Remarks on the Solution of Laplace’s Differential Equation and Fractional Differential Equation of That Type

We discuss the solution of Laplace’s differential equation by using operational calculus in the framework of distribution theory. We here study the solution of that differential Equation with an inhomogeneous term, and also a fractional differential equation of the type of Laplace’s differential equation.


Introduction
Yosida [1,2]  where l and l 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               is the Riemann-Liouville (R-L) fractional derivative defined in Section 2. We use to denote the set of all real numbers, and    to denote the least integer that is not less than x .
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 R   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 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 0 , 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 The discussion is done in the style of our preceding papers [4,5].

Formulas
We use Heaviside's step function, which we denote by is assumed to be equal to   f t when and to when .

Riemann-Liouville Fractional Integral and Derivative
Let be locally integrable on . We then define the R-L fractional integral b R of order is the gamma function. The thus-defined is locally integrable on , and if We define the R-L fractional derivative We now assume that the following condition is satisfied.
Condition A is locally integrable on , and there exists for , and We then assume that there exists a finite value   for every .
Because of this condition, the Taylor series expansion of is given by where is a function of as , so is also a regular distribution, and distribution , be continuous and differentiable on , for every By applying to this and using (2.6) and (2.8), we obtain (2.9).  We now adopt the following condition.

Condition B and
and they are related by Proof We obtain (2.11) from (2.4) by multiplying

 
H t from the right and then applying . We first Applying 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 (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 The last derivative with respect to is taken regarding as a variable.
The following lemma is a consequence of this lemma.

Lemma 5 Let satisfy Condition B. Then
Proof By using (2.10) and (2.13), we obtain

Recipe of Solving Laplace's DE and fDE of That Type
We now express the DE/fDE (1.2) to be solved, as follows: respectively.

Deform to DE/fDE for Distribution
Using Lemma 3, we express (3.1) as

Solution via Operational Calculus
By using (2.10) and (2.13), we express (3.2) as In order to solve the Equation (

Lemma 7 The complementary solution (C-solution) of
is an arbitrary constant and where the integral is the indefinite integral and is any constant.
where is any constant.

Neumann Series Expansion
Finally the obtained expression of   u D 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 is converted to the solution of (3.2) by using (2.10) and (2.9). It becomes a solution of (3.1) for .
, the C-solution of (3.1) is given by is given by

Solution of (3.1) from the Solution of (3.7)
In the above recipe, we first obtain the C-solution of

of (3.4) and hence the C-solutions
    u t H t of (3.2) and   u t of (3.1). We next obtain the P-solution of (3.7) when the inhomogeneous part is for    . As noted above, the P-solutions   u x of (3.7) for and for u t of (3.1) comes from the C-solution of (3.7) and the P-solution of (3.7) for .

Remarks
When we obtain   u D l b at the end of Section 3.2, we must examine whether it is compatible with Condition B. We will find that if 0  for > 1 l m    , the obtained   u D is not acceptable. Hence we have to solve the problem, assuming that for all

Laplace's and Kummer's DE
We now consider the case of 1
The C-solution of (3.4) is given by , Condition B is satisfied. Then by using (2.9), we obtain the C-solution of (3.2):  We now obtain the P-solution of (3.7) when the inhomogeneous part is equal to 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 We then confirm that the expression (4.8) agrees with one of the C-solutions of Kummer's DE given in those (4.20) By using (4.18) for 0 H t (4.21) Proposition 1 Let 0 0 u  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).
) (t H Remark 2 As stated in Remark 1, in [6,7], the result for 2  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.

Solution of fDE (3.1) for 1 2  σ
In this section, we consider the case of