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 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
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].
. When 
is defined on
, 
is assumed to be equal to 
when 
and to 
when
.
be locally integrable on
. We then define the R-L fractional integral 
of order 
by
is the gamma function. The thus-defined 
is locally integrable on
, and 
if
.
of order
, by
, and 
for
.
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
.
is given by
is a function of 
as
, so that 
as
. By comparing (2.2)
.
. 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.
and its fractional integral and derivative is given in Appendix A.
be a regular distribution. Then
for 
is also a regular distribution, and distribution 
is defined by
, and let 
be such a regular distribution that 
is continuous and differentiable on
, for every
. Then 
is defined by
, for 
and
, be continuous and differentiable on
, for every
. Then
is a regular distribution, 
is defined for all
.
, the index law:
.
is the distribution defined by
.
for
. Then
, 
, and 
in
. By (2.5), we then have
. By applying 
to this and using (2.6) and (2.8), we obtain (2.9). 
and 
are expressed as a linear combination of 
for
.
and 
are expressed as
exist for
. Then the products 
and 
belong to
and they are related by
from the right and then applying
. We first note 
due to (2.5).
to this, we obtain the lefthand side of (2.11), and hence from the lefthand side of (2.4). We next note that
as noted after
. Then
is taken regarding 
as a variable.
. Let
,
. Then by (2.9), we have
is given in Appendix B.
satisfy Condition B. Then
or
, and
. In Sections 4 and 5, we study this DE for 
and this fDE for
, respectively.
, we solve the following equation for function 
of real variable
:
, where 
is an arbitrary constant and
is any constant.
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
is any constant.
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
.
of (3.7), 
is obtained by substituting 
by
. Then we confirm that (3.4) is satisfied by that 
applied to
.
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
.
by (2.10), and
, 
and 
by (3.5) and (3.6).
by (3.8). If
, the Csolution of (3.1) is given by
or
, we obtain 
given by (3.9).
, the C-solution of (3.1) is given by
are constants.
, the P-solution of (3.1)
and 
are constants.
. It gives the C-solution 
of (3.4) and hence the C-solutions 
of (3.2) and 
of (3.1).
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
.
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
.
and
, we put
. When
and
, we put
. Discussion of this problem is given in Appendices C and D.
, 
, 
, 
, 
and
. Then (3.1) reduces to
, 
and 
are
.
of (3.7) by using (3.8), we express 
as follows:
is now expressed as
.
for 
and 
and
, and
.
, Condition B is satisfied. Then by using (2.9), we obtain the C-solution of (3.2):
, 
, 
and
. In this case,
for
.
given by (4.6), the P-solution of (3.7) is given by (3.9). By using (4.2) and (4.6), we obtain
defined by (4.11) is expressed as
of (3.7) is now expressed as
. 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
:
for
, the P-solution of (3.2) is given by
. When the inhomogeneous part is
, the P-solution of (3.7) is given by
, we obtain
and
.
, is given by the sum of the righthand sides of (4.8) and of (4.21).
, 
, 
and
, is given. In this case, 
and 
are given in (4.9), and
, 
,
, 
, 
, 
and
.
.
is expressed as
of (3.7) is given by
.
.
and 
given by (5.2) and (5.6) in (3.9), we obtain the P-solution of (3.7) when the inhomogeneous part is
:
is defined by (4.11) and is given by (4.12).
for
, the P-solution of (3.2) is given by
for 
gives the P-solution of (5.1)when the inhomogeneous part is 
for
.
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.
is a locally integrable function on
, which has a support bounded on the left, and 
is given by
. If
, the fractional integral 
is
, the fractional derivative 
is given by
. We set
, and
.
.
, 
for 
is defined by
is defined by
, as stated just below Lemma 1, and hence
and
. Then 
is defined by
, with the aid of notations explained in Appendix A.
, 
and
. Then
in the last member, we obtain
follows from this.
and
. Then (3.5) and (3.6) are expressed as
as
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
.
. Then (3.5) and (3.6) are expressed as
are given by (C.4).
as
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
.