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# Power series method and approximate linear differential equations of second order

- Soon-Mo Jung
^{1}and - Hamdullah Şevli
^{2}Email author

**2013**:76

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

© Jung and Şevli; licensee Springer. 2013

**Received:**14 December 2012**Accepted:**4 March 2013**Published:**26 March 2013

## 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

*X*be a normed space over a scalar field $\mathbb{K}$, and let $I\subset \mathbb{R}$ be an open interval, where $\mathbb{K}$ denotes either ℝ or ℂ. Assume that ${a}_{0},{a}_{1},\dots ,{a}_{n}:I\to \mathbb{K}$ and $g:I\to X$ are given continuous functions. If for every

*n*times continuously differentiable function $y:I\to X$ satisfying the inequality

*n*times continuously differentiable solution ${y}_{0}:I\to X$ of the differential equation

such that $\parallel y(x)-{y}_{0}(x)\parallel \le K(\epsilon )$ for any $x\in I$, where $K(\epsilon )$ is an expression of *ε* with ${lim}_{\epsilon \to 0}K(\epsilon )=0$, 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 [1–8].

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 ${y}^{\prime}(x)=y(x)$. It was further proved by Takahasi *et al.* that the Hyers-Ulam stability holds for the Banach space valued differential equation ${y}^{\prime}(x)=\lambda y(x)$ (see [12] and also [13–15]).

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. [17–25]).

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 [26–34]). 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.

for all $x\in (-{\rho}_{0},{\rho}_{0})$. Since $x=0$ is an ordinary point of (1), we remark that ${p}_{0}\ne 0$.

## 2 Inhomogeneous differential equation

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*${\sum}_{m=0}^{\mathrm{\infty}}{a}_{m}{x}^{m}$

*is*${\rho}_{1}>0$

*and that there exists a sequence*$\{{c}_{m}\}$

*satisfying the recurrence relation*

*for any*$m\in {\mathbb{N}}_{0}$.

*Let*${\rho}_{2}$

*be the radius of convergence of power series*${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$

*and let*${\rho}_{3}=min\{{\rho}_{0},{\rho}_{1},{\rho}_{2}\}$,

*where*$(-{\rho}_{0},{\rho}_{0})$

*is the domain of the general solution to*(1).

*Then every solution*$y:(-{\rho}_{3},{\rho}_{3})\to \mathbb{C}$

*of the linear inhomogeneous differential equation*(2)

*can be expressed by*

*for all* $x\in (-{\rho}_{3},{\rho}_{3})$, *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 ${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$ for $y(x)$ in (2) and use the formal multiplication of power series and consider (3) to get

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*$d\ge 0$.

*In particular*,

*let*${d}_{0}$

*be the degree of*$p(x)$.

*Assume that the radius of convergence of power series*${\sum}_{m=0}^{\mathrm{\infty}}{a}_{m}{x}^{m}$

*is*${\rho}_{1}>0$

*and that there exists a sequence*$\{{c}_{m}\}$

*satisfying the recurrence formula*

*for any*$m\in {\mathbb{N}}_{0}$,

*where*${m}_{0}=max\{0,m-d\}$.

*If the sequence*$\{{c}_{m}\}$

*satisfies the following conditions*:

- (i)
${lim}_{m\to \mathrm{\infty}}{c}_{m-1}/m{c}_{m}=0$,

- (ii)
*there exists a complex number**L**such that*${lim}_{m\to \mathrm{\infty}}{c}_{m}/{c}_{m-1}=L$*and*${p}_{{d}_{0}}+L{p}_{{d}_{0}-1}+\cdots +{L}^{{d}_{0}-1}{p}_{1}+{L}^{{d}_{0}}{p}_{0}\ne 0$,

*then every solution*$y:(-{\rho}_{3},{\rho}_{3})\to \mathbb{C}$

*of the linear inhomogeneous differential equation*(2)

*can be expressed by*

*for all* $x\in (-{\rho}_{3},{\rho}_{3})$, *where* ${\rho}_{3}=min\{{\rho}_{0},{\rho}_{1}\}$ *and* ${y}_{h}(x)$ *is a solution of the linear homogeneous differential equation* (1).

*Proof*Let

*m*be any sufficiently large integer. Since ${p}_{d+1}={p}_{d+2}=\cdots =0$, ${q}_{d+1}={q}_{d+2}=\cdots =0$ and ${r}_{d+1}={r}_{d+2}=\cdots =0$, if we substitute $m-d+k$ for

*k*in (4), then we have

which implies that the radius of convergence of the power series ${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$ is ${\rho}_{1}$. The rest of this corollary immediately follows from Theorem 2.1. □

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

**Corollary 2.3**

*Let*${\rho}_{3}$

*be a distance between the origin*0

*and the closest one among singular points of*$q(z)$, $r(z)$,

*or*${\sum}_{m=0}^{\mathrm{\infty}}{a}_{m}{z}^{m}$

*in a complex variable*

*z*.

*If there exists a sequence*$\{{c}_{m}\}$

*satisfying the recurrence relation*

*for any*$m\in {\mathbb{N}}_{0}$,

*then every solution*$y:(-{\rho}_{3},{\rho}_{3})\to \mathbb{C}$

*of the linear inhomogeneous differential equation*

*can be expressed by*

*for all* $x\in (-{\rho}_{3},{\rho}_{3})$, *where* ${y}_{h}(x)$ *is a solution of the linear homogeneous differential equation* (1) *with* $p(x)\equiv 1$.

