Solution of Laplace’s Differential Equation and Fractional Differential Equation of That Type ()
1. Introduction
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
(1.1)
where
and
for
are constants. His discussion is based on Mikusiński’s operational calculus [3]. Yosida [1,2] gave there only one of the solutions of the DE (1.1).
In the preceding paper [4], we discussed the solution of an fractional differential equation (fDE) of the type of DE (1.1), that is given by
(1.2)
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 [4], we adopt operational calculus in the framework of distribution theory developed for the solution of the fDE with constant coefficients in [5,6]. In [4], we give the recipe of obtaining the solution of the inhomogeneous equation as well as the homogeneous one, and we show how the set of two solutions of the homogeneous equation is attained.
In [4], we adopt the usual definition of the Riemann-Liouville fD, which defines
only for such a locally integrable function
on
that
is finite. Practically, we adopt Condition B in
[4], which is Condition IB
and
are expressed as a linear combination of
for
.
Here
is Heaviside’s step function, and when
is defined on
,
is assumed to be equal to
when
and to
when
.
is defined by
(1.3)
for
, where
is the gamma function.
In [4], we take up Kummer’s DE as an example, which is
(1.4)
where
are constants. If
, one of the solutions given in [7,8] is
(1.5)
where
for
and
and
. The other solution is
(1.6)
In [4], if
, we obtain both of the solutions. But when
, (1.6) does not satisfy Condition IB and we could not get it.
In a recent review [9], we discussed the analytic continuations of fD, where an analytic continuation of Riemann-Liouville fD,
, is such that the fD exists even for such a locally integrable function
on
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 [4] should be revised, with the change of definition of fD and the replacement of Condition IB with Condition A.
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 [4]. In Section 4, we apply the recipe to (1.2) where
and
, of which special one is Kummer’s DE. This is an example which Yosida [1,2] takes up. In Section 5, we apply the recipe to the fDE with
, assuming
.
For the Hermite DE with inhomogeneous term, Levine and Malek [10] showed that there exist particular solutions in the form of polynomial. In Appendices A and C, we show that such a solution exists for the DE and fDE studied in Sections 4 and 5, respectively. In Appendix B, we show how the results presented in [10] are derived from those in Appendix A.
2. Formulas
We now adopt Condition A. We then express
as follows;
(2.1)
where
are constants.
Lemma 1 For
,
(2.2)
Proof By (1.3), for
, we have
. ![](https://www.scirp.org/html/5-7401796\d077a68d-e43b-43aa-a6bd-23b0a2e72239.jpg)
2.1. Riemann-Liouville Fractional Integral and Derivative
Let
be locally integrable on
. We then define the Riemann-Liouville fractional integral,
, of order
by
(2.3)
We then define the Riemann-Liouville fD,
, of order
, by
(2.4)
if it exists, where
, and
for
.
For
, we have
(2.5)
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 [9].
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
. (2.6)
For
defined by (2.1), we note that
![](https://www.scirp.org/html/5-7401796\211b911a-4df1-403d-b6a9-4b1b474bfc4b.jpg)
is locally integrable on
.
2.2. Fractional Integral and Derivative of a Distribution
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 [4].
Let
be a regular distribution. Then
for
is also a regular distribution, and distribution
is defined by
(2.7)
Let
, and let
be such a regular distribution that
is continuous and differentiable on
, for every
. Then
is defined by
(2.8)
Let
,
, and let
![](https://www.scirp.org/html/5-7401796\31dd887a-0b13-4615-bfb9-636d2c031c73.jpg)
be continuous and differentiable on
for every
. Then
(2.9)
When
is a regular distribution,
is defined for all
.
Lemma 3 For
, the index law:
(2.10)
is valid for every
.
Dirac’s delta function
is the distribution defined by
.
Let
for
be defined by
(2.11)
Lemma 4 If
,
(2.12)
Proof By putting
in (2.7) and using (2.11) and (2.5), we obtain
![](https://www.scirp.org/html/5-7401796\c9018ffd-64af-46be-89df-06dd39e1d799.jpg)
By operating
to this and using (2.9) and (2.5), we obtain (2.12). ![](https://www.scirp.org/html/5-7401796\f830485c-8b48-45ec-9cfe-8e39f4b3a5b5.jpg)
Corresponding to
expressed by (2.1), we define
by
(2.13)
Then
and
are expressed as
(2.14)
where
(2.15)
Because of (2.11), we have
(2.16)
Lemma 5 Let
. Then
(2.17)
(2.18)
The last derivative with respect to
is taken regarding
as a variable.
