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In a preceding paper, we discussed the solution of Laplace’s differential equation by using operational calculus in the framework of distribution theory. We there studied the solution of that differential equation with an inhomogeneous term, and also a fractional differential equation of the type of Laplace’s differential equation. We there considered derivatives of a function

Yosida [1,2] discussed the solution of Laplace’s differential equation (DE), which is a linear DE, with coefficients which are linear functions of the variable. The DE which he takes up is

where and for are constants. His discussion is based on Mikusiński’s operational calculus [

In the preceding paper [

for and. Here for is the Riemann-Liouville fractional derivative (fD) defined in Section 2. We use to denote the set of all real numbers, and. When is equal to an integer,. When

, (1.2) is the inhomogeneous DE for (1.1). We use to denote the set of all integers, and, and

for satisfying.

We use for, to denote the least integer that is not less than.

In [

In [

is finite. Practically, we adopt Condition B in

[

Here is Heaviside’s step function, and when is defined on, is assumed to be equal to when and to when. is defined by

for, where is the gamma function.

In [

where are constants. If, one of the solutions given in [7,8] is

where for andand. The other solution is

In [

In a recent review [

that diverges. In the present paper, we adopt this analytic continuation of.

In place of the above Condition IB, we now adopt the following condition.

Condition A and are expressed as a linear combination of for, where is a set of for some.

As a consequence, we can now achieve ordinary solutions for (1.2) of. For (1.4), we obtain both solutions (1.5) and (1.6) if.

It is the purpose this paper to show how the presentation in [

In Section 2, we prepare the definition of RiemannLiouville fD and then explain how the function and its fD in (1.2) are converted into the corresponding distribution and its fD in distribution theory, and also how is converted back into. After these preparation, a recipe is given to be used in solving the fDE (1.2) with the aid of operational culculus in Section 3. In this recipe, the solution is obtained only when

and. When, is also required. An explanation of this fact is given in Appendices C and D of [

For the Hermite DE with inhomogeneous term, Levine and Malek [

We now adopt Condition A. We then express as follows;

where are constants.

Lemma 1 For,

Proof By (1.3), for, we have

.

Let be locally integrable on. We then define the Riemann-Liouville fractional integral, , of order by

We then define the Riemann-Liouville fD, , of order, by

if it exists, where, and for.

For, we have

If we assume that takes a complex value, by definition (2.3) is analytic function of in the domain, and defined by (2.4) is its analytic continuation to the whole complex plane. If we assume that also takes a complex value, defined by (2.4) is an analytic function of in the domain. The analytic continuation as a function of was also studied. The argument is naturally concluded that (2.5) should apply for the analytic continuation, even in except at the points where; see [

We now adopt this analytic continuation of to represent, and hence we accept the following lemma.

Lemma 2 (2.5) holds for every,.

By (2.1) and (2.5), we have

For defined by (2.1), we note that

is locally integrable on.

We consider distributions belonging to. When a function is locally integrable on and has a support bounded on the left, it belongs to and is called a regular distribution. The distributions in are called right-sided distributions.

A compact formal definition of a distribution in and its fractional integral and derivative is given in Appendix A of [

Let be a regular distribution. Then

for is also a regular distribution, and distribution is defined by

Let, and let be such a regular distribution that is continuous and differentiable on

, for every. Then is defined by

Let, , and let

be continuous and differentiable on for every. Then

When is a regular distribution, is defined for all.

Lemma 3 For, the index law:

is valid for every.

Dirac’s delta function is the distribution defined by.

Let for be defined by

Lemma 4 If,

Proof By putting in (2.7) and using (2.11) and (2.5), we obtain

By operating to this and using (2.9) and (2.5), we obtain (2.12).

Corresponding to expressed by (2.1), we define by

Then and are expressed as

where

Because of (2.11), we have

Lemma 5 Let. Then

The last derivative with respect to is taken regarding as a variable.

A proof of (2.17) for is given in Appendix B of [

Proof When, , by Lemmas 4 and 1,

The first equality in (2.18) is obtained from (2.17) and vice versa, by using (2.11).

