Power series method and approximate linear differential equations of second order

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 $\mathbb{K}$, 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

$$\parallel a_n(x)y^{(n)}(x)+a_{n-1}(x)y^{(n-1)}(x)+\cdots +a_1(x)y^{\prime }(x)+a_0(x)y(x)+g(x)\parallel \le \epsilon$$

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

$$a_n(x)y^{(n)}(x)+a_{n-1}(x)y^{(n-1)}(x)+\cdots +a_1(x)y^{\prime }(x)+a_0(x)y(x)+g(x)=0$$

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.

Throughout this paper, we assume that the linear differential equation of second order of the form

$$p(x)y^{″}(x)+q(x)y^{\prime }(x)+r(x)y(x)=0,$$

(1)

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

$$p(x)=\sum _{m=0}^{\mathrm{\infty }}{p}_m{x}^m,q(x)=\sum _{m=0}^{\mathrm{\infty }}{q}_m{x}^m\text{and}r(x)=\sum _{m=0}^{\mathrm{\infty }}{r}_m{x}^m$$

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

$$p(x)y^{″}(x)+q(x)y^{\prime }(x)+r(x)y(x)=\sum _{m=0}^{\mathrm{\infty }}{a}_m{x}^m$$

(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 ${\sum }_{m=0}^{\mathrm{\infty }}{a}_m{x}^m$ is ${\rho }_1>0$ and that there exists a sequence $\{c_m\}$ satisfying the recurrence relation

$$\sum _{k=0}^m[(k+2)(k+1)c_{k+2}{p}_{m-k}+(k+1)c_{k+1}{q}_{m-k}+c_k{r}_{m-k}]=a_m$$

(3)

for any $m\in {\mathbb{N}}_0$. 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

$$y(x)=y_h(x)+\sum _{m=0}^{\mathrm{\infty }}{c}_m{x}^m$$

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

$$\begin{array}{c}p(x)y^{″}(x)+q(x)y^{\prime }(x)+r(x)y(x)\hfill \\ =\sum _{m=0}^{\mathrm{\infty }}\sum _{k=0}^m{p}_{m-k}(k+2)(k+1)c_{k+2}{x}^m+\sum _{m=0}^{\mathrm{\infty }}\sum _{k=0}^m{q}_{m-k}(k+1)c_{k+1}{x}^m\hfill \\ +\sum _{m=0}^{\mathrm{\infty }}\sum _{k=0}^m{r}_{m-k}{c}_k{x}^m\hfill \\ =\sum _{m=0}^{\mathrm{\infty }}\sum _{k=0}^m[(k+2)(k+1)c_{k+2}{p}_{m-k}+(k+1)c_{k+1}{q}_{m-k}+c_k{r}_{m-k}]x^m\hfill \\ =\sum _{m=0}^{\mathrm{\infty }}{a}_m{x}^m\hfill \end{array}$$

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

$$y(x)=y_h(x)+\sum _{m=0}^{\mathrm{\infty }}{c}_m{x}^m,$$

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 $d\ge 0$. 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

$$\sum _{k=m_0}^m[(k+2)(k+1)c_{k+2}{p}_{m-k}+(k+1)c_{k+1}{q}_{m-k}+c_k{r}_{m-k}]=a_m$$

(4)

for any $m\in {\mathbb{N}}_0$, 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

$$y(x)=y_h(x)+\sum _{m=0}^{\mathrm{\infty }}{c}_m{x}^m$$

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

$$\begin{array}{rcl}{a}_m& =& \sum _{k=0}^d[(m-d+k+2)(m-d+k+1)c_{m-d+k+2}{p}_{d-k}\\ +(m-d+k+1)c_{m-d+k+1}{q}_{d-k}+c_{m-d+k}{r}_{d-k}].\end{array}$$

By (i) and (ii), we have

$$\begin{array}{c}\underset{m\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{|a_m|}^{1/m}\hfill \\ =\underset{m\to \mathrm{\infty }}{lim\hspace{0.17em}sup}|\sum _{k=0}^d(m-d+k+2)(m-d+k+1)c_{m-d+k+2}\hfill \\ \times (p_{d-k}+\frac{q_{d-k}}{(m-d+k+2)}\frac{c_{m-d+k+1}}{c_{m-d+k+2}}\hfill \\ +{\frac{r_{d-k}}{(m-d+k+2)(m-d+k+1)}\frac{c_{m-d+k}}{c_{m-d+k+1}}\frac{c_{m-d+k+1}}{c_{m-d+k+2}}\left)\right|}^{1/m}\hfill \\ =\underset{m\to \mathrm{\infty }}{lim\hspace{0.17em}sup}|\sum _{k=0}^d(m-d+k+2)(m-d+k+1)c_{m-d+k+2}{p}_{d-k}{|}^{1/m}\hfill \\ =\underset{m\to \mathrm{\infty }}{lim\hspace{0.17em}sup}|\sum _{k=d-d_0}^d(m-d+k+2)(m-d+k+1)c_{m-d+k+2}{p}_{d-k}{|}^{1/m}\hfill \\ =\underset{m\to \mathrm{\infty }}{lim\hspace{0.17em}sup}|(m-d_0+2)(m-d_0+1)c_{m-d_0+2}(p_{d_0}+L{p}_{d_0-1}+\cdots +L^{d_0}{p}_0){|}^{1/m}\hfill \\ =\underset{m\to \mathrm{\infty }}{lim\hspace{0.17em}sup}|(p_{d_0}+L{p}_{d_0-1}+\cdots +L^{d_0}{p}_0)(m-d_0+2)(m-d_0+1){|}^{1/m}\hfill \\ \times {\left({|c_{m-d_0+2}|}^{1/(m-d_0+2)}\right)}^{(m-d_0+2)/m}\hfill \\ =\underset{m\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{|c_{m-d_0+2}|}^{1/(m-d_0+2)},\hfill \end{array}$$

