** Applied Mathematics** Vol.4 No.11A(2013), Article ID:38844,11 pages DOI:10.4236/am.2013.411A1005

Solution of Laplace’s Differential Equation and Fractional Differential Equation of That Type

^{1}Tohoku University, Sendai, Japan

^{2}College of Engineering, Nihon University, Koriyama, Japan

Email: senmm@jcom.home.ne.jp

Copyright © 2013 Tohru Morita, Ken-ichi Sato. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received August 19, 2013; revised September 19, 2013; accepted September 26, 2013

**Keywords:** Laplace’s Differential Equation; Kummer’s Differential Equation; Fractional Differential Equation; Distribution Theory; Operational Calculus; Inhomogeneous Equation; Polynomial Solution

ABSTRACT

In a preceding paper, we discussed the solution of Laplace’s differential equation by using operational calculus in the framework of distribution theory. We there studied the solution of that differential equation with an inhomogeneous term, and also a fractional differential equation of the type of Laplace’s differential equation. We there considered derivatives of a function on, when is locally integrable on, and the integral converges. We now discard the last condition that should converge, and discuss the same problem. In Appendices, polynomial form of particular solutions are given for the differential equations studied and Hermite’s differential equation with special inhomogeneous terms.

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 andand. 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

.

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

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

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

By operating to this and using (2.9) and (2.5), we obtain (2.12).

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,

The first equality in (2.18) is obtained from (2.17) and vice versa, by using (2.11).

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).

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).

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

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

where and are constants. The P-solution of (3.1) with inhomogeneous term

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).

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 andrespectively.

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

In this section, we consider the case of, ,

, andThen 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

of (5.1), given by

(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)

REFERENCES

- K. Yosida, “The Algebraic Derivative and Laplace’s Differential Equation,” Proceedings of the Japan Academy, Vol. 59, Ser. A, 1983, pp. 1-4.
- K. Yosida, “Operational Calculus,” Springer-Verlag, New York, 1982, Chapter VII.
- J. Mikusiński, “Operational Calculus,” Pergamon Press, London, 1959.
- T. Morita and K. Sato, “Remarks on the Solution of Laplace’s Differential Equation and Fractional Differential Equation of That Type,” Applied Mathematics, Vol. 4, No. 11A, 2013, pp. 13-21.
- T. Morita and K. Sato, “Solution of Fractional Differential Equation in Terms of Distribution Theory,” Interdisciplinary Information Sciences, Vol. 12, No. 2, 2006, pp. 71-83.
- T. Morita and K. Sato, “Neumann-Series Solution of Fractional Differential Equation,” Interdisciplinary Information Sciences, Vol. 16, No. 1, 2010, pp. 127-137.
- M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables,” Dover Publ., Inc., New York, 1972, Chapter 13.
- M. Magnus and F. Oberhettinger, “Formulas and Theorems for the Functions of Mathematical Physics,” Chelsea Publ. Co., New York, 1949, Chapter VI.
- T. Morita and K. Sato, “Liouville and Riemann-Liouville Fractional Derivatives via Contour Integrals,” Fractional Calculus and Applied Analysis, Vol. 16, No. 3, 2013, pp. 630-653.
- L. Levine and R. Maleh, “Polynomial Solutions of the Classical Equations of Hermite, Legendre and Chebyshev,” International Journal of Mathematical Education in Science and Technology, Vol. 34, 2003, pp. 95-103.
- F. Riesz and B. Sz.-Nagy, “Functional Analysis,” Dover Publ., Inc., New York, 1990, p. 146.

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.

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.

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.

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.

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)