Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 923269, 13 pages
doi:10.1155/2011/923269
Research Article
Approximation of Analytic Functions by
Bessel’s Functions of Fractional Order
Soon-Mo Jung
Mathematics Section, College of Science and Technology, Hongik University,
Jochiwon 339-701, Republic of Korea
Correspondence should be addressed to Soon-Mo Jung, smjung@hongik.ac.kr
Received 31 March 2011; Accepted 6 June 2011
Academic Editor: Yong Zhou
Copyright q 2011 Soon-Mo Jung. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We
the inhomogeneous Bessel’s differential equation x2 y x xy x x2 − ν2 yx ∞will solve
m
a
x
,
where
ν is a positive nonintegral number and apply this result for approximating
m
m0
analytic functions of a special type by the Bessel functions of fractional order.
1. Introduction
The stability problem for functional equations starts from the famous talk of Ulam and the
partial solution of Hyers to the Ulam problem see 1, 2. Thereafter, Rassias 3 attempted
to solve the stability problem of the Cauchy additive functional equation in a more general
setting.
The stability concept introduced by Rassias’s theorem significantly influenced a
number of mathematicians to investigate the stability problems for various functional
equations see 4–14 and the references therein.
Assume that Y is a normed space and I is an open subset of R. If for any function
f : I → Y satisfying the differential inequality
an xyn x an−1 xyn−1 x · · · a1 xy x a0 xyx hx ≤ ε,
1.1
for all x ∈ I and for some ε ≥ 0, there exists a solution f0 : I → Y of the differential equation
an xyn x an−1 xyn−1 x · · · a1 xy x a0 xyx hx 0,
1.2
2
Abstract and Applied Analysis
such that fx − f0 x ≤ Kε for any x ∈ I, where Kε depends on ε only, then we say that
the above differential equation satisfies the Hyers-Ulam stability or the local Hyers-Ulam
stability if the domain I is not the whole space R. We may apply these terminologies for
other differential equations. For more detailed definition of the Hyers-Ulam stability, refer to
2, 3, 6, 8, 10–12, 14.
Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of
linear differential equations see 15, 16. Here, we will introduce a result of Alsina and Ger
see 17: If a differentiable function f : I → R is a solution of the differential inequality
|y x − yx| ≤ ε, where I is an open subinterval of R, then there exists a constant c such that
|fx − cex | ≤ 3ε for any x ∈ I.
This result of Alsina and Ger has been generalized by Takahasi et al. They proved in
18 that the Hyers-Ulam stability holds for the Banach space-valued differential equation
y x λyx see also 19.
Using the conventional power series method, the author has investigated the general
solution of the inhomogeneous Legendre differential equation under some specific condition,
and this result was applied to prove the Hyers-Ulam stability of the Legendre differential
equation see 20. In a recent paper, he has also investigated an approximation property
of analytic functions by the Legendre functions see 21. This study has been continued to
various special functions including the Airy functions, the exponential functions, the Hermite
functions, and the power functions see 22–25.
Recently, the author and Kim tried to prove the Hyers-Ulam stability of the Bessel
differential equation
x2 y x xy x x2 − ν2 yx 0.
1.3
However, the obtained theorem unfortunately does not describe the Hyers-Ulam stability of
the Bessel differential equation in a strict sense see 26.
In Section 2 of this paper, by using the ideas from 21, we will determine the general
solution of the inhomogeneous Bessel differential equation
∞
am xm ,
x2 y x xy x x2 − ν2 yx 1.4
m0
where the parameter ν is a positive nonintegral number. Section 3 will be devoted to the
investigation of an approximation property of the Bessel functions.
Throughout this paper, we denote by x the largest integer not exceeding x for any
x ∈ R, and we define Iρ −ρ, 0 ∪ 0, ρ for any ρ > 0.
2. Inhomogeneous Bessel’s Differential Equation
A function is called a Bessel function of fractional order if it is a solution of the Bessel
differential equation 1.3, where ν is a positive nonintegral number. The Bessel differential
equation plays a great role in physics and engineering. In particular, this equation is most
useful for treating the boundary value problems exhibiting cylindrical symmetries.
Abstract and Applied Analysis
3
m
The convergence of the power series ∞
m0 am x seems not to guarantee the existence
of solutions to the inhomogeneous Bessel differential equation 1.4. Thus, we adopt an
additional condition to ensure the existence of solutions to the equation.
