Remarks on the 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].
In our preceding papers [4,5], we discuss the initial-value problem of linear fractional differential equation (fDE) with constant coefficients, in terms of distribution theory. The formulation is given in the style of primitive operational calculus, solving a Volterra integral equation with the aid of Neumann series.
Yosida [1,2] studied the homogeneous Equation (1.1), where he gave only one of the solutions by that method. One of the purposes of the present paper is to give the recipe of obtaining the solution of the inhomogeneous equation as well as the homogeneous one, in the style of operational calculus in the framework of distribution theory. With the aid of that recipe, we show how the set of two solutions of the homogeneous equation is attained.
Another purpose of this paper is to discuss the solution of an fDE of the type of Laplace’s DE, which is a linear fDE with coefficients which are linear functions of the variable. In place of (1.1), we consider
(1.2)
for 
and
. Here 
for 
is the Riemann-Liouville (R-L) fractional derivative 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 Section 2, we prepare the definition of R-L fractional derivative and then explain how (1.2) is converted into a DE or an fDE of a distribution in distribution theory. A compact definition of distributions in the space 
and their fractional integral and derivative are described in Appendix A. A proof of a lemma in Section 2 is given in Appendix B. After these preparation, a recipe is given to be used in solving a DE 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. In Section 4, we apply the recipe to the DE where
, 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
.
The discussion is done in the style of our preceding papers [4,5].
2. Formulas
We use Heaviside’s step function, which we denote by
. When 
is defined on
, 
is assumed to be equal to 
when 
and to 
when
.
2.1. Riemann-Liouville Fractional Integral and Derivative
Let 
be locally integrable on
. We then define the R-L fractional integral 
of order 
by
(2.1)
where 
is the gamma function. The thus-defined 
is locally integrable on
, and 
if
.
We define the R-L fractional derivative 
of order
, by
(2.2)
if it exists, where
, and 
for
.
We now assume that the following condition is satisfied.
Condition A 
is locally integrable on
, and there exists 
for
, and 
for 
are continuous and differentiable at
, where
. We then assume that there exists a finite value
(2.3)
for every
.
Because of this condition, the Taylor series expansion of 
is given by
(2.4)
where 
is a function of 
as
, so that 
as
. By comparing (2.2)
and (2.4), we obtain
.
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.
Let 
be a regular distribution. Then
for 
is also a regular distribution, and distribution 
is defined by
(2.5)
Let
, and let 
be such a regular distribution that 
is continuous and differentiable on
, for every
. Then 
is defined by
(2.6)
Let
, for 
and
, be continuous and differentiable on
, for every
. Then
(2.7)
When 
is a regular distribution, 
is defined for all
.
Lemma 1 For
, the index law:
(2.8)
is valid for every
.
Dirac’s delta function 
is the distribution defined by
.
Lemma 2 Let 
for
. Then
(2.9)
Proof By putting
, 
, and 
in
(2.1), we obtain
. By (2.5), we then have
. By applying 
to this and using (2.6) and (2.8), we obtain (2.9). 
We now adopt the following condition.
Condition B 
and 
are expressed as a linear combination of 
for
.
Then 
and 
are expressed as
(2.10)
Lemma 3 Let 
exist for
. Then the products 
and 
belong to
and they are related by
(2.11)
Proof We obtain (2.11) from (2.4) by multiplying 
from the right and then applying
. We first note 
due to (2.5).
Applying 
to this, we obtain the lefthand side of (2.11), and hence from the lefthand side of (2.4). We next note that
due to (2.6) and 
as noted after
(2.4). Thus we obtain the first term on the righthand side of (2.11) from the last term of (2.4). As to the remaining terms, we only use (2.9). 
Lemma 4 Let
. Then
(2.12)
The last derivative with respect to 
is taken regarding 
as a variable.
Proof of Lemma 4 for
. Let
,
. Then by (2.9), we have
by using (2.9) repeatedly. 
A proof of this lemma for 
is given in Appendix B.
The following lemma is a consequence of this lemma.
Lemma 5 Let 
satisfy Condition B. Then
(2.13)
Lemma 6
(2.14)
Proof By using (2.10) and (2.13), we obtain
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 3, we express (3.1) as
(3.2)
where
(3.3)
3.2. Solution via Operational Calculus
By using (2.10) and (2.13), 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 7 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 8 Let 
be the C-solution of (3.7), and let the particular solution (P-solution) of (3.7) be 
when the inhomogeneous part is 
for
. Then
(3.9)
where 
is any constant.
Since 
satisfies Condition B and 
is given by (3.6), the P-solution 
of (3.7) is expressed as a linear combination of 
for 
and 
for
.
From the solution 
of (3.7), 
is obtained by substituting 
by
. Then we confirm that (3.4) is satisfied by that 
applied to
.
3.3. Neumann Series Expansion
Finally the obtained expression of 
is expanded into the sum of terms of negative powers of
, and then we obtain the solution 
of (3.4). If the obtained 
is a linear combination of 
for
, 
is converted to the solution 
of (3.2) by using (2.10) and (2.9). It becomes a solution 
of (3.1) for
.
3.4. Recipe of Obtaining the Solution of (3.1)
1) We prepare the data: 
by (2.10), and
, 
and 
by (3.5) and (3.6).
2) We obtain 
by (3.8). If
, the Csolution of (3.1) is given by
3) If 
or
, we obtain 
given by (3.9).
4) If
, the C-solution of (3.1) is given by
where 
are constants.
5) If
, the P-solution of (3.1)
is given by
where 
and 
are constants.
