Solution of Differential Equations with the Aid of an Analytic Continuation of Laplace Transform ()
1. Introduction
Yosida [1] [2] discussed the solution of Laplace’s differential equation, which is a linear differential equation, with coefficients which are linear functions of the variable. In recent papers [3] [4] , we discussed the solution of that equation, and fractional differential equation of that type. The differential equations are expressed as
(1.1)
for 
and
. Here 
for 
are constants, and 
are the Riemann-Liouville fractional derivatives to be defined in Section 2.
We use
, 
and
, to denote the sets of all real numbers, of all integers and of all complex numbers. We also use
, 
, 
and
for 
and 
satisfying
. If
, 
, and
. We use 
for
, to denote the least integer that is not less than x. In the present paper, the variable t is always assumed to take values on
.
Yosida [1] [2] studied the Equation (1.1) for 
with
, by using Mikusiński’s operational calculus [5] . In [3] [4] , operational calculus in terms of distribution theory is used, which was developed for the initial-value problem of fractional differential equation with constant coefficients in our preceding papers [6] [7] . In [3] , the derivative is the ordinary Riemann-Liouville fractional derivative, so that the fractional derivative of a function 
exists only when 
is locally integrable on
, and the integral 
converges.
Practically, we adopt Condition B in [3] , which is Condition 1. 
and 
in (1) are expressed as a linear combination of 
for
.
Here 
is defined by
(1.2)
for
, where 
is the gamma function.
We then express 
as follows:
(1.3)
where 
are constants, and 
is a set of
.
In a recent review [8] , we discussed the analytic continuations of fractional derivative, where an analytic continuation of Riemann-Liouville fractional derivative of function 
is such that the fractional derivative exists when 
is locally integrable on
, even when the integral 
diverges.
In [4] , we adopted this analytic continuation of Riemann-Liouville fractional derivative, and the following condition, in place of Condition 1.
Condition 2. 
and 
in (1.1) are expressed as a linear combination of 
for
, where S is a set of 
for some
.
We then express 
as follows;
(1.4)
In [3] [4] , we take up Kummer’s differential equation as an example, which is
(1.5)
where 
are constants. If
, one of the solutions given in [9] [10] is
(1.6)
where 
for 
and
, and
. The other solution is
(1.7)
In [3] , if
, we obtain both of the solutions. But when
, (1.7) does not satisfy Condition 1 and we could not get it in [3] . In [4] , we always obtain both of the solutions. In [1] [2] , Yosida obtained only the solution (1.7).
We now study the solution of a differential equation with the aid of Laplace transform. Then it is required that there exists the Laplace transform of the function 
to be determined.
When we consider the Laplace transform of a function 
which is locally integrable on
, we assume the following condition.
Condition 3. There exists some 
such that 
as
.
Let 
be locally integrable on 
and satisfy Condition 3, and the integral 
converge. We then denote its Laplace transform by
, so that
(1.8)
The Laplace transform, 
, of 
for 
is then given by
(1.9)
Let 
expressed by (1.3) satisfy Condition 3, and let its Laplace transform 
be given by
(1.10)
Then we can show that we are able to solve the problems solved in [3] , with the aid of Laplace transform.
When 
satisfies Conditions 2 and 3, Laplace transform is not applicable.
In [4] , we adopted an analytic continuation of Riemann-Liouville fractional derivative, by which we could solve the differential equation assuming Condition 2. The analytic continuation is achieved with the aid of Pochhammer’s contour, which is used in the analytic continuation of the beta function.
We now introduce the analytic continuation of Laplace transform with the aid of Hankel’s contour, which is used in the analytic continuation of the gamma function. We then show that (1.9) is valid for
, and that if 
expressed by (1.4) satisfies Condition 3, and its analytic continuation of Laplace transform of
, which we denote by
, is given by
(1.11)
then we can solve the problems solved in [4] , with the aid of the analytic continuation of Laplace transform.
In Section 2, we prepare the definition of analytic continuations of Riemann-Liouville fractional derivative and of Laplace transform, and then explain how the equation for the function 
and its fractional derivative in (1.1) are converted into the corresponding equation for the analytic continuation of Laplace transform, 
, of
, and also how 
is converted back into
. After these preparations, a recipe is given to be used in solving the fractional differential Equation (1.1) with the aid of the analytic continuation of Laplace transform 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 [3] . In Section 4, we apply the recipe to (1.1) where 
and
, of which special one is Kummer’s differential equation. In Section 5, we apply the recipe to the fractional differential equation with
, assuming
.
In Section 6, comments are given on the relation of the present method with the preceding one developed in [4] , and on the application of the analytic continuation of Laplace transform to the differential equations with constant coefficients.
2. Formulas
Lemma 1. Let 
be defined by (1.2). Then for
,
(2.1)
Proof. By (1.2), for
,
. ,
2.1. Analytic Continuation of Riemann-Liouville Fractional Derivative
Let a function 
be locally integrable on 
for
, and let 
exist. We then define the Riemann-Liouville fractional integral, 
, of order 
by
(2.2)
We then define the Riemann-Liouville fractional derivative, 
, of order
, by
(2.3)
if it exists, where
, and 
for
.