Theorem 2.1. Let ν be a positive nonintegral number, and let ρ be a positive constant. Assume that
m
the radius of convergence of power series ∞
m0 am x is at least ρ and there exists a constant σ > 0
satisfying the condition
|am2 | ≤
m2
|cm |,
σ2
2.1
for all sufficiently large integers m, where
cm ⎧ m/2 m/2
⎪
1
⎪
⎪
a2i
−
⎪
2
⎪
2
⎪
⎪
ji ν − 2j
⎨ i0
⎪
⎪
m/2
m/2
⎪
⎪
⎪
⎪
⎪
−
a
2i1
⎩
i0
ji
for even m,
2.2
1
2
ν2 − 2j 1
for odd m,
for all m ∈ N0 . Let ρ0 min{ρ, σ}. Then every solution y : Iρ0 → C of the Bessel’s differential
equation 1.4 can be expressed by
yx yh x ∞
cm x m ,
2.3
m0
for all x ∈ Iρ0 , where yh x is a solution of the homogeneous Bessel equation 1.3.
Proof. We assume that y : Iρ0 → C is a function given in the form 2.3, and we define
m
yp x yx − yh x ∞
m0 cm x . Then, it follows from 2.1 and 2.2 that
cm2 am2
1
≤ 1 ,
lim
lim −
1
σ2
m → ∞ cm
m → ∞ m 22 − ν 2
cm
2.4
since we can deduce the relation cm2 am2 − cm /m 22 − ν2 from 2.2 by some
manipulations. That is, the power series for yp x converges for all x ∈ Iρ0 . Hence, we see
that the domain of yx is well defined.
4
Abstract and Applied Analysis
We now prove that the function yp x satisfies the inhomogeneous equation 1.4.
Indeed, it follows from 2.2 that
x2 yp x xyp x x2 − ν2 yp x
∞
mm − 1cm xm m2
∞
mcm xm m1
c1 x − ν 2 c0 − ν 2 c1 x ∞
cm xm2 −
m0
∞
ν 2 cm x m
m0
2.5
∞ cm−2 m2 − ν2 cm xm
m2
a0 a1 x ∞
am xm ,
m2
since we obtain
c0 −
1
a0 ,
ν2
c1 cm−2 m2 − ν2 cm am ,
1
a1 ,
1 − ν2
for m ≥ 2,
2.6
which proves that yp x is a particular solution of the inhomogeneous equation 1.4.
On the other hand, since every solution to 1.4 can be expressed as a sum of a solution
yh x of the homogeneous equation and a particular solution yp x of the inhomogeneous
equation, every solution of 1.4 is certainly of the form 2.3.
3. Approximate Bessel’s Differential Equation
In this section, assume that ν is a positive nonintegral number and ρ is a positive constant.
For a given K ≥ 0, we denote by CK the set of all functions y : Iρ → C with the properties a
and b:
m
a yx is expressible by a power series ∞
m0 bm x whose radius of convergence is at
least ρ;
∞
m
m
2
2
b ∞
m0 |am x | ≤ K|
m0 am x | for any x ∈ Iρ , where am bm−2 m − ν bm for all
m ∈ N0 and set b−2 b−1 0.
For a positive nonintegral number ν, define
Me x max
Mo x max
⎧
k
⎨
⎩ ji
⎧
k
⎨
⎩ ji
⎫
⎬
x2
,
:
0
≤
i
≤
k
≤
μ
2 2 ⎭
ν − 2j ⎫
⎬
x2
:0≤i≤k≤μ ,
2
2
⎭
ν − 2j 1 Mx max{Me x, Mo x, 1},
3.1
Abstract and Applied Analysis
5
√
where μ ν2 x2 /2 and
Lν ∞
1
m0 m
− ν2
< ∞.
3.2
We remark that Mx → 1 as |x| → 0.
We will now solve the approximate Bessel differential equations in a special class of
analytic functions, CK .
Theorem 3.1. Let ν be a positive nonintegral number, and let p be a nonnegative integer with p <
ν < p 1. Assume that a function y ∈ CK satisfies the differential inequality
2 x y x xy x x2 − ν2 yx ≤ ε,
3.3
for all x ∈ Iρ and for some ε ≥ 0. If the sequence {bm } satisfies the condition
bm2 Obm as m −→ ∞
3.4
with a Landau constant C ≥ 0, then there exists a solution yh x of the Bessel differential equation
1.3 such that
yx − yh x ≤ KLν Mxε,
3.5
√
for any x ∈ Iρ0 , where ρ0 min{ρ, 1/ C∗ } and C∗ is a positive number larger than C. If C and ρ
are sufficiently small and large; respectively, then
Mx ≤ max
⎧
⎪
⎨
|x|2
|x|2
⎫
⎪
⎬
|x|
|x|
,
|x|/21 2 |x|/21 ⎪
⎪
2
2
⎭
⎩ |ν − p |
ν2 − p 1 3.6
for all sufficiently large |x|.
