1. Introduction
In the preceding papers [
1,
2], linear differential equations of order
, with polynomial coefficients, are studied. They take the form:
where
, and
for
and
are constants. We assume that a finite number of the constants are nonzero.
Here, , and are the sets of all integers, all real numbers and all complex numbers, respectively, and for , and for satisfying . We also use for , for , and .
We use
and
for
,
, which denote
if
, and
, as usual, and
and
.
We reassemble the terms of Equation (
1) as
where
We call
a block of classified terms. When
, Equation (
3) is expressed as
where
In Reference [
1,
2], discussions are focused on the solution of Equation (
3) with two blocks of classified terms. Kummer’s and the hypergeometric differential equations are special examples of them.
Equation (
3), which consists of only one block of classified terms for
, is expressed as
where
and
, which represent
in Equation (
6), are constants, among which
. This equation is called Euler’s differential equation (Section 6.3, Reference [
3]), (Chapter II, Section 7, Reference [
4]). In recent papers [
5,
6,
7], the solution of the equation in distribution theory, which corresponds to Equation (
7), is discussed. In Reference [
5], special attention is focused to the cases where the coefficients satisfy
for
, and
. In Reference [
1], a theorem is given on the solution of Equation (
7). It is the purpose of the present paper to present a theorem which provides
n solutions of Equation (7), by modifying Theorem 1.1 given in Reference [
1], and then the corresponding theorem in distribution theory. It is shown that the results in Reference [
5,
6,
7] are obtained as special results of that theorem given in
Section 4.
In the
Appendix A, a theorem is presented to show that there exist
n and only
n complementary solutions of a linear differential equation of order
n, with constant coefficients, in terms of distribution theory. It guarantees the corresponding theorem on (7).
In
Section 2, we present three theorems on the solution of Equation (7), two of which are related with the theorems given in Reference [
5,
6,
7]. In
Section 4, we give the corresponding theorems on the solution of the corresponding differential equation in distribution theory. In
Section 3, formulas in distribution theory, are presented, which are used in
Section 4. In
Section 5, an argument is given to show a relation of the solutions of Equation (7), and the solutions expressed by nonregular distributions, of the corresponding equation in distribution theory, on the basis of nonstandard analysis. In
Section 5.1, a brief discussion is given on the Laplace transform of Euler’s differential equation.
3. Preliminaries on Distribution Theory
Distributions in the space
are first introduced in Reference [
14,
15,
16,
17]. The distributions are either regular ones or their derivatives. A regular distribution in
corresponds to a function
which is locally integrable on
. We denote the distribution by
.
A distribution is a functional, to which is associated with every , where , that is dual to , is the space of testing functions, which are infinitely differentiable and have a compact support on .
If
is a regular distribution, we have
OperatorD is so defined that for , where . Because of this definition of D, we can confirm the following lemma.
Lemma 3. Let and be regular distributions in , which correspond to and , respectively. Then, .
Proof. In this condition, we have
□
If
is not a regular one, it is expressed as
, by
and a regular distribution
, and then we have
If , operator is so defined that ; hence, if exists, .
Lemma 4. Let the condition in Lemma 3 be satisfied. Then, , which corresponds to .
Lemma 5. Let and . Then, In particular, when , Proof. (i) We first give a proof for the case of
, in terms of mathematical induction. When
, we have
which gives (
27). If (
26) holds for a value of
, we have
which shows that (
26) holds also for
. (ii) We now assume
. We apply
to Equation (26), and put
or
in (26). We then obtain
This shows that (26) holds even when “” and are replaced by “” and , respectively. □
Lemma 6. Let and . Then,where in the last member, the notation defined by (2) is used for operator . Proof. A proof by mathematical induction is given. When
, by using Equation (
27) in Lemma 5, we have
which shows that (
30) holds for
. If (
30) holds for a value of
and for
, by using Lemma 5, we have
which shows that (
30) holds also for
. □
Lemma 7. Let and . Then, Proof. By applying
to Equation (
26) and then replacing
by
, we obtain (
32). □
3.1. Distributions in the Space
We now consider the space of distributions , which is a subspace of . A regular distribution in is such a distribution that it corresponds to a function which is locally integrable on and has a support bounded on the left. The space , that is dual to , is the space of testing functions, which are infinitely differentiable on and have a support bounded on the right.
The Heaviside step function is such that for , and for . The corresponding distribution is a regular distribution in the space , as well as in . Dirac’s function is the distribution, which is defined by .
In Reference [
18,
19], the solutions of special cases of Equation (
3) or (
1) were studied with the aid of Riemann-Liouville fractional integral and derivative, distribution theory and the AC-Laplace transform, that is the Laplace transform supplemented by its analytic continuation. In the study,
for
is defined by
where
is the gamma function, and the following condition was adopted.
Condition 1. and in (3) are expressed as a linear combination of for and , where S is a set of for some . As a consequence,
is expressed as follows:
where
are constants. Because of this condition, obtained solutions are expressed by a power series of
t multiplied by a power
:
where
,
and
.
A basic method of solving Equation (
1) is to assume the solution in the form (
35) with
. The solution is obtained by determining the coefficients
recursively; see e.g., Section 10.3 in Reference [
20].
