ISSN 0001-4346, Mathematical Notes, 2009, Vol. 85, No. 1, pp. 123–127. c© Pleiades Publishing, Ltd., 2009.Original Russian Text c© A. A. Ershov, 2009, published in Matematicheskie Zametki, 2009, Vol. 85, No. 1, pp. 134–138.

SHORTCOMMUNICATIONS

Asymptotic Expansion of the Solutionof a Second-Order Equation

A. A. Ershov*

Chelyabinsk State UniversityReceived June 6, 2008

DOI: 10.1134/S0001434609010131

Key words: asymptotic expansion, second-order differential equation, algebraic polynomial,asymptotic series, WKB method.

1. Introduction. We consider the second-order differential equation

d2u

dt2+ Q(t)u = 0. (1)

It is well known that the asymptotic expansions of the solutions of this equation for large values ofthe argument t can be obtained under rather wide assumptions about the complex-valued function Q(t)of a real variable. The so-called WKB method yields such expansions, for example, for the case in whichthe absolute values of the coefficient Q(t) and of its derivatives of any arbitrary order tend to infinity ast → ∞ [1].

2. In this section, we consider the case in which

Q(t) = Pn(t) + ϕ(t),

where Pn(t) is an algebraic polynomial of degree n and ϕ(t) is an almost periodic function of the form∞∑

k=1

ckeiλkt, ck, λk ∈ R.

The problem is to prove that, for a large argument, there is an asymptotic expansion of the solution ofthe form

u(t) as= Q(t)−1/4 exp(iS(t))(

1 +∞∑

j=1

wj(t)tj

), where S(t) =

∫ t

0

√Q(s) ds

and wj(t) are almost periodic functions of the form∑∞

j=1 bj,keiλj,kt.

For the further search of the solution, we introduce a function class V closed with respect todifferentiation, multiplication, and the operation of taking linear combinations. We also demand thatthe integral with variable upper limit, whose integrand average is equal to zero, belong to V .

An example of such a class is given by functions of the form

f(t) =∞∑

k=1

ckeiλkt,

where λk > ε >0 for any positive integer k and∞∑

k=1

|ckλnk | < ∞

*E-mail: ale10919@yandex.ru

123

124 ERSHOV

for any integer n. For example, the function

f(t) = (3 − eit − ei√

2t)−1

belongs to this class. It is easy to prove that the set of such functions is indeed closed with respect to theabove operations.

We assume that the function ϕ belongs to the class V .

To study the formal asymptotic expansion of the solution, we perform the following change of theunknown function:

u(t) = w(t)Q(t)−1/4 exp(iS(t)).

Then the function w(t) satisfies the equation

d2w

dt2+

dw

dt

(2iQ(t)1/2 − 1

2Q′(t)Q(t)

)+ w

(516

[Q′(t)]2

[Q(t)]2− 1

4Q′′(t)Q(t)

)= 0. (2)

We consider a polynomial P2m(t) of an even degree and assume that its highest-order coefficient isequal to 1 (this does not violate the generality of our argument, the study in the case of a polynomial ofany odd degree does not actually differ from the study given below). Let

P2m(t) =2m∑

k=0

aktk, where a2m = 1

and m is a positive integer. Then we can obtain the following expansions of the coefficients of Eq. (2)into series. For example,

Q(t)1/2 =

√√√√t2m +2m−1∑

k=0

aktk + ϕ(t)

= tm(

1 +12

(2m−1∑

k=0

aktk−2m + t−2mϕ(t)

)− 1

8

(2m−1∑

k=0

aktk−2m + t−2mϕ(t)

)2

+ · · ·)

.

Since the class V is closed with respect to multiplication and the operation of taking linearcombinations, we can rewrite Eq. (2) as

d2w

dt2+

dw

dt

(2itm +

∞∑

k=−m+1

1tk

ϕk(t))

+ w

( ∞∑

k=2

1tk

ψk(t))

= 0,

where ϕk(t) and ψk(t) are functions from the class V and the series converge. The first terms of theseries ϕ−m+1, . . . , ϕm−1 and ψ2, . . . , ψ2m−1 are even constant.

We seek the solution as a formal series

w = 1 +wm+1

tm+1+

wm+2

tm+2+

wm+3

tm+3+

wm+4

tm+4+ · · · ,

where wi are bounded functions.Although all the functions wk are constant for k < 3m, for the sake of uniformity of the proof, we

generalize all wi to almost periodic functions of the class V .

After formal differentiation of this series, we obtain

w′ =w′

m+1

tm+1− (m + 1)

wm+1

tm+2+

w′m+2

tm+2− (m + 2)

wm+2

tm+3+

w′m+3

tm+3− (m + 3)

wm+3

tm+4+ · · · ,

w′′ =w′′

m+1

tm+1− 2(m + 1)

w′m+1

tm+2+ (m + 1)(m + 2)

wm+1

tm+3+

w′′m+2

tm+2− 2(m + 2)

w′m+2

tm+3+ · · · .

