## 1. Introduction

Stimulated by Yosida’s work [

1,

2], in which the solution of Laplace’s differential equation (DE) is obtained with the aid of the operational calculus of Mikusiński [

3], we are concerned in [

4,

5,

6], with the DE or fractional DE of the form:

where

$\sigma =\frac{1}{2}$ or

$\sigma =1$,

$m=2$, and

${a}_{l}\in \mathbb{C}$ and

${b}_{l}\in \mathbb{C}$ are constants.

In solving the DE, we assume that the solution

$u(t)$ and the inhomogeneous part

$f(t)$ for

$t>0$ are expressed as a linear combination of

for

$\nu \in \mathbb{C}\setminus {\mathbb{Z}}_{<1}$, where

$\Gamma (\nu )$ is the gamma function.

In [

5,

6],

${}_{0}{D}_{R}^{\beta}u(t)$ is the analytic continuation (AC) of Riemann–Liouville fractional derivative (fD), which was introduced in [

7,

8] and is reviewed in [

9]. It is defined for

$u(t)$ and

$f(t)$ satisfying the following conditions.

**Condition A.** $u(t)H(t)$ and $f(t)H(t)$ are expressed as a linear combination of ${g}_{\nu}(t)H(t)$ for $\nu \in S$, where S is an enumerable set of $\nu \in \mathbb{C}\setminus {\mathbb{Z}}_{<1}$ satisfying $\mathrm{Re}\phantom{\rule{0.277778em}{0ex}}\nu >-M$ for some $M\in {\mathbb{Z}}_{>-1}$.

We now adopt Condition A. We then express

$u(t)$ as follows:

where

${u}_{\nu -1}\in \mathbb{C}$ are constants.

For

$\nu \in \mathbb{C}\setminus {\mathbb{Z}}_{<1}$,

${}_{0}{D}_{R}^{\beta}{g}_{\nu}(t)$ is defined such that

When

$\beta =n\in {\mathbb{Z}}_{>-1}$,

${}_{0}{D}_{R}^{n}{g}_{\nu}(t)=\frac{{d}^{n}}{d{t}^{n}}{g}_{\nu}(t)$. Throughout the present paper, the equations involving

β are valid for

$\beta \in \mathbb{C}$, but in the applications given in

Section 4,

Section 5,

Section 6 and

Section 7, we use them only for

$\beta =n\in {\mathbb{Z}}_{>-1}$, when

${}_{0}{D}_{R}^{n}{g}_{\nu}(t)=\frac{{d}^{n}}{d{t}^{n}}{g}_{\nu}(t)$.

We use $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{Z}$ to denote the sets of all real numbers, of all complex numbers and of all integers, respectively. We also use ${\mathbb{R}}_{>r}:=\{x\in \mathbb{R}|x>r\}$ for $r\in \mathbb{R}$, ${}_{+}\mathbb{C}:=\{z\in \mathbb{C}|\mathrm{Re}\phantom{\rule{0.277778em}{0ex}}z>0\}$, ${\mathbb{Z}}_{>a}:=\{n\in \mathbb{Z}|n>a\}$, ${\mathbb{Z}}_{<b}:=\{n\in \mathbb{Z}|n<b\}$ and ${\mathbb{Z}}_{[a,b]}:=\{n\in \mathbb{Z}|a\le n\le b\}$ for $a,b\in \mathbb{Z}$ satisfying $a<b$. We use Heaviside’s step function $H(t)$, which is defined such that (i) $H(t)=1$ for $t>0$ and $=0$ for $t\le 0$, and (ii) when $f(t)$ is defined on ${\mathbb{R}}_{>r}$, $f(t)H(t-r)$ is equal to $f(t)$ when $t>r$ and to 0 when $t\le r$.

In [

4,

5,

6], we take up a modified Kummer’s DE as an example, which is

where

$a\in \mathbb{C}$,

$b\in \mathbb{C}$ and

$c\in \mathbb{C}$ are constants. Kummer’s DE is this DE with

$b=1$ [

10,

11]. If

$c\notin \mathbb{Z}$, the basic solutions of Equation (

5) are given by

Here

${}_{1}{F}_{1}(a;c;z)={\sum}_{k=0}^{\infty}\frac{{(a)}_{k}}{k!{(c)}_{k}}{z}^{k}$ is the confluent hypergeometric series, where

$z\in \mathbb{C}$,

${(a)}_{n}={\prod}_{k=0}^{n-1}(a+k)$ for

$a\in \mathbb{C}$ and

$n\in {\mathbb{Z}}_{>0}$, and

${(a)}_{0}=1$. These solutions are expressed as linear combinations of

${g}_{\nu}(t)$.