The following lemma is a consequence of this lemma.

Lemma 6 Let be expressed as a linear combination of for. Then

Lemma 7 Let, satisfy. Then

Proof Formula (2.20) is derived by applying (2.3), (2.12) and (2.16) to the righthand. Formula (2.21) follows from (2.20) by replacing and by, and, respectively, by using (2.2) and (2.17).

By using Lemma 7 to (2.6), we obtain

Lemma 8 Let, satisfy. Then

This follows from (2.20).

Condition B is expressed as a linear combination of for, where is a set of, for some.

When this condition is satisfied, is expressed as (2.13) with replaced by.

Lemma 9 Let satisfy Condition B. Then the corresponding is obtained from, by

and is expressed by (2.1) with replaced by.

Lemma 10 Let and be given by (2.13) and (2.1), respectively. Then and are related by

if satisfies.

Proof By (2.13) and (2.16), we have

Using (2.22) in the first term on the righthand side, we obtain (2.26). Multiplying (2.28) by and noting that the first term on the righthand side is then equal to (2.23), we obtain (2.27).

We now express the DE/fDE (1.2) to be solved, as follows:

where or, and. In Sections 4 and 5, we study this DE for and this fDE for, respectively.

Using Lemma 10, we express (3.1) as

where

By using (2.14) and (2.19), we express (3.2) as

where

In order to solve the Equation (3.4) for

we solve the following equation for function of real variable:

Lemma 11 The complementary solution (C-solution) of equation (3.7) is given by, where is an arbitrary constant and

where the integral is the indefinite integral and is any constant.

Lemma 12 Let be the C-solution of (3.7), and be the particular solution (P-solution) of (3.7), when the inhomogeneous term is for. Then

where is any constant.

Since satisfies Condition A and is given by (3.6), the P-solution of (3.7) is expressed as a linear combination of for, and of for, respectively.

From the solution of (3.7), is obtained by substituting by. Then we confirm that (3.4) is satisfied by that operated to.

Finally the obtained expression of is expanded into Neumann series [

1) We prepare the data: by (2.14), and, and by (3.5) and (3.6).

2) We obtain by (3.8). The C-solution of (3.2) is given by

If, the C-solution of (3.1) is obtained from this with the aid of Lemma 9.

3) If or, we obtain given by (3.9).

4) If and, the solution of (3.2) is given by

where are constants. The C-solution of (3.1) is then obtained from this with the aid of Lemma 9.

5) If, the P-solution of (3.2)

is given by

where and are constants. The P-solution of (3.1) with inhomogeneous term

is obtained from this with the aid of Lemma 9.

In the above recipe, we first obtain the C-solution of (3.7), that is. It gives the C-solution of (3.4) and hence the C-solutions of (3.2). A C-solution of (3.1) is then obtained with the aid of Lemma 9.

We next obtain the P-solution of (3.7), when the inhomogeneous part is for. As noted above, the P-solutions of (3.7) for and for, are expressed as a linear combination of for, and of for, respectively. The sum of the P-solutions of (3.7) for and for gives the P-solution of (3.4) and hence the P-solution of (3.2). The C-solution of (3.1) comes from the C-solution of (3.7) and the P-solution of (3.7) for.

When we obtain at the end of Section 3.2, we must examine whether it is compatible with Condition B. We will find that if for, the obtained is not acceptable. Hence we have to solve the problem, assuming that for all.

When and, we put. When

and, we put. Discussion of this problem is given in Appendices C and D of [

We now consider the case of σ = 1, m = 2, , and. Then (3.1) reduces to

By (3.5) and (3.6), , and are

where.

In order to obtain the C-solution of (3.7) by using (3.8), we express as follows:

where

B(x) is now expressed as.

By using (3.8), we obtain

where for and are the binomial coefficients.

The C-solution of (3.2) is given by

If, we obtain a C-solution of (4.1), by using Lemma 9:

Remark 1 In Introduction, Kummer’s DE is given by (1.4). It is equal to (4.1) for, , and. In this case,

We then confirm that the expression (4.8) for agrees with (1.6), which is one of the C-solutions of Kummer’s DE given in [7,8].