*Proof* If we put ${p}_{0}=1$ and ${p}_{i}=0$ for each $i\in \mathbb{N}$, then the recurrence relation (3) reduces to (5). As we did in the proof of Theorem 2.1, we can show that ${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$ 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 ${\rho}_{3}$. Moreover, since ${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$ 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 $p(x)\equiv 1$. Hence, the radius of convergence of ${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$ is at least ${\rho}_{3}$.

where ${y}_{h}(x)$ is a solution of the linear differential equation (1) with $p(x)\equiv 1$. □

## 3 Approximate differential equation

- (a)
$y(x)$ is expressible by a power series ${\sum}_{m=0}^{\mathrm{\infty}}{b}_{m}{x}^{m}$ whose radius of convergence is at least ${\rho}_{1}$;

- (b)There exists a constant $K\ge 0$ such that ${\sum}_{m=0}^{\mathrm{\infty}}|{a}_{m}{x}^{m}|\le K|{\sum}_{m=0}^{\mathrm{\infty}}{a}_{m}{x}^{m}|$ for any $x\in (-{\rho}_{1},{\rho}_{1})$, where${a}_{m}=\sum _{k=0}^{m}[(k+2)(k+1){b}_{k+2}{p}_{m-k}+(k+1){b}_{k+1}{q}_{m-k}+{b}_{k}{r}_{m-k}]$

for all $m\in {\mathbb{N}}_{0}$ and ${p}_{0}\ne 0$.

**Lemma 3.1** *Given a sequence* $\{{a}_{m}\}$, *let* $\{{c}_{m}\}$ *be a sequence satisfying the recurrence formula* (3) *for all* $m\in {\mathbb{N}}_{0}$. *If* ${p}_{0}\ne 0$ *and* $n\ge 2$, *then* ${c}_{n}$ *is a linear combination of* ${a}_{0},{a}_{1},\dots ,{a}_{n-2}$, ${c}_{0}$, *and* ${c}_{1}$.

*Proof*We apply induction on

*n*. Since ${p}_{0}\ne 0$, if we set $m=0$ in (3), then

*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 ${a}_{0},\dots ,{a}_{i-2}$, ${c}_{0}$, ${c}_{1}$ for all $i\in \{2,3,\dots ,n\}$, namely,

*m*in (3) with $n-1$, then

where ${\alpha}_{n+1}^{0},\dots ,{\alpha}_{n+1}^{n-1}$, ${\beta}_{n+1}$, ${\gamma}_{n+1}$ are complex numbers. That is, ${c}_{n+1}$ is a linear combination of ${a}_{0},{a}_{1},\dots ,{a}_{n-1}$, ${c}_{0}$, ${c}_{1}$, 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 ${p}_{0}\ne 0$.

**Theorem 3.2**

*Let*$\{{c}_{m}\}$

*be a sequence of complex numbers satisfying the recurrence relation*(3)

*for all*$m\in {\mathbb{N}}_{0}$,

*where*(b)

*is referred for the value of*${a}_{m}$,

*and let*${\rho}_{2}$

*be the radius of convergence of the power series*${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$.

*Define*${\rho}_{3}=min\{{\rho}_{0},{\rho}_{1},{\rho}_{2}\}$,

*where*$(-{\rho}_{0},{\rho}_{0})$

*is the domain of the general solution to*(1).

*Assume that*$y:(-{\rho}_{1},{\rho}_{1})\to \mathbb{C}$

*is an arbitrary function belonging to*$\mathcal{C}$

*and satisfying the differential inequality*

*for all*$x\in (-{\rho}_{3},{\rho}_{3})$

*and for some*$\epsilon >0$.

*Let*${\alpha}_{n}^{0},{\alpha}_{n}^{1},\dots ,{\alpha}_{n}^{n-2}$, ${\beta}_{n}$, ${\gamma}_{n}$

*be the complex numbers satisfying*

*for any integer*$n\ge 2$.

*If there exists a constant*$C>0$

*such that*

*for all integers*$n\ge 2$,

*then there exists a solution*${y}_{h}:(-{\rho}_{3},{\rho}_{3})\to \mathbb{C}$

*of the linear homogeneous differential equation*(1)

*such that*

*for all* $x\in (-{\rho}_{3},{\rho}_{3})$, *where* *K* *is the constant determined in* (b).

*Proof*By the same argument presented in the proof of Theorem 2.1 with ${\sum}_{m=0}^{\mathrm{\infty}}{b}_{m}{x}^{m}$ instead of ${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$, we have

for all $x\in (-{\rho}_{1},{\rho}_{1})$.

for any $x\in (-{\rho}_{3},{\rho}_{3})$. (That is, the radius of convergence of power series ${\sum}_{m=0}^{\mathrm{\infty}}{a}_{m}{x}^{m}$ is at least ${\rho}_{3}$.)

for all $x\in (-{\rho}_{3},{\rho}_{3})$, 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 $n\ge 2$.

for all $x\in (-{\rho}_{3},{\rho}_{3})$. □

## 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

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