For 
and
, we have
(2.4)
If we assume that 
takes a complex value, 
by definition (2.2) is analytic function of 
in the domain
, and 
defined by (2.3) is its analytic continuation to the whole complex plane. If we assume that 
also takes a complex value, 
defined by (2.3) is an analytic function of 
in the domain
. The analytic continuation as a function of 
was also studied. The argument is concluded that (2.4) should apply for the analytic continuation via Pochhammer’s contour, even in 
except at the points where
; see [8] .
We now adopt this analytic continuation of 
to represent
, and hence we accept the following lemma.
Lemma 2. Formula (2.4) holds for every
,
.
By (1.4) and (2.4), we have
(2.5)
For 
defined by (1.4), we note that 
is locally integrable on
.
2.2. Analytic Continuation of Laplace Transform
The gamma function 
for 
satisfying
, is defined by Euler’s second integral:
(2.6)
The analytic continuation of 
for 
is given by Hankel’s formula:
(2.7)
where 
is Hankel’s contour shown in Figure 1(a).
We now define an integral transform 
of a function 
which satisfies the following condition.
Condition 4. 
is expressed as 
on a neighborhood of
, for
, where
, and 
is analytic on the neighborhood of
.
Let 
satisfy Conditions 3 and 4. Then we define 
for
, by
, where
(2.8)
for
, and
(2.9)
for
.
Lemma 3. Let 
satisfy Condition 4. Then 
defined by (2.8) is an analytic continuation of
, which is defined by (1.8) for
, as a function of
.
Proof. The equality 
when 
is proved in the same way as the equality of 
given by (2.6) and by (2.7) for
; see e.g. ([11] , Section 12.22). The analyticity of 
and of 
is proved as in ([11] , Sections 5.31, 5.32). Lemma 4. For
,
(a)
(b)
Figure 1. (a) Hankel’s contour CH, and (b) contours CI, −CH and CB which appear in (2.12), (2.15) and (2.16).
(2.10)
(2.11)
(2.12)
where 
and 
are two of the contours shown in Figure 1(b).
Proof. Formula (2.8) for 
gives
(2.13)
for
. The last equality in (2.13) is due to (1.9). By using (2.1) and (2.10), the lefthand side of (2.11) is expressed as
. By replacing
, 
and 
in (2.13), by
, 
and
, respectively, we obtain the first equality of (2.12) for
, with the aid of the formula
. The equality for 
is obtained by continuity. Theorem 1. Let 
satisfy Conditions 3 and 4 for
, and 
be expressed as
(2.14)
where 
is a finite set of
. Then the Laplace inversion formula is given by the contour integral:
(2.15)
where CI is a contour shown in Figure 1(b). Here it is assumed that 
is analytic to the right of the vertical line on CI, and is so above and below the upper and lower horizontal lines, respectively, on CI.
Proof. For
, the usual Laplace inversion formula applies, so that
(2.16)
where CB is a contour shown in Figure 1(b). Here it is assumed that 
is analytic to the right of the contour CB. By using this with (2.10) and (2.12), we confirm (2.15). Lemma 5. Let 
satisfy Conditions 3 and 4 with an entire function
. Then
(2.17)
Lemma 6. Let 
be expressed in the form of (1.11). Then the Laplace inversion 
is given by (1.4), provided that the obtained 
satisfies the conditions for 
in Lemma 5, or it is a linear combination of such functions.
Lemma 7. Let 
satisfy the conditions for 
in Lemma 5. Then
(2.18)
(2.19)
Proof. By using (2.5) and Lemma 4, we obtain these results. ,
3. Recipe of Solving Laplace’s Differential Equation and Fractional Differential Equation of That Type
We now express the differential Equation (1.1) to be solved, as follows:
(3.1)
where 
or
, and
. In Sections 4 and 5, we study this differential equation for 
and this fractional differential equation for
, respectively.
We now apply the integral transform 
to (3.1). By using (2.18) and (2.19), we then obtain
(3.2)
where
(3.3)
(3.4)
Here
.
Lemma 8. The complementary solution (C-solution) of Equation (3.2) is given by
, where C1 is an arbitrary constant and
(3.5)
where the integral is the indefinite integral and C2 is any constant.
Lemma 9. Let 
be the C-solution of (3.2), and 
be the particular solution (P-solution) of (3.2), when the inhomogeneous term is 
for
. Then
(3.6)
where C3 is any constant.
Since 
in (3.1) satisfies Condition 2 and 
is given by (3.4), the P-solution 
of (3.2) is expressed as a linear combination of 
for 
for
, and of 
for
, respectively.
The solution 
of (3.2) is converted to a solution 
of (3.1) for
, with the aid of Lemma 6.
