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Power series method and approximate linear differential equations of second order

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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.

1 Introduction

Let X be a normed space over a scalar field K, and let IR be an open interval, where K denotes either or . Assume that a 0 , a 1 ,, a n :IK and g:IX are given continuous functions. If for every n times continuously differentiable function y:IX 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 xI and for a given ε>0, there exists an n times continuously differentiable solution y 0 :IX 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 xI, 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)=λy(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 andr(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 d0. 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,md}. 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 md+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 iN, 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 K0 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 n2, 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 n1, 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 n2. 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 n2, 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 ) | CKε

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 K0 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 |Kε

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 n2.

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 | CKε

for all x( ρ 3 , ρ 3 ). □

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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.

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Correspondence to Hamdullah Şevli.

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Keywords

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