ, y
t= –ω0a sin ϕ, where a = a(t), ϕ = ϕ(t),
to an equivalent system of two first-order differential equations:
a
t= – ω ε
0f (a cos ϕ, –ω0a sin ϕ) sin ϕ,
ϕ
t= ω0– ε
ω0a f (a cos ϕ, –ω0a sin ϕ) cos ϕ. (12.3.5.15)
The right-hand sides of equations (12.3.5.15) are periodic in ϕ, with the amplitude a being
a slow function of time t The amplitude and the oscillation character are changing little during the time the phase ϕ changes by 2π.
At the second stage, the right-hand sides of equations (12.3.5.15) are being averaged
with respect to ϕ This procedure results in an approximate system of equations:
a
t= – ω ε
0fs(a),
ϕ
t= ω0– ω ε0a fc(a),
(12.3.5.16)
where
fs(a) = 1
2π
2π
0 sin ϕ f (a cos ϕ, –ω0a sin ϕ) dϕ,
fc(a) = 1
2π
2π
0 cos ϕ f (a cos ϕ, –ω0a sin ϕ) dϕ.
System (12.3.5.16) is substantially simpler than the original system (12.3.5.15)—the first
equation in (12.3.5.16), for the oscillation amplitude a, is a separable equation and, hence, can readily be integrated; then the second equation in (12.3.5.16), which is linear in ϕ, can
also be integrated.
Note that the Krylov–Bogolyubov–Mitropolskii method (see the fourth row in Ta-ble 12.3) generalizes the above approach and allows obtaining subsequent asymptotic terms
as ε → 0.
2◦ Below we outline the general scheme of the averaging method We consider the
second-order nonlinear equation with a small parameter:
y
tt+ g(t, y, yt ) = εf (t, y, y t). (12.3.5.17) Equation (12.3.5.17) should first be transformed to the equivalent system of equations
y
t= u,
u
t= –g(t, y, u) + εf (t, y, u). (12.3.5.18)
Suppose the general solution of the “truncated” system (12.3.5.18), with ε = 0, is known:
y0= ϕ(t, C1, C2), u0= ψ(t, C1, C2), (12.3.5.19)
Trang 5where C1 and C2 are constants of integration Taking advantage of the method of variation
of constants, we pass from the variables y, u in (12.3.5.18) to Lagrange’s variables x1, x2
according to the formulas
y = ϕ(t, x1, x2), u = ψ(t, x1, x2), (12.3.5.20)
where ϕ and ψ are the same functions that define the general solution of the “truncated”
system (12.3.5.19) Transformation (12.3.5.20) allows the reduction of system (12.3.5.18)
to the standard form
x
1= εF1(t, x1, x2),
x
2= εF2(t, x1, x2) (12.3.5.21)
Here, the prime denotes differentiation with respect to t and
F1 = ϕ2f (t, ϕ, ψ)
ϕ2ψ1– ϕ1ψ2, F2 = –
ϕ1f (t, ϕ, ψ)
ϕ2ψ1– ϕ1ψ2; ϕk=
∂ϕ
∂xk, ψk =
∂ψ
∂xk,
ϕ = ϕ(t, x1, x2), ψ = ψ(t, x1, x2).
It is significant to note that system (12.3.5.21) is equivalent to the original
equa-tion (12.3.5.17) The unknowns x1 and x2 are slow functions of time.
As a result of averaging, system (12.3.5.21) is replaced by a simpler, approximate autonomous system of equations:
x
1= ε F1(x1, x2),
x
2= εF2(x1, x2), (12.3.5.22) where
Fk(x1, x2) = 1
T
T
0 Fk(t, x1, x2) dt if Fk is a T -periodic function of t;
Fk(x1, x2) = lim
T →∞
1
T
T
0 Fk(t, x1, x2) dt if Fk is not periodic in t.
Remark 1 The averaging method is applicable to equations (12.3.5.14) and (12.3.5.17) with nonsmooth right-hand sides
Remark 2 The averaging method has rigorous mathematical substantiation There is also a procedure that allows finding subsequent asymptotic terms For this procedure, e.g., see the books by Bogolyubov and Mitropolskii (1974), Zhuravlev and Klimov (1988), and Arnold, Kozlov, and Neishtadt (1993)
12.3.5-5 Method of two-scale expansions (Cole–Kevorkian scheme).
1◦ We illustrate the characteristic features of the method of two-scale expansions with a
specific example Thereafter we outline possible generalizations and modifications of the method.
Example 3 Consider the Van der Pol equation
y tt + y = ε(1– y2)y t (12.3.5.23) The solution is sought in the form (see the fifth row in Table 12.3):
y = y0(ξ, η) + εy1(ξ, η) + ε2y2(ξ, η) +· · · ,
ξ = εt, η= 1+ ε2ω2+· · · t (12.3.5.24)
Trang 6On substituting (12.3.5.24) into (12.3.5.23) and on matching the coefficients of like powers of ε, we obtain the
following system for two leading terms:
∂2y0
∂2y1
∂η2 + y1= –2∂2y0
∂ξ∂η + (1– y2)∂y0
The general solution of equation (12.3.5.25) is given by
y0= A(ξ) cos η + B(ξ) sin η. (12.3.5.27)
The dependence of A and B on the slow variable ξ is not being established at this stage.