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

$$(m+2)(m+1)c_{m+2}+\sum _{k=0}^m[(k+1)c_{k+1}{q}_{m-k}+c_k{r}_{m-k}]=a_m$$

(5)

for any $m\in {\mathbb{N}}_0$, then every solution $y:(-{\rho }_3,{\rho }_3)\to \mathbb{C}$ of the linear inhomogeneous differential equation

$$y^{″}(x)+q(x)y^{\prime }(x)+r(x)y(x)=\sum _{m=0}^{\mathrm{\infty }}{a}_m{x}^m$$

(6)

can be expressed by

$$y(x)=y_h(x)+\sum _{m=0}^{\mathrm{\infty }}{c}_m{x}^m$$

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

$$y(x)=y_h(x)+\sum _{m=0}^{\mathrm{\infty }}{c}_m{x}^m,$$

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

  $$a_m=\sum _{k=0}^m[(k+2)(k+1)b_{k+2}{p}_{m-k}+(k+1)b_{k+1}{q}_{m-k}+b_k{r}_{m-k}]$$

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

$$c_2=\frac{1}{2p_0}{a}_0-\frac{r_0}{2p_0}{c}_0-\frac{q_0}{2p_0}{c}_1,$$

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 $a_0,\dots ,a_{i-2}$, $c_0$, $c_1$ for all $i\in \{2,3,\dots ,n\}$, namely,

$$c_i={\alpha }_i^0{a}_0+{\alpha }_i^1{a}_1+\cdots +{\alpha }_i^{i-2}{a}_{i-2}+{\beta }_i{c}_0+{\gamma }_i{c}_1,$$

where ${\alpha }_i^0,\dots ,{\alpha }_i^{i-2}$, ${\beta }_i$, ${\gamma }_i$ are complex numbers. If we replace m in (3) with $n-1$, then

$$\begin{array}{rcl}{a}_{n-1}& =& 2c_2{p}_{n-1}+c_1{q}_{n-1}+c_0{r}_{n-1}\\ +6c_3{p}_{n-2}+2c_2{q}_{n-2}+c_1{r}_{n-2}\\ +\cdots \\ +n(n-1)c_n{p}_1+(n-1)c_{n-1}{q}_1+c_{n-2}{r}_1\\ +(n+1)n{c}_{n+1}{p}_0+n{c}_n{q}_0+c_{n-1}{r}_0\\ =& (n+1)n{p}_0{c}_{n+1}+[n(n-1)p_1+n{q}_0]c_n+\cdots \\ +(2p_{n-1}+2q_{n-2}+r_{n-3})c_2+(q_{n-1}+r_{n-2})c_1+r_{n-1}{c}_0,\end{array}$$

which implies

$$\begin{array}{rcl}{c}_{n+1}& =& \frac{1}{(n+1)n{p}_0}{a}_{n-1}-\frac{n(n-1)p_1+n{q}_0}{(n+1)n{p}_0}{c}_n-\cdots \\ -\frac{2p_{n-1}+2q_{n-2}+r_{n-3}}{(n+1)n{p}_0}{c}_2-\frac{q_{n-1}+r_{n-2}}{(n+1)n{p}_0}{c}_1-\frac{r_{n-1}}{(n+1)n{p}_0}{c}_0\\ =& {\alpha }_{n+1}^0{a}_0+{\alpha }_{n+1}^1{a}_1+\cdots +{\alpha }_{n+1}^{n-1}{a}_{n-1}+{\beta }_{n+1}{c}_0+{\gamma }_{n+1}{c}_1,\end{array}$$

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

$$|p(x)y^{″}(x)+q(x)y^{\prime }(x)+r(x)y(x)|\le \epsilon$$

(7)

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

$$c_n={\alpha }_n^0{a}_0+{\alpha }_n^1{a}_1+\cdots +{\alpha }_n^{n-2}{a}_{n-2}+{\beta }_n{c}_0+{\gamma }_n{c}_1$$

(8)

for any integer $n\ge 2$. If there exists a constant $C>0$ such that

$$|{\alpha }_n^0{a}_0+{\alpha }_n^1{a}_1+\cdots +{\alpha }_n^{n-2}{a}_{n-2}|\le C|a_n|$$

(9)

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

$$|y(x)-y_h(x)|\le CK\epsilon$$

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

$$p(x)y^{″}(x)+q(x)y^{\prime }(x)+r(x)y(x)=\sum _{m=0}^{\mathrm{\infty }}{a}_m{x}^m$$

(10)

for all $x\in (-{\rho }_3,{\rho }_3)$. In view of (b), there exists a constant $K\ge 0$ such that

$$\sum _{m=0}^{\mathrm{\infty }}\left|a_m{x}^m\right|\le K\left|\sum _{m=0}^{\mathrm{\infty }}{a}_m{x}^m\right|$$

(11)

for all $x\in (-{\rho }_1,{\rho }_1)$.

Moreover, by using (7), (10), and (11), we get

$$\sum _{m=0}^{\mathrm{\infty }}\left|a_m{x}^m\right|\le K\left|\sum _{m=0}^{\mathrm{\infty }}{a}_m{x}^m\right|\le K\epsilon$$

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

$$y(x)=y_h(x)+\sum _{n=0}^{\mathrm{\infty }}{c}_n{x}^n$$

(12)

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

$$|y(x)-y_h(x)|\le \sum _{n=0}^{\mathrm{\infty }}\left|c_n{x}^n\right|\le CK\epsilon$$

for all $x\in (-{\rho }_3,{\rho }_3)$. □