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Power series method and approximate linear differential equations of second order
Advances in Difference Equations volume 2013, Article number: 76 (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 HyersUlam stability.
MSC:34A05, 39B82, 26D10, 34A40.
1 Introduction
Let 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
for all $x\in I$ and for a given $\epsilon >0$, there exists an 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 HyersUlam stability. For more detailed definitions of the HyersUlam stability, we refer the reader to [1–8].
Obłoza seems to be the first author who has investigated the HyersUlam stability of linear differential equations (see [9, 10]). Thereafter, Alsina and Ger [11] proved the HyersUlam stability of the differential equation ${y}^{\prime}(x)=y(x)$. It was further proved by Takahasi et al. that the HyersUlam 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 HyersUlam stability of an n thorder linear differential equation. The first author also proved the HyersUlam stability of various linear differential equations of first order (ref. [17–25]).
Recently, the first author applied the power series method to studying the HyersUlam 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 HyersUlam 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 HyersUlam stability.
Throughout this paper, we assume that the linear differential equation of second order of the form
for which $x=0$ is an ordinary point, has the general solution ${y}_{h}:({\rho}_{0},{\rho}_{0})\to \mathbb{C}$, where ${\rho}_{0}$ is a constant with $0<{\rho}_{0}\le \mathrm{\infty}$ and the coefficients $p,q,r:({\rho}_{0},{\rho}_{0})\to \mathbb{C}$ are analytic at 0 and have power series expansions
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
In the following theorem, we solve the linear inhomogeneous differential equation of second order of the form
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
for all $x\in ({\rho}_{3},{\rho}_{3})$. That is, ${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$ is a particular solution of the linear inhomogeneous differential equation (2), and hence every solution $y:({\rho}_{3},{\rho}_{3})\to \mathbb{C}$ of (2) can be expressed by
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 ${m}_{0}=max\{0,md\}$. If the sequence $\{{c}_{m}\}$ satisfies the following conditions:

(i)
${lim}_{m\to \mathrm{\infty}}{c}_{m1}/m{c}_{m}=0$,

(ii)
there exists a complex number L such that ${lim}_{m\to \mathrm{\infty}}{c}_{m}/{c}_{m1}=L$ 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 $md+k$ for k in (4), then we have
By (i) and (ii), 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 ${\rho}_{3}$.
Now, every solution $y:({\rho}_{3},{\rho}_{3})\to \mathbb{C}$ of (6) can be expressed by
where ${y}_{h}(x)$ is a solution of the linear differential equation (1) with $p(x)\equiv 1$. □
3 Approximate differential equation
In this section, let ${\rho}_{1}>0$ be a constant. We denote by $\mathcal{C}$ the set of all functions $y:({\rho}_{1},{\rho}_{1})\to \mathbb{C}$ with the following properties:

(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
$${a}_{m}=\sum _{k=0}^{m}[(k+2)(k+1){b}_{k+2}{p}_{mk}+(k+1){b}_{k+1}{q}_{mk}+{b}_{k}{r}_{mk}]$$
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 ${a}_{0},{a}_{1},\dots ,{a}_{n2}$, ${c}_{0}$, 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 ${a}_{0},\dots ,{a}_{i2}$, ${c}_{0}$, ${c}_{1}$ for all $i\in \{2,3,\dots ,n\}$, namely,
where ${\alpha}_{i}^{0},\dots ,{\alpha}_{i}^{i2}$, ${\beta}_{i}$, ${\gamma}_{i}$ are complex numbers. If we replace m in (3) with $n1$, then
which implies
where ${\alpha}_{n+1}^{0},\dots ,{\alpha}_{n+1}^{n1}$, ${\beta}_{n+1}$, ${\gamma}_{n+1}$ are complex numbers. That is, ${c}_{n+1}$ is a linear combination of ${a}_{0},{a}_{1},\dots ,{a}_{n1}$, ${c}_{0}$, ${c}_{1}$, which ends the proof. □
In the following theorem, we investigate a kind of HyersUlam 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 ${\alpha}_{n}^{0},{\alpha}_{n}^{1},\dots ,{\alpha}_{n}^{n2}$, ${\beta}_{n}$, ${\gamma}_{n}$ 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 $x\in ({\rho}_{3},{\rho}_{3})$. In view of (b), there exists a constant $K\ge 0$ such that
for all $x\in ({\rho}_{1},{\rho}_{1})$.
Moreover, by using (7), (10), and (11), we get
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 ${\rho}_{3}$.)
According to Theorem 2.1 and (10), $y(x)$ can be written as
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 $n\ge 2$.
Since ${\sum}_{n=0}^{\mathrm{\infty}}{c}_{n}{x}^{n}$ is a particular solution of (2), if we set ${c}_{0}={c}_{1}=0$, then it follows from (8), (9), and (12) that
for all $x\in ({\rho}_{3},{\rho}_{3})$. □
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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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Keywords
 power series method
 approximate linear differential equation
 simple harmonic oscillator equation
 HyersUlam stability
 approximation