**Remark 1.** In [

4,

5,

6],

a,

b and

c in Equation (

5) are expressed as

${\gamma}_{1}+1$,

$-\alpha $ and

${\gamma}_{1}+{\gamma}_{2}+2$, respectively.

In [

4,

5], we consider the theory of distributions in the space

${\mathcal{D}}_{R}^{\prime}$, which is presented in [

12,

13] and is explained briefly in

Section 3.3. The solution of Equation (

1) with the aid of distributions in

${\mathcal{D}}_{R}^{\prime}$ is presented in [

5], assuming that the solution satisfies Condition A. Both of the solutions given by Equations (

6) and (

7), of Equation (

5), satisfy Condition A, and we can obtain both of them, by solving Equation (

5) by this method.

In [

4], we adopt the following condition.

**Condition B.** $u(t)H(t)$ and $f(t)H(t)$ are expressed as a linear combination of ${g}_{\nu}(t)H(t)$ for $\nu \in {S}_{1}$, where ${S}_{1}$ is a set of $\nu \in {}_{+}\mathbb{C}$.

When Condition B is satisfied,

${}_{0}{D}_{R}^{\beta}u(t)$ of a function

$u(t)$ denotes the Riemann–Liouville fD which is defined when

$u(t)H(t)$ is locally integrable on

$\mathbb{R}$, and hence

${}_{0}{D}_{R}^{\beta}{g}_{\nu}(t)$ is defined only for

$\nu \in {}_{+}\mathbb{C}$, satisfying Equation (

4). In this case, the DE given by Equation (

1) in terms of the distribution theory in space

${\mathcal{D}}_{R}^{\prime}$ is presented in [

4]. The solutions of fractional DE with constant coefficients are presented in [

12,

13]. In [

4], the solution given by Equation (

6) of Equation (

5), satisfies Condition B, and hence we can obtain it by solving Equation (

5) by this method. However, the solution given by Equation (

7) satisfies Condition B only when

$1-c>-1$, and hence we can obtain it only when

$1-c>-1$, by this method.

**Condition C.** There exists $\lambda \in {\mathbb{R}}_{>0}$ such that $u(t){e}^{-\lambda t}\to 0$ as $t\to \infty $.

In [

6], it was mentioned that, when Conditions B and C are satisfied, the Laplace transform of

$u(t)$ exists and the DE is solved with the aid of Laplace transform, and that the solutions of Equation (

5), satisfying Condition B, satisfy Condition C and hence are obtained by using the Laplace transform.

In [

6], the AC of Laplace (AC-Laplace) transform is introduced as in

Section 1.1 given below, and it is shown that, when Conditions A and C are satisfied, the AC-Laplace transform of

$u(t)$, which is denoted by

$\widehat{u}(s)={\mathcal{L}}_{H}[u(t)]={\mathcal{L}}_{H}[u(t)](s)$, exists and the DE given by Equation (

1) is solved with the aid of the AC-Laplace transform. In fact, the AC-Laplace transform of

${g}_{\nu}(t)$ and of

$u(t)$ given by Equation (

3) are expressed as

We review the solution in terms of the AC-Laplace transform in

Section 2, and the solution with the aid of the distribution theory in

Section 3. In

Section 4 we confirm the following lemma.

**Lemma 1.** Both of the solutions given by Equations (

6) and (

7), of Equation (

5), satisfy Conditions A and C, and hence we can obtain both of them, by solving Equation (

5) by using distribution theory in the space

${\mathcal{D}}_{R}^{\prime}$ and also by using the AC-Laplace transform.

In

Section 5, we consider the hypergeometric DE, which is given by

where

$a\in \mathbb{C}$,

$b\in \mathbb{C}$ and

$c\in \mathbb{C}$ are constants.

If

$c\notin \mathbb{Z}$, the basic solutions of Equation (

10) in [

10,

11] are given by

where

${}_{2}{F}_{1}(a,b;c;z)={\sum}_{n=0}^{\infty}\frac{{(a)}_{n}{(b)}_{n}}{n!{(c)}_{n}}{z}^{n}$ of

$z\in \mathbb{C}$ is the hypergeometric series.

