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Power series method and approximate linear differential equations of second order

Advances in Difference Equations20132013:76

https://doi.org/10.1186/1687-1847-2013-76

  • Received: 14 December 2012
  • Accepted: 4 March 2013
  • Published:

Abstract

In this paper, we will establish a theory for the power series method that can be applied to various types of linear differential equations of second order to prove the Hyers-Ulam stability.

MSC:34A05, 39B82, 26D10, 34A40.

Keywords

  • power series method
  • approximate linear differential equation
  • simple harmonic oscillator equation
  • Hyers-Ulam stability
  • approximation

1 Introduction

Let X be a normed space over a scalar field K , and let I R be an open interval, where K denotes either or . Assume that a 0 , a 1 , , a n : I K and g : I X are given continuous functions. If for every n times continuously differentiable function y : I X satisfying the inequality
a n ( x ) y ( n ) ( x ) + a n 1 ( x ) y ( n 1 ) ( x ) + + a 1 ( x ) y ( x ) + a 0 ( x ) y ( x ) + g ( x ) ε
for all x I and for a given ε > 0 , there exists an n times continuously differentiable solution y 0 : I X of the differential equation
a n ( x ) y ( n ) ( x ) + a n 1 ( x ) y ( n 1 ) ( x ) + + a 1 ( x ) y ( x ) + a 0 ( x ) y ( x ) + g ( x ) = 0

such that y ( x ) y 0 ( x ) K ( ε ) for any x I , where K ( ε ) is an expression of ε with lim ε 0 K ( ε ) = 0 , then we say that the above differential equation has the Hyers-Ulam stability. For more detailed definitions of the Hyers-Ulam stability, we refer the reader to [18].

Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [9, 10]). Thereafter, Alsina and Ger [11] proved the Hyers-Ulam stability of the differential equation y ( x ) = y ( x ) . It was further proved by Takahasi et al. that the Hyers-Ulam stability holds for the Banach space valued differential equation y ( x ) = λ y ( x ) (see [12] and also [1315]).

Moreover, Miura et al. [16] investigated the Hyers-Ulam stability of an n th-order linear differential equation. The first author also proved the Hyers-Ulam stability of various linear differential equations of first order (ref. [1725]).

Recently, the first author applied the power series method to studying the Hyers-Ulam stability of several types of linear differential equations of second order (see [2634]). However, it was inconvenient that he had to alter and apply the power series method with respect to each differential equation in order to study the Hyers-Ulam stability. Thus, it is inevitable to develop a power series method that can be comprehensively applied to different types of differential equations.

In Sections 2 and 3 of this paper, we establish a theory for the power series method that can be applied to various types of linear differential equations of second order to prove the Hyers-Ulam stability.

Throughout this paper, we assume that the linear differential equation of second order of the form
p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = 0 ,
(1)
for which x = 0 is an ordinary point, has the general solution y h : ( ρ 0 , ρ 0 ) C , where ρ 0 is a constant with 0 < ρ 0 and the coefficients p , q , r : ( ρ 0 , ρ 0 ) C are analytic at 0 and have power series expansions
p ( x ) = m = 0 p m x m , q ( x ) = m = 0 q m x m and r ( x ) = m = 0 r m x m

for all x ( ρ 0 , ρ 0 ) . Since x = 0 is an ordinary point of (1), we remark that p 0 0 .

2 Inhomogeneous differential equation

In the following theorem, we solve the linear inhomogeneous differential equation of second order of the form
p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = m = 0 a m x m
(2)

under the assumption that x = 0 is an ordinary point of the associated homogeneous linear differential equation (1).