3.5. Solution of (3.1) from the Solution of (3.7)
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) and 
of (3.1).
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.
4. Laplace’s and Kummer’s DE
We now consider the case of
, 
, 
, 
, 
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), (3.4) and (3.2)
In order to obtain the C-solution 
of (3.7) by using (3.8), we express 
as follows:
(4.4)
where
(4.5)
is now expressed as
.
By using (3.8), we obtain
(4.6)
where 
for 
and 
are the binomial coefficients. Here 
for 
and
, and
.
The C-solution of (3.4) is given by
(4.7)
If
, Condition B is satisfied. Then by using (2.9), we obtain the C-solution of (3.2):
(4.8)
Remark 1 In [6,7], Kummer’s DE is given, which is equal to the DE (4.1) for
, 
, 
and
. In this case,
(4.9)
We then confirm that the expression (4.8) agrees with one of the C-solutions of Kummer’s DE given in those books.
4.2. Particular Solution of (3.7)
We now obtain the P-solution of (3.7) when the inhomogeneous part is equal to 
for
.
When the C-solution of (3.7) is 
given by (4.6), the P-solution of (3.7) is given by (3.9). By using (4.2) and (4.6), we obtain
(4.10)
where
(4.11)
Lemma 9 
defined by (4.11) is expressed as
(4.12)
Proof Equation (4.10) shows that the P-solution 
of (3.7) is now expressed as
(4.13)
where
. Substituting this into (3.7), we obtain an equation which states that a power series of 
is equal to 0. By the condition that the coefficient of every power must be 0, we obtain a recurrence equation for the coefficients
:
(4.14)
(4.15)
By using this repeatedly, we have
(4.16)
By comparing (4.10), (4.13) and (4.16), we obtain (4.12). 
4.3. Particular Solution of (3.2)
Equation (4.10) shows that if the inhomogeneous part is 
for
, the P-solution of (3.2) is given by
(4.17)
By using (4.12) in (4.17), we obtain
(4.18)
(4.19)
4.4. Complementary Solution of (4.1)
By (4.3),
. When the inhomogeneous part is
, the P-solution of (3.7) is given by
(4.20)
By using (4.18) for
, we obtain
(4.21)
Proposition 1 Let 
and
.
Then the complementary solution of (4.1), multiplied by
, is given by the sum of the righthand sides of (4.8) and of (4.21).
Remark 2 As stated in Remark 1, in [6,7], the result for
, 
, 
and
, is given. In this case, 
and 
are given in (4.9), and
(4.22)
We then confirm that the set of (4.8) and (4.21) agrees with the set of two C-solutions of Kummer’s DE given in those books.
5. Solution of fDE (3.1) for 
In this section, we consider the case of
, 
,
, 
, 
, 
and
.
Then the Equation (3.1) to be solved is
(5.1)
Then (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) or (5.1)
The C-solution of (3.2) is given by
(5.7)
By Condition B, we have to require
.
Then by using (2.9) in (5.7), we obtain
(5.8)
The C-solution of (5.1) is equal to this for
.
5.3. Particular Solution of (3.2) or (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 part is
:
(5.9)
where 
is defined by (4.11) and is given by (4.12).
By using (4.12) in (5.9), we can show that if the inhomogeneous part is 
for
, the P-solution of (3.2) is given by
(5.10)
This 
for 
gives the P-solution of (5.1)when the inhomogeneous part is 
for
.
Appendix A: Definition of a Distribution in 
and Its Fractional Integral and Derivative
A right-sided distribution 
is a functional for which a number 
is associated with all
, where 
is the space of infinitely differentiable functions which is defined on 
and has a support bounded on the right.
A regular right-sided distribution 
is a locally integrable function on
, which has a support bounded on the left, and 
is given by
(A.1)
Let
. If
, the fractional integral 
is
(A.2)
and if
, the fractional derivative 
is given by
(A.3)
where
. We set
, and
for
.
In this place, we can confirm that the index law
(A.4)
is valid for every
.
For a distribution
, 
for 
is defined by
(A.5)
The index law (2.8) follows from (A.4) by (A.5).
Dirac’s delta function 
is defined by
, as stated just below Lemma 1, and hence
(A.6)
It is customary to use the notation:
(A.7)
Let 
and
. Then 
is defined by
(A.8)
Appendix B: Proof of Lemma 4 for 
Here we give a proof of Lemma 4 for
, with the aid of notations explained in Appendix A.
Let
, 
and
. Then
Using Lemma 4 for 
in the last member, we obtain
Formula (2.12) for 
follows from this.
Appendix C: Solution of Laplace’s DE (3.1) for 
We now consider the DE (3.1) for 
and
. Then (3.5) and (3.6) are expressed as
(C.1)
(C.2)
(C.3)
where
(C.4)
In solving (3.7), we express 
as
(C.5)
where 
and 
are constants. In Section 4.1, we assume that 
and obtain the C-solution given by (4.8) which satisfies Condition B. In the presence of the first term on the righthand side of (C.5), we will see that we cannot obtain a solution satisfying Condition B. Hence we have to assume
.
Appendix D: Solution of fDE (3.1) for 
In this section, we consider the fDE (3.1) for 
and
. Then (3.5) and (3.6) are expressed as
(D.1)
(D.2)
(D.3)
where 
are given by (C.4).
In solving (3.7), we express 
as
(D.4)
where 
and 
are constants. In Section 5.2, we assume that 
and obtain the C-solution given by (5.8) which satisfies Condition B. In the presence of the first two terms on the righthand side of (D.4), we will see that we cannot obtain a solution satisfying Condition B. Hence we have to assume
.