Proof. Since y belongs to CK , it follows from a and b that
∞ ∞
bm−2 m2 − ν2 bm xm x2 y x xy x x2 − ν2 yx am xm ,
m0
3.7
m0
for all x ∈ Iρ . By considering 3.3 and 3.7, we get
∞
m
am x ≤ ε,
m0
3.8
6
Abstract and Applied Analysis
for any x ∈ Iρ . This inequality, together with b, yields
∞
m
|am x | ≤ K am x ≤ Kε,
m0
m0
∞
m
3.9
for each x ∈ Iρ .
Now, it follows from b that
n
i0
a2i
n
ji
1
2
2
ν − 2j
n
n
b2i−2
i0
ji
n−1
b2i
i−1
b−2
n
n
1
1
−
b
2i
2
2
2
2
ν − 2j
i0
ji1 ν − 2j
n
n
1
1
− b2i
2
2
2
2
i0
ji1 ν − 2j
ji1 ν − 2j
n
n
j0
3.10
1
2 − b2n
ν2 − 2j
−b2n ,
since b−2 0. Similarly, we obtain
n
n
a2i1
i0
ji
1
2 −b2n1 ,
ν2 − 2j 1
3.11
for all n ∈ N0 , that is,
bm ⎧ m/2 m/2
⎪
1
⎪
⎪
a
−
⎪
2i
2
⎪
2
⎪
⎪
ji ν − 2j
⎨ i0
⎪
⎪
m/2
m/2
⎪
⎪
⎪
⎪
⎪
a2i1
⎩−
i0
ji
ν2
1
2
− 2j 1
for even m,
3.12
for odd m,
for all m ∈ N0 .
On the other hand, by b and 3.4, we have
|am2 | bm m 22 − ν2 bm2 ≤ |bm | m 22 |bm2 |
3.13
m2
≤ √ 2 |bm |,
1/ C∗
for all sufficiently large integers m, where C∗ is a positive number larger
√ than C. Hence,
in view of 3.12 and 3.13, the condition 2.1 is satisfied with σ 1/ C∗ . Moreover, we
Abstract and Applied Analysis
7
m
know that the radius of convergence of power series ∞
m0 am x is at least ρ because the
convergence radius of the power series expression for yx is at least ρ see a and 3.7.
According to 3.7 and Theorem 2.1, there exists a solution yh x of the homogeneous
Bessel differential equation 1.3 satisfying 2.3 for all x ∈ Iρ0 . Thus, it follows from 2.2 and
3.12 that
∞
∞
2n
2n1
yx − yh x c2n x c2n1 x
n0
n0
n
n
∞
n
n
∞ 2n 1
1
2n1
− x
a2i
−
x
a
2i1
2
2
2
2
n0
n0
i0
i0
ji ν − 2j
ji ν − 2j 1 n−1
n
∞
1
x2
2i
a
x
2i
2
2
2
n0 ν2 − 2n i0
ji ν − 2j
n−1
n
∞
2
1
x
2i1
,
a2i1 x
2
2
2
2 − 2j 1 ν
−
1
2n
ν
n0
i0
ji
3.14
for any x ∈ Iρ0 . Moreover, we have
μ1
n−1
n 1
x2
2i yx − yh x ≤
x
a
2i
2 2
2
n0 ν − 2n i0
ji ν 2 − 2j μ n−1
1
x2
2i a
2i x 2 2
nμ2 ν 2 − 2n i0
ji ν 2 − 2j n−1
n 1
x2
2i a
2i x 2 2
nμ2 ν 2 − 2n iμ1
ji ν 2 − 2j ∞
∞
n−1
n 1
x2
a2i1 x2i1 2 2
n0 ν 2 − 2n 1 i0
ji ν 2 − 2j 1 μ1
μ n−1
1
x2
a2i1 x2i1 2 2
nμ2 ν 2 − 2n 1 i0
ji ν 2 − 2j 1 n−1
n 1
x2
2i1 a
2i1 x
2 ,
2
nμ2 ν 2 − 2n 1 iμ1
ji ν 2 − 2j 1 ∞
∞
√
for all x ∈ Iρ0 , where μ ν2 x2 /2.