We substitute (12.3.5.27) into the right-hand side of equation (12.3.5.26) and perform elementary manip-ulations to obtain
∂2y1
∂η2 + y1=
–2B
ξ+14B(4– A2– B2)
cos η +
2A
ξ–14A(4– A2– B2)
sin η
+14(B3–3A2B) cos3η+14(A3–3AB2) sin3η (12.3.5.28)
The solution of this equation must not contain unbounded terms as η → ∞; otherwise the necessary condition
y1/y0= O(1) is not satisfied Therefore the coefficients of cos η and sin η must be set equal to zero:
–2B
ξ+14B(4– A2– B2) =0,
2A
ξ–14A(4– A2– B2) =0 (12.3.5.29)
Equations (12.3.5.29) serve to determine A = A(ξ) and B = B(ξ) We multiply the first equation in (12.3.5.29)
by –B and the second by A and add them together to obtain
r ξ–18r(4– r2) =0, where r2= A2+ B2 (12.3.5.30) The integration by separation of variables yields
r2= 4r2
r2+ (4– r2)e–ξ, (12.3.5.31)
where r0 is the initial oscillation amplitude
On expressing A and B in terms of the amplitude r and phase ϕ, we have A = r cos ϕ and B = –r sin ϕ.
Substituting these expressions into either of the two equations in (12.3.5.29) and using (12.3.5.30), we find that
ϕ ξ=0 or ϕ = ϕ0= const Therefore the leading asymptotic term can be represented as
y0= r(ξ) cos(η + ϕ0),
where ξ = εt and η = t, and the function r(ξ) is determined by (12.3.5.31).
2◦ The method of two-scale expansions can also be used for solving boundary value
problems where the small parameter appears together with the highest derivative as a factor (such problems for 0 ≤ x ≤ a are indicated in the seventh row of Table 12.3 and in Paragraph
12.3.5-6) In the case where a boundary layer arises near the point x = 0 (and its thickness
has an order of magnitude of ε), the solution is sought in the form
y = y0(ξ, η) + εy1(ξ, η) + ε2y2(ξ, η) + · · · ,
ξ = x, η = ε–1
g0(x) + εg1(x) + ε2g2(x) + · · · ,
where the functions yk = yk(ξ, η) and gk = gk(x) are to be determined The derivative with respect to x is calculated in accordance with the rule
d
dx = ∂
∂ξ + ηx ∂
∂η = ∂
∂ξ + 1
ε g
0+ εg1 + ε2g2 + · · · ∂
∂η Additional conditions are imposed on the asymptotic terms in the domain under
consider-ation; namely, yk+1/yk= O(1) and gk+1/gk= O(1) for k = 0, 1, , and g0(x) → x as
x → 0.
Remark The two-scale method is also used to solve problems that arise in mechanics and physics and are described by partial differential equations
Trang 712.3.5-6 Method of matched asymptotic expansions.
1◦ We illustrate the characteristic features of the method of matched asymptotic expansions
with a specific example (the form of the expansions is specified in the seventh row of Table 12.3) Thereafter we outline possible generalizations and modifications of the method.
Example 4 Consider the linear boundary value problem
εy xx + y x + f (x)y =0, (12.3.5.32)
y(0) = a, y(1) = b, (12.3.5.33) where0< f (0) <∞.
At ε =0 equation (12.3.5.32) degenerates; the solution of the resulting first-order equation
y x + f (x)y =0 (12.3.5.34) cannot meet the two boundary conditions (12.3.5.33) simultaneously It can be shown that the condition at
x=0has to be omitted in this case (a boundary layer arises near this point)
The leading asymptotic term of the outer expansion, y = y0(x)+O(ε), is determined by equation (12.3.5.34).
The solution of (12.3.5.34) that satisfies the second boundary condition in (12.3.5.33) is given by
y0(x) = b exp
1
x
f (ξ) dξ
We seek the leading term of the inner expansion, in the boundary layer adjacent to the left boundary, in the following form (see the seventh row and third column in Table 12.3):
2y = 2y0(z) + O(ε), z = x/ε, (12.3.5.36)
where z is the extended variable Substituting (12.3.5.36) into (12.3.5.32) and extracting the coefficient of ε– 1,
we obtain
2y
0 +2y
where the prime denotes differentiation with respect to z The solution of equation (12.3.5.37) that satisfies
the first boundary condition in (12.3.5.33) is given by
2y0= a – C + Ce–z (12.3.5.38)
The constant of integration C is determined from the condition of matching the leading terms of the outer and
inner expansions:
y0(x →0) =2y0(z → ∞). (12.3.5.39) Substituting (12.3.5.35) and (12.3.5.38) into condition (12.3.5.39) yields
C = a – be〈f〉, where 〈f〉=
1
0 f (x) dx. (12.3.5.40) Taking into account relations (12.3.5.35), (12.3.5.36), (12.3.5.38), and (12.3.5.40), we represent the ap-proximate solution in the form
y=
⎧
⎨
⎩
be〈f〉+ a – be〈f〉
e–x/ε for 0 ≤x≤O (ε),
bexp* 1
x f (ξ) dξ+
for O(ε)≤x≤ 1 (12.3.5.41)
It is apparent that inside the thin boundary layer, whose thickness is proportional to ε, the solution rapidly
changes by a finite value,Δ = be〈f〉– a.
To determine the function y on the entire interval x[0,1] using formula (12.3.5.41), one has to “switch”
at some intermediate point x = x0 from one part of the solution to the other Such switching is not convenient
and, in practice, one often resorts to a composite solution instead of using the double formula (12.3.5.41) In
similar cases, a composite solution is defined as
y = y0(x) + 2y0(z) – A, A= lim
x→0 y0(x) = lim
z→∞ 2y0(z).
In the problem under consideration, we have A = be〈f〉 and hence the composite solution becomes
y= a – be〈f〉
e–x/ε + b exp
* 1
x f (ξ) dξ
+
For ε ≤ 1, this solution transforms to the outer solution y0(x) and for0 ≤x
thus providing an approximate representation of the unknown over the entire domain