Theorem 2.1 Assume that the radius of convergence of power series m = 0 a m x m is ρ 1 > 0 and that there exists a sequence { c m } satisfying the recurrence relation
k = 0 m [ ( k + 2 ) ( k + 1 ) c k + 2 p m k + ( k + 1 ) c k + 1 q m k + c k r m k ] = a m
(3)
for any m N 0 . Let ρ 2 be the radius of convergence of power series m = 0 c m x m and let ρ 3 = min { ρ 0 , ρ 1 , ρ 2 } , where ( ρ 0 , ρ 0 ) is the domain of the general solution to (1). Then every solution y : ( ρ 3 , ρ 3 ) C of the linear inhomogeneous differential equation (2) can be expressed by
y ( x ) = y h ( x ) + m = 0 c m x m

for all x ( ρ 3 , ρ 3 ) , where y h ( x ) is a solution of the linear homogeneous differential equation (1).

Proof Since x = 0 is an ordinary point, we can substitute m = 0 c m x m for y ( x ) in (2) and use the formal multiplication of power series and consider (3) to get
p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = m = 0 k = 0 m p m k ( k + 2 ) ( k + 1 ) c k + 2 x m + m = 0 k = 0 m q m k ( k + 1 ) c k + 1 x m + m = 0 k = 0 m r m k c k x m = m = 0 k = 0 m [ ( k + 2 ) ( k + 1 ) c k + 2 p m k + ( k + 1 ) c k + 1 q m k + c k r m k ] x m = m = 0 a m x m
for all x ( ρ 3 , ρ 3 ) . That is, m = 0 c m x m is a particular solution of the linear inhomogeneous differential equation (2), and hence every solution y : ( ρ 3 , ρ 3 ) C of (2) can be expressed by
y ( x ) = y h ( x ) + m = 0 c m x m ,

where y h ( x ) is a solution of the linear homogeneous differential equation (1). □

For the most common case in applications, the coefficient functions p ( x ) , q ( x ) , and r ( x ) of the linear differential equation (1) are simple polynomials. In such a case, we have the following corollary.

Corollary 2.2 Let p ( x ) , q ( x ) , and r ( x ) be polynomials of degree at most d 0 . In particular, let d 0 be the degree of p ( x ) . Assume that the radius of convergence of power series m = 0 a m x m is ρ 1 > 0 and that there exists a sequence { c m } satisfying the recurrence formula
k = m 0 m [ ( k + 2 ) ( k + 1 ) c k + 2 p m k + ( k + 1 ) c k + 1 q m k + c k r m k ] = a m
(4)
for any m N 0 , where m 0 = max { 0 , m d } . If the sequence { c m } satisfies the following conditions:
  1. (i)

    lim m c m 1 / m c m = 0 ,

     
  2. (ii)

    there exists a complex number L such that lim m c m / c m 1 = L and p d 0 + L p d 0 1 + + L d 0 1 p 1 + L d 0 p 0 0 ,

     
then every solution y : ( ρ 3 , ρ 3 ) C of the linear inhomogeneous differential equation (2) can be expressed by
y ( x ) = y h ( x ) + m = 0 c m x m

for all x ( ρ 3 , ρ 3 ) , where ρ 3 = min { ρ 0 , ρ 1 } and y h ( x ) is a solution of the linear homogeneous differential equation (1).

Proof Let m be any sufficiently large integer. Since p d + 1 = p d + 2 = = 0 , q d + 1 = q d + 2 = = 0 and r d + 1 = r d + 2 = = 0 , if we substitute m d + k for k in (4), then we have
a m = k = 0 d [ ( m d + k + 2 ) ( m d + k + 1 ) c m d + k + 2 p d k + ( m d + k + 1 ) c m d + k + 1 q d k + c m d + k r d k ] .
By (i) and (ii), we have
lim sup m | a m | 1 / m = lim sup m | k = 0 d ( m d + k + 2 ) ( m d + k + 1 ) c m d + k + 2 × ( p d k + q d k ( m d + k + 2 ) c m d + k + 1 c m d + k + 2 + r d k ( m d + k + 2 ) ( m d + k + 1 ) c m d + k c m d + k + 1 c m d + k + 1 c m d + k + 2 ) | 1 / m = lim sup m | k = 0 d ( m d + k + 2 ) ( m d + k + 1 ) c m d + k + 2 p d k | 1 / m = lim sup m | k = d d 0 d ( m d + k + 2 ) ( m d + k + 1 ) c m d + k + 2 p d k | 1 / m = lim sup m | ( m d 0 + 2 ) ( m d 0 + 1 ) c m d 0 + 2 ( p d 0 + L p d 0 1 + + L d 0 p 0 ) | 1 / m = lim sup m | ( p d 0 + L p d 0 1 + + L d 0 p 0 ) ( m d 0 + 2 ) ( m d 0 + 1 ) | 1 / m × ( | c m d 0 + 2 | 1 / ( m d 0 + 2 ) ) ( m d 0 + 2 ) / m = lim sup m | c m d 0 + 2 | 1 / ( m d 0 + 2 ) ,