3.15
8
Abstract and Applied Analysis
We know that x2 /|ν2 −2j2 | < 1 and x2 /|ν2 −2j 12 | < 1 for j ≥ μ1 and x2 /|ν2 −2j2 |
or x /|ν2 − 2j 12 | is not perhaps less than 1 for j ≤ μ. Then, we have
2
n−1
ji
⎛
⎞⎛
⎞
μ
n−1
2
2
x2
x
x
⎠⎝
⎠
⎝
2 2 ≤ Me x ≤ Mx,
2 2 ji ν 2 − 2j jμ1 ν 2 − 2j ν − 2j 3.16
for all n ≥ μ 2 and i 0, 1, . . . , μ. Similarly, we get
n−1
ji
x2
2 ≤ Mx,
2
ν − 2j 1 3.17
for all n ≥ μ 2 and i 0, 1, . . . , μ.
Thus, it follows from 3.9 and 3.15 that
μ1
μ1 n−1 a2n x2n Mx 2i yx − yh x ≤
a2i x 2
2
n0 ν 2 − 2n i0
n0 ν 2 − 2n μ
Mx 2i a
2i x 2
nμ2 ν 2 − 2n i0
∞
n 1
2i a
2i x 2
nμ2 ν 2 − 2n iμ1
∞
μ1 n−1 a2n1 x2n1 Mx
2i1 a2i1 x 2
2
n0 ν 2 − 2n 1 i0
n0 ν 2 − 2n 1 μ1
μ Mx
a2i1 x2i1 2
nμ2 ν 2 − 2n 1 i0
n 1
2i1 a
2i1 x
2
nμ2 ν 2 − 2n 1 iμ1
∞
∞
⎛
⎞
n ∞
n 1
a2i x2i a2i1 x2i1 ⎠
≤ Mx⎝ 2
2
n0 ν 2 − 2n i0
n0 ν 2 − 2n 1 i0
∞
≤ MxKε
1
∞
m0
|ν2
1
− m2 |
≤ KLν Mxε,
3.18
for x ∈ Iρ0 .
Abstract and Applied Analysis
9
Finally, assume that C is sufficiently small and ρ is sufficiently large. Then ρ0 is
sufficiently large. For any j ≥ 0 and x ∈ Iρ0 , we have
⎧
⎫
⎨ x2
⎬
x2
x2
≤ max , .
2
2
⎩ ν2 − p2 ν2 − p 1 2 ⎭
ν − 2j 3.19
If |x| is so large that
μ
1
|x|
|x|
≤
,
ν 2 x2 2
2
2
3.20
then it follows from the definition of Mx that
Mx ≤ max
⎧
⎪
⎨
|x|2
|x|2
⎫
⎪
⎬
|x|
|x|
,
|x|/21 , 2 |x|/21 ⎪
⎪
2
2
⎩ ν −p
⎭
2
ν − p 1 3.21
for all sufficiently large |x|.
If ν is large enough, then we can prove the local Hyers-Ulam stability of the Bessel
differential equation 1.3 as we see in the following corollary.
Corollary 3.2. Let ν be a positive nonintegral number and let p be a nonnegative integer with p <
ν < p 1. Assume that a function y ∈ CK satisfies the differential inequality 3.3 for all x ∈ Iρ and
for some ε ≥ 0. Suppose the√sequence{bm } satisfies the condition 3.4 with a Landau constant C ≥ 0
and define ρ0 min{ρ, 1/ C∗ } for a positive number C∗ > C. If
x2
≤ 1,
2
ν − p2 x2
2 ≤ 1,
2
ν − p 1 3.22
for all x ∈ Iρ0 , then there exists a solution yh x of the Bessel differential equation 1.3 such that
yx − yh x ≤ KLν ε,
3.23
for any x ∈ Iρ0 .
Proof. For any j ≥ 0 and x ∈ Iρ0 , we have
⎧
⎫
⎫
⎨ x2
⎬
⎬
x2
x2
x2
,
≤
max
≤ 1.
max ,
⎩ ν2 − p2 ν2 − p 12 ⎭
⎩ ν2 − 2j 2 ν2 − 2j 12 ⎭
⎧
⎨
3.24
10
Abstract and Applied Analysis
Thus, we get
⎫
⎬
x
x
, 1,
,
1
Mx ≤ max 2
⎩ ν − p2 ν2 − p 12 ⎭
⎧
⎨
2
2
3.25
and the assertion is true due to Theorem 3.1.