**Remark 2.** These solutions of Equation (

10) converge only at

t satisfying

$\left|t\right|<1$, and do not satisfy Condition A, and they are not obtained by the methods stated above.

We introduce the theory of distributions in the space

${\mathcal{D}}_{r,R}^{\prime}$, in

Section 3.4. We now use the step function

${H}_{r}(t)$, which is defined for

$r\in {\mathbb{R}}_{>0}$ such that (i)

${H}_{r}(t)=1$ for

$0<t<r$ and

$=0$ for

$t\le 0$ or

$t\ge r$, and (ii) when

$f(t)$ is defined on

${\mathbb{R}}_{>0}\cap {\mathbb{R}}_{<r}$,

$f(t){H}_{r}(t)$ is equal to

$f(t)$ for

$0<t<r$ and to 0 for

$t\le 0$ or

$t\ge r$.

**Condition D.** Condition A with $H(t)$ replaced by ${H}_{r}(t)$ is valid.

In

Section 5, we show that when

$u(t)$ satisfies Condition D, we can solve Equation (

10) with the aid of distributions in

${\mathcal{D}}_{r,R}^{\prime}$.

**Definition 1.** Let

$u(t)$ be given by Equation (

3) and satisfy Condition D. Then, we define its AC of Riemann–Liouville fD of order

$\beta \in \mathbb{C}$, by

which satisfies Condition D.

**Definition 2.** Let

$u(t)$ be given by Equation (

3) and satisfy Condition D. Then, we define

$\widehat{u}(s)={\mathcal{L}}_{S}[u(t)]={\mathcal{L}}_{S}[u(t)](s)$ by

We call this the AC-Laplace transform series of

$u(t)$.

For the solutions of Equation (

10), the existence of the AC-Laplace transform is not guaranteed, but we can define

$\widehat{u}(s)$ by Equation (

14). We can then set up a DE satisfied by the thus-defined

$\widehat{u}(s)$. In

Section 5, we write the DE for the

$\widehat{u}(s)$, and its solution in the form of Equation (

14) is obtained. We find that the obtained series converges for no value of

s, and yet we obtain the solution

$u(t)$ by the term-by-term inverse Laplace transform of the series

$\widehat{u}(s)$.

The solutions given by Equations (

11) and (

12), of Equation (

10), satisfy Condition D for

$r=1$, and we show that they are obtained by using the distribution theory in the space

${\mathcal{D}}_{r,R}^{\prime}$ and also by using the AC of Laplace transform series, in

Section 5.

In

Section 6, we show that the Bessel functions

${J}_{\pm \nu}(t)$ are the solusions of Bessel’s DE with the aid of the AC-Laplace transform. In

Section 7, some discussions are given on Hermite’s DE. Concluding remarks are given in

Section 8.

#### 1.1. Definition of the AC-Laplace Transform

The AC-Laplace transform

$\widehat{f}(s)={\mathcal{L}}_{H}[f(t)]$ of a function

$f(t)$ is defined in [

6] as follows.

**Condition E.** ${f}_{\gamma}(z)$ is expressed as ${f}_{\gamma}(z)={z}^{\gamma -1}{f}_{1}(z)$ on a neighborhood of ${\mathbb{R}}_{>0}$, for $0\le argz<2\pi $, where $\gamma \in \mathbb{C}\setminus {\mathbb{Z}}_{<1}$, and ${f}_{1}(z)$ is analytic on the neighborhood of ${\mathbb{R}}_{>0}$.

**Definition 3.** Let

${f}_{\gamma}(z)$ and

$u(t)={f}_{\gamma}(t)$ satisfy Conditions E and C, respectively. Then, we define the AC-Laplace transform

${\widehat{f}}_{\gamma}(s)$ for

$\gamma \in \mathbb{C}\setminus {\mathbb{Z}}_{<1}$, by

${\widehat{f}}_{\gamma}(s)={\mathcal{L}}_{H}[{f}_{\gamma}(t)]$, where

When

$\gamma =n\in {\mathbb{Z}}_{>0}$, we put

${\mathcal{L}}_{H}[{f}_{n}(t)]={lim}_{{\gamma}_{i}\to n}{\mathcal{L}}_{H}[{t}^{{\gamma}_{i}-1}\xb7{f}_{1}(t)]$, where

${\gamma}_{i}\in \mathbb{C}\setminus \mathbb{Z}$. Here,

${C}_{H}$ is the contour which appears in Hankel’s formula giving the AC of the gamma function

$\Gamma (z)$, so that

${C}_{H}$ is the contour which starts from

$\infty +i\u03f5$, goes to

$\delta +i\u03f5$, encircles the origin counterclockwise, goes to

$\delta -i\u03f5$, and then to

$\infty -i\u03f5$, where

$\delta \in {\mathbb{R}}_{>0}$ and

$\u03f5\in {\mathbb{R}}_{>0}$ satisfy

$\u03f5\leqq 1$ and

$\delta <1$, see [

14] (Section 12.22).