which implies that the radius of convergence of the power series m = 0 c m x m is ρ 1 . The rest of this corollary immediately follows from Theorem 2.1. □

In many cases, it occurs that p ( x ) 1 in (1). For this case, we obtain the following corollary.

Corollary 2.3 Let ρ 3 be a distance between the origin 0 and the closest one among singular points of q ( z ) , r ( z ) , or m = 0 a m z m in a complex variable z. If there exists a sequence { c m } satisfying the recurrence relation
( m + 2 ) ( m + 1 ) c m + 2 + k = 0 m [ ( k + 1 ) c k + 1 q m k + c k r m k ] = a m
(5)
for any m N 0 , then every solution y : ( ρ 3 , ρ 3 ) C of the linear inhomogeneous differential equation
y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = m = 0 a m x m
(6)
can be expressed by
y ( x ) = y h ( x ) + m = 0 c m x m

for all x ( ρ 3 , ρ 3 ) , where y h ( x ) is a solution of the linear homogeneous differential equation (1) with p ( x ) 1 .

Proof If we put p 0 = 1 and p i = 0 for each i N , then the recurrence relation (3) reduces to (5). As we did in the proof of Theorem 2.1, we can show that m = 0 c m x m is a particular solution of the linear inhomogeneous differential equation (6).

According to [[35], Theorem 7.4] or [[36], Theorem 5.2.1], there is a particular solution y 0 ( x ) of (6) in a form of power series in x whose radius of convergence is at least ρ 3 . Moreover, since m = 0 c m x m is a solution of (6), it can be expressed as a sum of both y 0 ( x ) and a solution of the homogeneous equation (1) with p ( x ) 1 . Hence, the radius of convergence of m = 0 c m x m is at least ρ 3 .

Now, every solution y : ( ρ 3 , ρ 3 ) C of (6) can be expressed by
y ( x ) = y h ( x ) + m = 0 c m x m ,

where y h ( x ) is a solution of the linear differential equation (1) with p ( x ) 1 . □

3 Approximate differential equation

In this section, let ρ 1 > 0 be a constant. We denote by C the set of all functions y : ( ρ 1 , ρ 1 ) C with the following properties:
  1. (a)

    y ( x ) is expressible by a power series m = 0 b m x m whose radius of convergence is at least ρ 1 ;

     
  2. (b)
    There exists a constant K 0 such that m = 0 | a m x m | K | m = 0 a m x m | for any x ( ρ 1 , ρ 1 ) , where
    a m = k = 0 m [ ( k + 2 ) ( k + 1 ) b k + 2 p m k + ( k + 1 ) b k + 1 q m k + b k r m k ]
     

for all m N 0 and p 0 0 .

Lemma 3.1 Given a sequence { a m } , let { c m } be a sequence satisfying the recurrence formula (3) for all m N 0 . If p 0 0 and n 2 , then c n is a linear combination of a 0 , a 1 , , a n 2 , c 0 , and c 1 .