A proof of (2.17) for
is given in Appendix B of [4].
Proof When
,
, by Lemmas 4 and 1,
![](https://www.scirp.org/html/5-7401796\62060bc6-0d5a-4012-85a5-bb3bbec82492.jpg)
The first equality in (2.18) is obtained from (2.17) and vice versa, by using (2.11). ![](https://www.scirp.org/html/5-7401796\5c368e8c-9789-440b-8617-744c8d221430.jpg)
The following lemma is a consequence of this lemma.
Lemma 6 Let
be expressed as a linear combination of
for
. Then
(2.19)
2.3. From
to
and Vice Versa
Lemma 7 Let
,
satisfy
. Then
(2.20)
(2.21)
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). ![](https://www.scirp.org/html/5-7401796\a1436ed6-8bcb-445f-87a1-63f726d407b7.jpg)
By using Lemma 7 to (2.6), we obtain
(2.22)
(2.23)
Lemma 8 Let
,
satisfy
. Then
(2.24)
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
(2.25)
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
(2.26)
(2.27)
if
satisfies
.
Proof By (2.13) and (2.16), we have
(2.28)
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). ![](https://www.scirp.org/html/5-7401796\b42db5cc-af3b-4399-936c-49913f62cfe6.jpg)
3. Recipe of Solving Laplace’s DE and fDE of That Type
We now express the DE/fDE (1.2) to be solved, as follows:
(3.1)
where
or
, and
. In Sections 4 and 5, we study this DE for
and this fDE for
, respectively.
3.1. Deform to DE/fDE for Distribution
Using Lemma 10, we express (3.1) as
(3.2)
where
(3.3)
3.2. Solution Via Operational Calculus
By using (2.14) and (2.19), we express (3.2) as
(3.4)
where
(3.5)
(3.6)
In order to solve the Equation (3.4) for
we solve the following equation for function
of real variable
:
(3.7)
Lemma 11 The complementary solution (C-solution) of equation (3.7) is given by
, where
is an arbitrary constant and
(3.8)
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
(3.9)
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
.
3.3. Neumann Series Expansion
Finally the obtained expression of
is expanded into Neumann series [11]. Practically we expand it into the sum of terms of negative powers of D, and then we obtain the solution
of (3.4). If the obtained
is a linear combination of
for
with some
, then
is the solution
of (3.2). If it satisfies Condition B, it is converted to a solution
of (3.1) for
, with the aid of Lemma 9.
3.4. Recipe of Obtaining the Solution of (3.1)
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
![](https://www.scirp.org/html/5-7401796\7ec74ebd-ea72-4516-8f36-d0173bf4e284.jpg)
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
(3.10)
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
![](https://www.scirp.org/html/5-7401796\6ff28700-b843-437f-bea6-b1dc13774ca8.jpg)
where
and
are constants. The P-solution of (3.1) with inhomogeneous term
![](https://www.scirp.org/html/5-7401796\03f1dd3c-a4d7-4037-9213-759ed321ebdb.jpg)
is obtained from this with the aid of Lemma 9.
3.5. Comments on the Recipe
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
.
3.6. Remarks
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 [4]. In the present case, the discussion must be read taking Condition B there to represent the present Condition B.
4. Laplace’s and Kummer’s DE
We now consider the case of σ = 1, m = 2,
, and
. Then (3.1) reduces to
(4.1)
By (3.5) and (3.6),
,
and
are
(4.2)
(4.3)
where
.
4.1. Complementary Solution of (3.7), (3.2) and (4.1)
In order to obtain the C-solution
of (3.7) by using (3.8), we express
as follows:
(4.4)
where
(4.5)
B(x) is now expressed as
.
By using (3.8), we obtain
(4.6)
where
for
and
are the binomial coefficients.
The C-solution of (3.2) is given by
(4.7)
If
, we obtain a C-solution of (4.1), by using Lemma 9:
(4.8)
Remark 1 In Introduction, Kummer’s DE is given by (1.4). It is equal to (4.1) for
,
,
and
. In this case,
(4.9)
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].
4.2. Particular Solution of (3.7)
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 [4]:
(4.10)
where
(4.11)
Lemma 13 When
,
defined by (4.11) is expressed as
(4.12)
This lemma is proved in [4].