We now obtain the P-solution of (3.7), when the inhomogeneous term is equal to for.

When the C-solution of (3.7) is, the P-solution of (3.7) is given by (3.9). By using (4.2) and (4.6), the following result is obtained in [

where

Lemma 13 When, defined by (4.11) is expressed as

This lemma is proved in [

Equation (4.10) shows that if the inhomogeneous term is for, the P-solution of (3.2) is given by

Theorem 1 Let, , and. Then we have a P-solution of (4.1), given by

where

Proof Applying Lemma 9 to (4.13), we obtain

By using (4.12) in (4.16), we obtain (4.14) with (4.15).

We note that is expressed as

By (4.3) and (4.5),. When

and, the P-solution of (4.7) is given by

By using (4.14) for, if, we obtain a C-solution of (4.1):

In Section 4.1, we have (4.8), that is another C-solution of (4.1). If we compare (4.8) with (4.15), when, it can be expressed as

Proposition 1 When, the complementary solution of (4.1), multiplied by, is given by the sum of the righthand sides of (4.8) and of (4.20), which are equal to andrespectively.

Remark 2 As stated in Remark 1, for Kummer’s DE, and are given in (4.9), and

We then confirm that if, the set of (4.8) and (4.20) agrees with the set of (4.5) and (4.6).

In [

In this section, we consider the case of, ,

, andThen the Equation (3.1) to be solved is

Now (3.5) and (3.6) are expressed as

where.

By using (5.2), is expressed as

where

By (3.8), the C-solution of (3.7) is given by

The C-solution of (3.2) is given by

If, by applying Lemma 9 to this, we obtain the C-solution of (5.1):

By using the expressions of and given by (5.2) and (5.6) in (3.9), we obtain the P-solution of (3.7), when the inhomogeneous term is for:

where is defined by (4.11) and is given by

(4.12), if.

By using (4.12) in (5.9), we can show that if the inhomogeneous term is for, the P-solution of (3.2) is. By applying Lemma 9 to this, we obtain the following theorem.

Theorem 2 Let, and . Then we have a P-solution

of (5.1), given by

where

In Appendix C, discussion is given to show that there exist P-solutions in the form of polynomial for (5.1).

We obtain the solution only for. Even though we have P-solutions of (3.2) for, when is given by (5.3) with nonzero values of, it does not satisfy Condition B, and does not give a solution of (5.1). Hence given by (5.8) is the only C-solution of (5.1).

If we compare (5.8) with (5.11), we obtain the following proposition.

Proposition 2 Let. Then the C-solution of (5.1) is given by

Let and. Then (4.15) gives

where

We obtain the following theorems from (A.2) with the aid of Proposition 1.

Theorem 3 Let, , and. Then we have the polynomial form of P-solution of (4.1):

Theorem 4 Let, , and for. Then we have the polynomial form of P-solution of (4.1):

We now consider the inhomogeneous Hermite DE given by

for and. We put and. Then the equation for is given by

This is Laplace’s DE (4.1) with parameters

and the inhomogeneous term.

Theorem 5 Let, , and,. Then we have the polynomial form of Psolution of (B.2):

Proof In this case, , , and

. By Theorem 3, we obtain this result.

Theorem 6 Let, , and,. Then we have the polynomial form of Psolution of (B.2):

Proof In this case, , , and. By Theorem 4, we obtain this result.

Theorem 7 Let, , and,. Then we have the polynomial form of Psolution of (B.2):

Proof In this case, , , and. By Theorem 4, we obtain this result.

Theorem 8 Let, , and,. Then we have the polynomial form of Psolution of (B.2):

Proof In this case, , , and. By Theorem 3, we obtain this result.

Remark 3 We confirm that Theorems 7 and 5, respectively, agree with Theorems 1 and 2 in [

Let and. Then (5.11) gives

where

We obtain the following theorem from (C.2) with the aid of Proposition 2.

Theorem 9 Let, , and for. Then we have the polynomial form of P-solution of (5.1):