Proof We apply induction on n. Since p 0 0 , if we set m = 0 in (3), then
c 2 = 1 2 p 0 a 0 r 0 2 p 0 c 0 q 0 2 p 0 c 1 ,
i.e., c 2 is a linear combination of a 0 , c 0 , and c 1 . Assume now that n is an integer not less than 2 and c i is a linear combination of a 0 , , a i 2 , c 0 , c 1 for all i { 2 , 3 , , n } , namely,
c i = α i 0 a 0 + α i 1 a 1 + + α i i 2 a i 2 + β i c 0 + γ i c 1 ,
where α i 0 , , α i i 2 , β i , γ i are complex numbers. If we replace m in (3) with n 1 , then
a n 1 = 2 c 2 p n 1 + c 1 q n 1 + c 0 r n 1 + 6 c 3 p n 2 + 2 c 2 q n 2 + c 1 r n 2 + + n ( n 1 ) c n p 1 + ( n 1 ) c n 1 q 1 + c n 2 r 1 + ( n + 1 ) n c n + 1 p 0 + n c n q 0 + c n 1 r 0 = ( n + 1 ) n p 0 c n + 1 + [ n ( n 1 ) p 1 + n q 0 ] c n + + ( 2 p n 1 + 2 q n 2 + r n 3 ) c 2 + ( q n 1 + r n 2 ) c 1 + r n 1 c 0 ,
which implies
c n + 1 = 1 ( n + 1 ) n p 0 a n 1 n ( n 1 ) p 1 + n q 0 ( n + 1 ) n p 0 c n 2 p n 1 + 2 q n 2 + r n 3 ( n + 1 ) n p 0 c 2 q n 1 + r n 2 ( n + 1 ) n p 0 c 1 r n 1 ( n + 1 ) n p 0 c 0 = α n + 1 0 a 0 + α n + 1 1 a 1 + + α n + 1 n 1 a n 1 + β n + 1 c 0 + γ n + 1 c 1 ,

where α n + 1 0 , , α n + 1 n 1 , β n + 1 , γ n + 1 are complex numbers. That is, c n + 1 is a linear combination of a 0 , a 1 , , a n 1 , c 0 , c 1 , which ends the proof. □

In the following theorem, we investigate a kind of Hyers-Ulam stability of the linear differential equation (1). In other words, we answer the question whether there exists an exact solution near every approximate solution of (1). Since x = 0 is an ordinary point of (1), we remark that p 0 0 .

Theorem 3.2 Let { c m } be a sequence of complex numbers satisfying the recurrence relation (3) for all m N 0 , where (b) is referred for the value of a m , and let ρ 2 be the radius of convergence of the power series m = 0 c m x m . Define ρ 3 = min { ρ 0 , ρ 1 , ρ 2 } , where ( ρ 0 , ρ 0 ) is the domain of the general solution to (1). Assume that y : ( ρ 1 , ρ 1 ) C is an arbitrary function belonging to C and satisfying the differential inequality
| p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) | ε
(7)
for all x ( ρ 3 , ρ 3 ) and for some ε > 0 . Let α n 0 , α n 1 , , α n n 2 , β n , γ n be the complex numbers satisfying
c n = α n 0 a 0 + α n 1 a 1 + + α n n 2 a n 2 + β n c 0 + γ n c 1
(8)
for any integer n 2 . If there exists a constant C > 0 such that
| α n 0 a 0 + α n 1 a 1 + + α n n 2 a n 2 | C | a n |
(9)
for all integers n 2 , then there exists a solution y h : ( ρ 3 , ρ 3 ) C of the linear homogeneous differential equation (1) such that
| y ( x ) y h ( x ) | C K ε

for all x ( ρ 3 , ρ 3 ) , where K is the constant determined in (b).

Proof By the same argument presented in the proof of Theorem 2.1 with m = 0 b m x m instead of m = 0 c m x m , we have
p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = m = 0 a m x m
(10)
for all x ( ρ 3 , ρ 3 ) . In view of (b), there exists a constant K 0 such that
m = 0 | a m x m | K | m = 0 a m x m |
(11)

for all x ( ρ 1 , ρ 1 ) .

Moreover, by using (7), (10), and (11), we get
m = 0 | a m x m | K | m = 0 a m x m | K ε

for any x ( ρ 3 , ρ 3 ) . (That is, the radius of convergence of power series m = 0 a m x m is at least ρ 3 .)