4. Laplace’s and Kummer’s Differential Equations
We now consider the case of
, 
, 
, 
, and
. Then (3.1) reduces to
(4.1)
By (3.3) and (3.4), 
, 
and 
are
(4.2)
(4.3)
4.1. Complementary Solution of (3.2) and (4.1)
In order to obtain the C-solution 
of (3.2) by using (3.5), we express 
as follows:
(4.4)
where
(4.5)
By using (3.5), we obtain
(4.6)
in the form of (2.17) or (1.11), where 
for 
and 
are the binomial coefficients.
If
, we obtain a C-solution of (4.1), by using Lemma 6:
(4.7)
(4.8)
Remark 1. In Introduction, Kummer’s differential equation is given by (1.5). It is equal to (4.1) for
, 
, 
and
. In this case,
(4.9)
We then confirm that the expression (4.8) for 
agrees with (1.7), which is one of the C-solutions of Kummer’s differential equation given in [9] [10] .
4.2. Particular Solution of (3.2)
We now obtain the P-solution of (3.2), when the inhomogeneous term is equal to 
for
.
When the C-solution of (3.2) is
, the P-solution of (3.2) is given by (3.6). By using (4.2) and (4.6), the following result is obtained in [3] :
(4.10)
where
(4.11)
Lemma 10. When
, 
defined by (4.11) is expressed as
(4.12)
where
(4.13)
This lemma is proved in [3] . In fact, (4.11) is the partial fraction expansion of 
given by (4.12) as a function of
.
Applying Lemma 6 to (4.10), and using (4.12), we obtain Theorem 2. Let
, 
, and let 
for
. Then we have a P-solution 
of (4.1), given by
(4.14)
where
(4.15)
(4.16)
Here
.
4.3. Complementary Solution of (4.1)
By (4.3) and (4.5),
. When 
and
, the P-solution of (3.2) is given by
(4.17)
By using (4.14) for
, if
, we obtain a C-solution of (4.1):
(4.18)
In Section 4.1, we have (4.7), that is another C-solution of (4.1). If we compare (4.7) with (4.15), when
, it can be expressed as
(4.19)
Proposition 1. When
, the complementary solution of (4.1) is given by the sum of the righthand sides of (4.8) and of (4.18), which are equal to 
and
, respectively.
Remark 2. As stated in Remark 1, for Kummer’s differential equation, 
and 
are given in (4.9), and
(4.20)
We then confirm that if
, the set of (4.8) and (4.18) agrees with the set of (1.6) and (1.7).
5. Solution of Fractional Differential Equation (3.1) for 
In this section, we consider the case of
, 
, 
, 
, 
and
. Then the Equation (3.1) to be solved is
(5.1)
Now (3.3) and (3.4) are expressed as
(5.2)
(5.3)
5.1. Complementary Solution of (3.2)
By using (5.2), 
is expressed as
(5.4)
where
(5.5)
By (3.5), the C-solution 
of (3.2) is given by
(5.6)
If
, by applying Lemma 6 to this, we obtain the C-solution of (5.1):
(5.7)
5.2. Particular Solution of (3.2) and (5.1)
By using the expressions of 
and 
given by (5.2) and (5.6) in (3.6), we obtain the P-solution of (3.2), when the inhomogeneous term is 
for 
satisfying
:
(5.8)
where 
is defined by (4.11) and is given by (4.12).
By applying Lemma 6 to this, we obtain the following theorem.
Theorem 3. Let
, 
, and let
. Then we have a P-solution 
of (5.1), given by
(5.9)
where
(5.10)
5.3. Complementary Solution of (5.1)
We obtain the solution 
only for
. When 
is given by (5.3) with nonzero values of
, Theorem 2 does not give a solution of (5.1). Hence 
given by (5.7) is the only C-solution of (5.1).
If we compare (5.7) with (5.10), we obtain the following proposition.
Proposition 2. Let
. Then the C-solution of (5.1) is given by
(5.11)
6. Concluding Remarks
6.1. Solution with the Aid of Distribution Theory
In [4] , the solution of (3.1) is assumed to be expressed as (1.4). In distribution theory, the differential equation for the distribution 
is set up, where 
for 
and 
for
. Then it is expressed as
, where 
is the derivative of order 
in the space of
of distributions. In [4] , after obtaining
, 
is obtained by using Neumann series expansion. In the present paper, 
is the analytic continuation of Laplace transform of
. After obtaining
, we obtain 
for 
by Laplace inversion.
The steps of solution in [4] and the present paper are closely related with each other, and one may use a favorite one. One difference is that Condition 3 is assumed in the present paper but is not required in [4] .
6.2. Solutions of Differential Equations with Constant Coefficients
We now consider the differential equation given by (3.1), where
. Then assuming that the solution 
and the inhomogeneous term 
satisfy Conditions 2 and 3, we show that the analytic continuation of Laplace transform of that equation is given by (3.2) with
. We then obtain the analytic continuation of Laplace transform of
,
. If it can be expressed as (1.11), then 
is given by its Laplace inverse (1.4). If we take account of Section 6.1, we confirm that the results obtained in [7] are obtained by Laplace inversion.