4. Examples
We will show that there exist functions yx which satisfy all the conditions given in
Theorem 3.1 and Corollary 3.2. Let us define a function y : I10 → R by
yx J100.5 x cx2 ∞
bm xm ,
4.1
m0
where J100.5 x is the Bessel function of the first kind of order 100.5, n is a positive integer,
and c is a constant satisfying
c
ε
,
999625
4.2
m
for some ε > 0. It is obvious that the convergence radius of the power series ∞
m0 bm x is
∞
m
infinity. So we can set ρ 10. In fact, the infinite series m0 bm 11 converges. So we have
11m1 b
m1 lim sup
≤ 1,
m
m → ∞ 11 bm 11m2 b
m2 lim sup m1
≤ 1.
bm1 m → ∞ 11
4.3
Thus, it holds true that
bm2 bm2 bm1 ≤ 1
lim sup
lim sup
lim sup
bm
bm1 m → ∞ bm 121
m→∞
m→∞
4.4
which implies that the sequence {bm } satisfies the condition
3.4 with a Landau constant
√
C 1/121. If we take C∗ 1/100, then ρ0 min{ρ, 1/ C∗ } 10.
Since J100.5 x is a particular solution of the Bessel differential equation 1.3 with ν 100.5, it follows from 4.1 that
40385 2
40401
yx −
cx cx4 ,
x y x xy x x −
4
4
2 for any x ∈ I10 .
2
4.5
Abstract and Applied Analysis
11
If we set
⎧ 40385
⎪
⎪
c
−
⎪
⎪
4
⎪
⎨
am c
⎪
⎪
⎪
⎪
⎪
⎩
0
for m 2,
for m 4,
4.6
otherwise,
then we have
∞
40401
x2 y x xy x x2 −
yx am xm ,
4
m0
∞
2 x y x xy x x2 − 40401 yx am xm c 40385 x2 − x4 < 999625c ε,
4
4
m0
4.7
for all x ∈ I10 .
Moreover, we have
am xm | 40385/4 cx2 − cx4 39985
| ∞
∞m0
,
>
m
40785
40385/4cx2 cx4
m0 |am x |
4.8
∞
8157 m
|am x | <
am x ,
7997 m0
m0
4.9
and hence, we get
∞
m
for all x ∈ I10 . That is, {am } satisfies the property b with K 8157/7997.
It holds true that
x2
< 1,
ν2 − p2 x2
2 < 1,
2 ν − p 1 4.10
12
Abstract and Applied Analysis
for all x ∈ I10 , and since
L100.5 ∞
1
m0 m
− 100.52
1
−100.5
2
1
−99.5
2
1
−98.5
2
··· 1
−0.52
1
1
1
1
···
0.52 1.52 2.52 3.52
1
1
1
1
1
1
≤
·
·
·
1002 992 982
12
0.52 0.52
4.11
1
1
1
2 2 ···
2
1
2
3
≤ 2ζ2 8 π2
8,
3
it follows from Corollary 3.2 that there exists a solution yh x of the Bessel differential
equation 1.3 such that
yx − yh x ≤ 8157 L100.5 ε < 8157
7997
7997
π2
8 ε,
3
4.12
for any x ∈ I10 .
Acknowledgment
This research was supported by the Basic Science Research Program through the National
Research Foundation of Korea NRF funded by the Ministry of Education, Science, and
Technology no. 2010-0007143.
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19 T. Miura, S.-M. Jung, and S.-E. Takahasi, “Hyers-Ulam-Rassias stability of the Banach space valued
linear differential equations y λy,” Journal of the Korean Mathematical Society, vol. 41, no. 6, pp.
995–1005, 2004.
20 S.-M. Jung, “Legendre’s differential equation and its Hyers-Ulam stability,” Abstract and Applied
Analysis, vol. 2007, Article ID 56419, 14 pages, 2007.
21 S.-M. Jung, “Approximation of analytic functions by Legendre functions,” Nonlinear Analysis, vol. 71,
no. 12, pp. 103–108, 2009.
22 S.-M. Jung, “Approximation of analytic functions by Airy functions,” Integral Transforms and Special
Functions, vol. 19, no. 11-12, pp. 885–891, 2008.
23 S.-M. Jung, “An approximation property of exponential functions,” Acta Mathematica Hungarica, vol.
124, no. 1-2, pp. 155–163, 2009.
24 S.-M. Jung, “Approximation of analytic functions by Hermite functions,” Bulletin des Sciences
Mathématiques, vol. 133, no. 7, pp. 756–764, 2009.
25 S.-M. Jung and S. Min, “On approximate Euler differential equations,” Abstract and Applied Analysis,
vol. 2009, Article ID 537963, 8 pages, 2009.
26 B. Kim and S.-M. Jung, “Bessel’s differential equation and its Hyers-Ulam stability,” Journal of
Inequalities and Applications, vol. 2007, Article ID 21640, 8 pages, 2007.
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