According to Theorem 2.1 and (10), y ( x ) can be written as
y ( x ) = y h ( x ) + n = 0 c n x n
(12)

for all x ( ρ 3 , ρ 3 ) , where y h ( x ) is a solution of the homogeneous differential equation (1). In view of Lemma 3.1, the c n can be expressed by a linear combination of the form (8) for each integer n 2 .

Since n = 0 c n x n is a particular solution of (2), if we set c 0 = c 1 = 0 , then it follows from (8), (9), and (12) that
| y ( x ) y h ( x ) | n = 0 | c n x n | C K ε

for all x ( ρ 3 , ρ 3 ) . □

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

This research was completed with the support of The Scientific and Technological Research Council of Turkey while the first author was a visiting scholar at Istanbul Commerce University, Istanbul, Turkey.

Authors’ Affiliations

(1)
Mathematics Section, College of Science and Technology, Hongik University, Sejong, 339-701, Republic of Korea
(2)
Department of Mathematics, Faculty of Sciences and Arts, Istanbul Commerce University, Uskudar, Istanbul, 34672, Turkey

References

  1. Brillouet-Belluot N, Brzdȩk J, Cieplinski K: On some recent developments in Ulam’s type stability. Abstr. Appl. Anal. 2012., 2012: Article ID 716936Google Scholar
  2. Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge; 2002.View ArticleGoogle Scholar
  3. Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar
  4. Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhäuser, Boston; 1998.View ArticleGoogle Scholar
  5. Hyers DH, Rassias TM: Approximate homomorphisms. Aequ. Math. 1992, 44: 125–153. 10.1007/BF01830975MathSciNetView ArticleGoogle Scholar
  6. Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York; 2011.View ArticleGoogle Scholar
  7. Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1View ArticleGoogle Scholar
  8. Ulam SM: Problems in Modern Mathematics. Wiley, New York; 1964.Google Scholar
  9. Obłoza M: Hyers stability of the linear differential equation. Rocznik Nauk.-Dydakt. Prace Mat. 1993, 13: 259–270.Google Scholar
  10. Obłoza M: Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocznik Nauk.-Dydakt. Prace Mat. 1997, 14: 141–146.Google Scholar
  11. Alsina C, Ger R: On some inequalities and stability results related to the exponential function. J. Inequal. Appl. 1998, 2: 373–380.MathSciNetGoogle Scholar
  12. Takahasi S-E, Miura T, Miyajima S:On the Hyers-Ulam stability of the Banach space-valued differential equation y = λ y . Bull. Korean Math. Soc. 2002, 39: 309–315. 10.4134/BKMS.2002.39.2.309MathSciNetView ArticleGoogle Scholar
  13. Miura T: On the Hyers-Ulam stability of a differentiable map. Sci. Math. Jpn. 2002, 55: 17–24.MathSciNetGoogle Scholar
  14. Miura T, Jung S-M, Takahasi S-E:Hyers-Ulam-Rassias stability of the Banach space valued differential equations y = λ y . J. Korean Math. Soc. 2004, 41: 995–1005. 10.4134/JKMS.2004.41.6.995MathSciNetView ArticleGoogle Scholar
  15. Miura T, Miyajima S, Takahasi S-E: A characterization of Hyers-Ulam stability of first order linear differential operators. J. Math. Anal. Appl. 2003, 286: 136–146. 10.1016/S0022-247X(03)00458-XMathSciNetView ArticleGoogle Scholar
  16. Miura T, Miyajima S, Takahasi S-E: Hyers-Ulam stability of linear differential operator with constant coefficients. Math. Nachr. 2003, 258: 90–96. 10.1002/mana.200310088MathSciNetView ArticleGoogle Scholar