MATHEMATICAL NOTES Vol. 85 No. 1 2009

ASYMPTOTIC EXPANSION OF SOLUTIONS 125

We equate the coefficients of like powers of t:

t−1 : 2iw′m+1 = 0, wm+1 = αm+1,

t−2 : 2i(−(m + 1)wm+1 + w′m+2) + w′

m+1ϕ1(t) + ψ2(t) = 0,

− 2i(m + 1)αm+1 + 2iw′m+2 + ψ2(t) = 0,

w′m+2 = (m + 1)αm+1 −

12i

ψ2(t) =i

2ψ2(t) + (m + 1)αm+1 = χm+2(t) + (m + 1)αm+1,

wm+2 =∫ t

0χm+2(τ) dτ + (m + 1)αm+1t + αm+2.

We find αm+1 from the condition

(m + 1)αm+1t +∫ t

0

i

2ψ2(θ) dθ = O(1), t → ∞.

It follows from the choice of αm+1 that the mean value of the function χm+2(t) is equal to zero, andhence wm+2(t) belongs to the class V .

The constant αm+2 is still undetermined, we find it from the following expansions:

t−3 : 2i(−(m + 2)wm+2 + w′m+3) + w′

m+1ϕ2(t) + w′m+2ϕ1(t) + ψ3(t) = 0,

w′m+3 = (m + 2)wm+2 +

i

2(w′

m+1ϕ2 + w′m+2ϕ1 + ψ3) = χm+3(t) + (m + 2)αm+2

(by χm+3 we denote the corresponding function).Similarly, we have

wm+3 =∫ t

0χm+3(τ) dτ + (m + 2)αm+2t + αm+3.

The function χm+3 consists of products of ϕ, ϕ′, ϕ′′, the antiderivatives of these functions, and theirlinear combinations, but since the function ϕ is equal to

∞∑

k=1

(pk sin λ̃kt + qk cos λ̃kt),

the function χm+3 has the same form and hence belongs to the class V . This means that the constant(m + 2)αm+2 can also be found as the mean value of the function (−χm+3), i.e.,

αm+2 = − 1m + 2

limT→∞

1T

∫ T

0χm+3(τ) dτ.

We note that, for almost periodic functions, this limit is equal to the mean value of the function−(1/(m + 2))χm+3. The function wm+3 also belongs to the class V .

We consider an arbitrary power −p, where p is a positive integer:

t−p : w′′p − 2(p − 1)w′

p−1 + (p − 2)(p − 1)wp−2 + 2i(w′m+p − (m + p − 1)wm+p−1)

+p+m−1∑

j=m+1

(w′j − (j − 1)wj−1)ϕp−j +

p−m−1∑

k=2

ψkwp−k + ψp = 0,

w′m+p = (m + p − 1)wm+p−1 +

i

2

(w′′

p − 2(p − 1)w′p−1 + (p − 2)(p − 1)wp−2

+p+m−1∑

j=m+1

(w′j − (j − 1)wj−1)ϕp−j +

p−m−1∑

k=2

ψkwp−k + ψp

)

= χm+p(t) + (m + p − 1)αm+p−1,

wm+p(t) =∫ t

0χm+p(τ) dτ + (m + p − 1)αm+p−1t + αm+p,

MATHEMATICAL NOTES Vol. 85 No. 1 2009

126 ERSHOV

where

αm+p−1 = − 1m + p − 1

limT→∞

1T

∫ T

0χm+p(τ) dτ.

This formula holds for any function wi. In this case, for all wi, we assume that wi ≡ 0 if i < m + 1.Thus, following this algorithm, we find an infinite asymptotic series, which is a formal series. We can

prove that the equation has a solution for which this series is an asymptotic series.

3. We note that other cases of a fast increasing Q(t) can also be considered. For example, if we takeQ(t) = exp(2λt) + ϕ(t), where λ ∈ R, λ > 0, and ϕ(t) is a smooth function of a slower growth rate.But no resonance already occurs in this case. For a large argument, we shall prove the existence of anasymptotic expansion of the solution in the form

u(t) as= Q(t)−1/4 exp(iS(t))(

1 +∞∑

j=1

wj(t)ejλt

), where S(t) =

∫ t

0

√Q(s) ds

and the wj(t) are functions increasing slower than the exponential function cjejμt, 0 < μ < λ.

To this end, we consider the following family of function classes Wk. We define the class Wk by thefollowing condition: a smooth function f(t) belongs to the class Wk if, for any integer nonnegative n,there exists a constant Cn depending on k and satisfying the condition |f (n)(t)| < Cnekμt for any t > 0.It follows from the definition of the class Wk that the integral with a variable upper limit of a functionfrom Wk belongs to Wk. We also note that if functions f and g belong to the respective classes Wi

and Wj , then the product f ∗ g also belongs to the class Wi+j .Algebraic or trigonometric polynomials can serve as such functions.In what follows, we assume that the function ϕ belongs to the class W1.To find a formal asymptotic expansion of the solution, we perform the same change of the function:

u(t) = w(t)Q(t)−1/4 exp(iS(t)).