**Remark 3.** ${\widehat{f}}_{\gamma}(s)$ defined by Definition 3 is an analytic continuation of the Laplace transform defined by ${\widehat{f}}_{\gamma}(s)={\int}_{0}^{\infty}{f}_{\gamma}(t){e}^{-st}dt$ for $\mathrm{Re}\phantom{\rule{0.277778em}{0ex}}\gamma >0$, as a function of γ.

#### 1.2. Remarks on Recent Developments

Here, we call attention to recent developments on the solutions of differential equations related with fractional calculus and perturbation method, which are based on He’s variational iteration method (VIM) [

15]. By using the VIM, He gave the fD and fI which involve the terms determined by the initial or boundary condition. Liu

et al. [

16] discussed the solution of heat conduction in a fractal medium with the aid of He’s fD. Kumar

et al. discussed the solution of partial differential equations involving time-fD by using Laplace transform and perturbation method based on the VIM; see [

17,

18] and references in them. In [

19,

20], discussions are given on the fractional complex transform, which reduces an equation involving fD to an equation involving only integer-order derivatives.

## 8. Conclusions

In the present paper, we are concerned with the problem of obtaining a solution $u(t)$ of a DE with polynomial coefficients.

We know that by the basic method of solution [

14] (Section 10.4), we usually obtain solutions in the form of Equation (

54) which are a power of

t multiplied by a power series in

t. In the present paper, we are interested in obtaining the solutions with the aid of AC of Riemann–Liouville fD along with distribution theory or the Laplace transform or its AC.

We then set up the DE satisfied by the AC-Laplace transform, $\widehat{u}(s)$, of $u(t)$. The obtained DE for $\widehat{u}(s)$ is found to be a DE with polynomial coefficients.

We now obtain the solution in the form of Equation (

55) which is a power of

s multiplied by a power series in

${s}^{-1}$, by solving the DE for

$\widehat{u}(s)$. When it converges at large

$\left|s\right|$, it is the Laplace transform of a solution of the DE for

$u(t)$ or its AC.

In

Section 4, we obtain such a solution

$\widehat{u}(s)$ that the series in it converges at

$\left|s\right|>\left|b\right|$ for

$b\ne 0$. Then, we obtain the solution

$u(t)$ by term-by-term inverse Laplace transform by writing the distribution associated with the solution

$u(t)$, by using the obtained

$\widehat{u}(s)$. In

Section 6, another example is given.

In

Section 5, we obtain such a solution

$\widehat{u}(s)$ that the series in it converges for no value of

s. Then, we obtain the solution

$u(t)$ by writing the distribution

$\tilde{u}(t)=\widehat{u}(D){\delta}_{1}(t)$ associated with the solution

$u(t)$, by using the obtained non-convergent series of

${s}^{-1}$ for

$\widehat{u}(s)$. The result is seen to be obtained by term-by-term inverse Laplace transform of

$\widehat{u}(s)$, as mentioned in

Section 3.2.

We may conclude the study in this paper as follows. When we desire to obtain the solution

$u(t)$ of a linear DE with polynomial coefficients, in the form of Equation (

3), we can obtain it, by setting up the DE for the AC-Laplace transform

$\widehat{u}(s)$, obtaining its solution in the form of Equation (

9), and then taking its term-by-term inverse Laplace transform.

We can obtain the solution of the DE for

$\widehat{u}(s)$ by the basic method, obtaining it in the form of Equation (

55). Comparing the solutions of the DE for

$u(t)$ and

$\widehat{u}(s)$, obtained by the basic method of solution, we find that the solution

$\widehat{u}(s)$ is the term-by-term Laplace transform of

$u(t)$. Thus, we obtain

$u(t)$ by the term-by-term inverse Laplace transform of

$\widehat{u}(s)$, when the latter is obtained.