  17. Cimpean DS, Popa D: On the stability of the linear differential equation of higher order with constant coefficients. Appl. Math. Comput. 2010, 217(8):4141–4146. 10.1016/j.amc.2010.09.062MathSciNetView ArticleGoogle Scholar
  18. Cimpean DS, Popa D: Hyers-Ulam stability of Euler’s equation. Appl. Math. Lett. 2011, 24(9):1539–1543. 10.1016/j.aml.2011.03.042MathSciNetView ArticleGoogle Scholar
  19. Jung S-M: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 2004, 17: 1135–1140. 10.1016/j.aml.2003.11.004MathSciNetView ArticleGoogle Scholar
  20. Jung S-M: Hyers-Ulam stability of linear differential equations of first order, II. Appl. Math. Lett. 2006, 19: 854–858. 10.1016/j.aml.2005.11.004MathSciNetView ArticleGoogle Scholar
  21. Jung S-M: Hyers-Ulam stability of linear differential equations of first order, III. J. Math. Anal. Appl. 2005, 311: 139–146. 10.1016/j.jmaa.2005.02.025MathSciNetView ArticleGoogle Scholar
  22. Jung S-M: Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients. J. Math. Anal. Appl. 2006, 320: 549–561. 10.1016/j.jmaa.2005.07.032MathSciNetView ArticleGoogle Scholar
  23. Lungu N, Popa D: On the Hyers-Ulam stability of a first order partial differential equation. Carpath. J. Math. 2012, 28(1):77–82.MathSciNetGoogle Scholar
  24. Lungu N, Popa D: Hyers-Ulam stability of a first order partial differential equation. J. Math. Anal. Appl. 2012, 385(1):86–91. 10.1016/j.jmaa.2011.06.025MathSciNetView ArticleGoogle Scholar
  25. Popa D, Rasa I: The Frechet functional equation with application to the stability of certain operators. J. Approx. Theory 2012, 164(1):138–144. 10.1016/j.jat.2011.09.009MathSciNetView ArticleGoogle Scholar
  26. Jung S-M: Legendre’s differential equation and its Hyers-Ulam stability. Abstr. Appl. Anal. 2007., 2007: Article ID 56419. doi:10.1155/2007/56419Google Scholar
  27. Jung S-M: Approximation of analytic functions by Airy functions. Integral Transforms Spec. Funct. 2008, 19(12):885–891. 10.1080/10652460802321287MathSciNetView ArticleGoogle Scholar
  28. Jung S-M: Approximation of analytic functions by Hermite functions. Bull. Sci. Math. 2009, 133: 756–764. 10.1016/j.bulsci.2007.11.001MathSciNetView ArticleGoogle Scholar
  29. Jung S-M: Approximation of analytic functions by Legendre functions. Nonlinear Anal. 2009, 71(12):e103-e108. 10.1016/j.na.2008.10.007View ArticleGoogle Scholar
  30. Jung S-M:Hyers-Ulam stability of differential equation y + 2 x y 2 n y = 0 . J. Inequal. Appl. 2010., 2010: Article ID 793197. doi:10.1155/2010/793197Google Scholar
  31. Jung S-M: Approximation of analytic functions by Kummer functions. J. Inequal. Appl. 2010., 2010: Article ID 898274. doi:10.1155/2010/898274Google Scholar
  32. Jung S-M: Approximation of analytic functions by Laguerre functions. Appl. Math. Comput. 2011, 218(3):832–835. doi:10.1016/j.amc.2011.01.086 10.1016/j.amc.2011.01.086MathSciNetView ArticleGoogle Scholar
  33. Jung S-M, Rassias TM: Approximation of analytic functions by Chebyshev functions. Abstr. Appl. Anal. 2011., 2011: Article ID 432961. doi:10.1155/2011/432961Google Scholar
  34. Kim B, Jung S-M: Bessel’s differential equation and its Hyers-Ulam stability. J. Inequal. Appl. 2007., 2007: Article ID 21640. doi:10.1155/2007/21640Google Scholar
  35. Ross CC: Differential Equations - An Introduction with Mathematica. Springer, New York; 1995.Google Scholar
  36. Kreyszig E: Advanced Engineering Mathematics. 9th edition. Wiley, New York; 2006.Google Scholar

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