The function w(t) satisfies Eq. (2).We consider the exponential function with an arbitrary exponent and represent it as e2λt, where λ is

an arbitrary positive number. We expand the coefficients of Eq. (2) in series. For example,

Q(t)1/2 = eλt

√1 +

ϕ(t)e2λt

= eλt

(1 +

12

ϕ(t)e2λt

− 18

ϕ(t)2

e4λt+ · · ·

),

1Q(t)

=1

e2λt

11 + ϕ(t)/e2λt

=1

e2λt

(1 − ϕ(t)

e2λt+

(ϕ(t))2

e4λt− · · ·

).

Since all classes Wk are closed with respect to the operation of taking linear combinations, we takethe property of the product of functions from different classes into account and reduce Eq. (2) to the form

d2w

dt2+

dw

dt

(2ieλt +

∞∑

k=0

1ekλt

ϕk(t))

+ w

( ∞∑

k=0

1ekλt

ψk(t))

= 0,

where ϕ0(t), ψ0(t) ∈ W1, ϕk(t), ψk(t) ∈ Wk for k > 0, and the series converge.We seek the solution as the formal series

w′ =w1

eλt+

w2

e2λt+

w3

e3λt+

w4

e4λt+ · · · ,

where wi are functions belonging to the classes Wi.We formally differentiate this series and obtain the relations

w′ =w′

1 − λw1

eλt+

w′2 − 2λw2

e2λt+

w′3 − 3λw3

e3λt+ · · · ,

w′′ =w′′

1 − 2λw′1 + λ2w1

eλt+

w′′2 − 4λw′

1 + 4λ2w1

e2λt+ · · · .

MATHEMATICAL NOTES Vol. 85 No. 1 2009

ASYMPTOTIC EXPANSION OF SOLUTIONS 127

Equating the functions at equal exponents, we obtain

e0 : 2i(w′1 − λw1) + ψ0(t) = 0,

w′1 = λw1 +

i

2ψ0(t), w′

1 = λw1, w1odn. = ceλt, w1 = c(t)eλt,

c′(t)eλt =i

2ψ0(t), c(t) =

i

2

∫ t

αψ0e

−λt dτ + C,

w1(t) =(

i

2

∫ t

αψ0e

−λτ dτ + C

)eλt.

For this series to be meaningful, the function w1 must increase slower than the denominator or,which is practically the same, must belong to the class W1. If, starting from some t, the function ψ0 islarger than a certain positive constant but belongs to the class W1, then the only way to preserve thewell-posedness of the series is to set C = 0 and α = +∞. In this case, the function w1 belongs to theclass W1. We obtain

e−λt : w′′1 − 2λw′

1 + λ2w1 + 2i(w′2 − 2λw2) + (w′

1 − λw1)ϕ0 + ψ1 + w1ψ0 = 0,

w′2 = 2λw2 +

i

2(w′′

1 − 2λw′1 + λ2w1 + (w′

1 − λw1)ϕ0 + ψ1 + w1ψ0) = 2λw2 + χ2(t),

where

χ2(t) =i

2(w′′

1 − 2λw′1 + λ2w1 + (w′

1 − λw1)ϕ0 + ψ1 + w1ψ0), w2 = c(t)e2λt,

c(t) =∫ t

+∞χ2e

−2λτ dτ + C, w2(t) =(

i

2

∫ t

+∞χ2e

−2λτ dτ + C

)e2λt, C = 0.

All the functions constituting χ2 belong to the class W1. It is easy to show that any derivative of χ2

increases slower than the exponential function e2μt, and hence χ2 and w2 belong to the class W2.We consider an arbitrary exponent −kλ, where k is positive integer:

e−kλt : w′′k − 2kλw′

k + k2λ2wk + 2i(w′k+1 − kλwk) +

k∑

j=1

(w′j − jλwj)ϕk−j + ψk +

k∑

j=1

wjψk−j = 0,

w′k+1 = kλwk +

i

2

(w′′

k − 2kλw′k + k2λ2wk +

k∑

j=1

(w′j − jλwj)ϕk−j + ψk +

k∑

j=1

wjψk−j

)

= kλwk + χm+1(t),

wk+1 = c(t)ekλt, c(t) =∫ t

+∞χk+1e

−kλτ dτ + C, wk+1(t) =i

2ekλt

∫ t

+∞χk+1e

−kλτ dτ, C = 0.

This means that, in this case, we have also constructed an infinite asymptotic series. We can alsoprove that there exists a solution such that it can be expanded into this series.

ACKNOWLEDGMENTS

The author wishes to express gratitude to A. M. Il’in for stating the problem and constant attentionto this work.

This work was supported by the Russian Foundation for Basic Research (grant no. 07-01-96002r-ural-a).

REFERENCES1. M. V. Fedoryuk, Asymptotic Methods for Linear Ordinary Differential Equations, in Reference Mathe-

matical Library (Nauka, Moscow, 1983) [in Russian].

MATHEMATICAL NOTES Vol. 85 No. 1 2009