4.3. Particular Solutions of (3.2) and (4.1)
Equation (4.10) shows that if the inhomogeneous term is
for
, the P-solution of (3.2) is given by
(4.13)
Theorem 1 Let
,
, and
. Then we have a P-solution
of (4.1), given by
(4.14)
where
(4.15)
Proof Applying Lemma 9 to (4.13), we obtain
(4.16)
By using (4.12) in (4.16), we obtain (4.14) with (4.15). ![](https://www.scirp.org/html/5-7401796\e6f8eede-aa8c-49c8-b76d-5a77dd4b086e.jpg)
We note that
is expressed as
(4.17)
(4.18)
4.4. Complementary Solution of (4.1)
By (4.3) and (4.5),
. When
and
, the P-solution of (4.7) is given by
(4.19)
By using (4.14) for
, if
, we obtain a C-solution of (4.1):
(4.20)
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
(4.21)
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
and
respectively.
Remark 2 As stated in Remark 1, for Kummer’s DE,
and
are given in (4.9), and
(4.22)
We then confirm that if
, the set of (4.8) and (4.20) agrees with the set of (4.5) and (4.6).
4.5. Remarks
In [10], it was shown that there exist P-solutions expressed by a polynomial for inhomogeneous Hermite’s DE, et al. We can obtain the corresponding result for Laplace’s DE. We discuss this problem in Appendix A, and then discuss the P-solution of inhomogeneous Hermite’s DE in the present formulation in Appendix B.
5. Solution of fDE (3.1) for ![](https://www.scirp.org/html/5-7401796\63a0a436-070e-44df-8547-740d57dd918b.jpg)
In this section, we consider the case of
,
,
, and
Then the Equation (3.1) to be solved is
(5.1)
Now (3.5) and (3.6) are expressed as
(5.2)
(5.3)
where
.
5.1. Complementary Solution of (3.7)
By using (5.2),
is expressed as
(5.4)
where
(5.5)
By (3.8), the C-solution
of (3.7) is given by
(5.6)
5.2. Complementary Solution of (3.2) and (5.1)
The C-solution of (3.2) is given by
(5.7)
If
, by applying Lemma 9 to this, we obtain the C-solution of (5.1):
(5.8)
5.3. Particular Solution of (3.2) and (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
:
(5.9)
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 ![](https://www.scirp.org/html/5-7401796\43349673-dcfc-4ed9-bec3-90af9b272420.jpg)
of (5.1), given by
![](https://www.scirp.org/html/5-7401796\28ed9fda-b5f3-423d-8043-ec2eb20e5e5d.jpg)
(5.10)
where
(5.11)
In Appendix C, discussion is given to show that there exist P-solutions in the form of polynomial for (5.1).
5.4. Complementary Solution of (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
(5.12)
Appendix A: Polynomial Form of P-Solution of (4.1)
Let
and
. Then (4.15) gives
(A.1)
(A.2)
where
(A.3)
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):
(A.4)
Theorem 4 Let
,
,
and
for
. Then we have the polynomial form of P-solution of (4.1):
(A.5)
Appendix B: Polynomial Form of P-Solution of Hermite DE
We now consider the inhomogeneous Hermite DE given by
(B.1)
for
and
. We put
and
. Then the equation for
is given by
(B.2)
This is Laplace’s DE (4.1) with parameters
(B.3)
and the inhomogeneous term
.
Theorem 5 Let
,
, and
,
. Then we have the polynomial form of Psolution of (B.2):
(B.4)
Proof In this case,
,
, and
. By Theorem 3, we obtain this result. ![](https://www.scirp.org/html/5-7401796\89379e02-142c-4b50-a7b6-08614f01e20c.jpg)
Theorem 6 Let
,
, and
,
. Then we have the polynomial form of Psolution of (B.2):
(B.5)
Proof In this case,
,
, and
. By Theorem 4, we obtain this result. ![](https://www.scirp.org/html/5-7401796\a63808fa-66dc-4fbd-a349-9f5cd1376a5f.jpg)
Theorem 7 Let
,
, and
,
. Then we have the polynomial form of Psolution of (B.2):
(B.6)
Proof In this case,
,
, and
. By Theorem 4, we obtain this result. ![](https://www.scirp.org/html/5-7401796\7ca52a1d-1c42-477d-915e-cbacee1964da.jpg)
Theorem 8 Let
,
, and
,
. Then we have the polynomial form of Psolution of (B.2):
(B.7)
Proof In this case,
,
, and
. By Theorem 3, we obtain this result. ![](https://www.scirp.org/html/5-7401796\50907048-03b1-486c-8cb0-680111a8d4e3.jpg)
Remark 3 We confirm that Theorems 7 and 5, respectively, agree with Theorems 1 and 2 in [10].
Appendix C: Polynomial Form of P-Solution of (5.1)
Let
and
. Then (5.11) gives
(C.1)
(C.2)
where
(C.3)
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):
(C.4)