In the space
, we define regular distribution
for
, which corresponds to function
, and then define operator of fractional integral and derivative
and distribution
for
such that
Lemma 8. for represents Proof. The equation for
is due to (
36) for
. □
3.2. Distributions in the Space , as well as in the Space
Remark 4. When , is a regular distribution, and when , it is a non-regular distribution, which are given bywhere and for which . Remark 5. In (38), for satisfying , which is not a regular distribution, is represented by , in accordance with the definition. Proof. If
, by using (
37) or (
38), we have
If
, we adopt (
38), and then by using Lemma 5 and also Equation (
40), we have
where
and
for which
. This equation and Equation (
40) show that (
39) holds for all
. □
Lemma 10. Let and . Then, In particular, when , we have Proof. Lemma 9 shows that, if
,
, which gives (
43). By using (
43) in (
30), we obtain (
42). □
3.3. Regular Distributions in the Space
In the present paper, we study the solution of the equation which corresponds to Equation (7), in distribution theory. When function , we introduce .
Lemma 11. Let , and function be such that for . Then, Proof. When
, we have
which gives (
44) for
. When
, by using it, we obtain
which gives (
44) for
. Equation (
44) can be proved by mathematical induction. □
Lemma 12. Let and satisfy . Then, Lemma 13. Let the condition of Lemma 11 be satisfied. Then, Proof. By multiplying
to Equation (
44) and then using (
47), we obtain (
49). □
Lemma 14. Let , function be such that for , and . Then,where are constants. Lemma 15. Let , function be such that , , and . Then,where the condition may be replaced by , since 4. Euler’s Equation in the Space of Distributions
Lemma 14 shows that the equation which corresponds to (7), in distribution theory, is
Remark 6. Let the l-th solution of (7) given in Theorem 1 be expressed by for . Then, Lemma 14 shows that if , is a solution of (53).
By using (
30), (
9), and then (
10) with
in (53), we obtain
As a consequence, Equation (53) is expressed by
Lemma 16. Let and be a solution of (55). Then, is a solution of This shows that, if we choose μ such that for all , is expressed by the regular distribution .
Proof. This is confirmed with the aid of Lemma 7. □
Lemma 17. Let , be the solution of Equation (43) shows that is a solution of (57). Lemma 8 or Remark 4 shows that if , the solution is given by . If , Lemma 16 shows that, if we choose μ such that , the solution is given by . Remark 7. An alternative proof of Lemma 17 is given for the case of . Then, we put and , and, we confirm The first and the second equalities are due to Lemmas 3 and 1, respectively.
Lemma 18. Let , and If , we choose or , and, if , we choose which satisfies , where is the least integer which is not less than p. We then put , and obtain m solutions of (59) given by Proof. Lemma 17 shows that
is a solution of (
59). If
,
and
, we note that, if
,
, and
By using this formula
l times and Lemma 17 for
, we obtain
which shows that we have
solutions of (
59), which involve
and are given in (
60) for
and
. If
,
satisfies
, and then Lemma 16 shows that, if
is a solution of the following equation:
is a solution of (
59). As a consequence of this fact, we obtain
m solutions of (
59) given by (
60). □
The theorem which corresponds to Theorem 1 is as follows.
Theorem 4. Let the condition of Theorem 1 be satisfied. Then, Equation (53) is expressed by (55), and we have series of solutions of Equation (53). In the kth series, if , we have one solution given by , and, if , we have solutions given bywhere and satisfies , so that for are regular distributions. See Lemma 8 and Remark 4 for the expression . Remark 8. If , we may choose and in (
64)
. Remark 9. In (64), we may choose such that satisfies for all . Proof of Theorem 4. (i) With the aid of Lemmas 17 and 18, we confirm that the
n solutions given in Theorem 4 are the solutions of Equation (
53) or (
55). (ii) If
for all
, we may choose
and
for all
k, and then this theorem is proved with the aid of Lemma 15 and Theorem 1. (iii) When we choose
, as in Remark 9, this theorem is proved with the aid of Lemma 16 and Theorem 1. □
Example 3. Let , , , andwhere the second equality is justified by Lemma 6. Then, if , ; hence, we have two solutions of (65) given by and if ; see Lemma 8 or Remark 4 for their expressions. If , we have two solutions and , as shown in Lemma 18. Remark 10. We now consider the inhomogeneous differential equation which corresponds to (65):If , and , a particular solution of this equation is . Example 4. Let , andwhere the second equality is justified by Lemma 6. Equation (42) shows that, if , ; hence, if , we have two solutions of (67), given by and . The case of is Example 3 for , so that we have two solutions and . Here, we present a theorem which corresponds to Theorem 2, Theorem 3.1 given in Reference [
5], and Theorems 1 and 2 given in Reference [
7].
Theorem 5. Let and for in Equation (53) be given, and then in Equation (53) be chosen to be , where is given by (21). Then, is a solution of (53). See Lemma 8 and Remark 4 for the expression .
Proof. This theorem is proved with the aid of Theorem 4. □
Remark 11. Equation (67), for and , is taken up in Reference [5]. When , we have two solutions and , but is not mentioned there, and when , we have two solutions and , but is not mentioned there. Example 5. Let , , for which , and . We choose . Then, Equation (53) becomes (67), and Theorem 5 gives two solutions given in Example 4. Here, we present a theorem which corresponds to Theorem 3 and Theorems 1 and 2 given in Reference [
6].
Theorem 6. Let the condition of Theorem 1 be satisfied. Then, we have solutions of Equation (53), which are expressed by . See Lemma 8 and Remark 4 for the expression .
Remark 12. In this theorem, when , the solutions involving are not mentioned, among the n solutions given by (64) in Theorem 4. When , we have all the solutions by this theorem. Such is the case for Theorem 2 given in Reference [6]. We recall two examples from Reference [
6].
Example 6. We put , , and in Theorem 6. Then, Equation (53) becomes We have three solutions , and . In Reference [6], the last two are not mentioned. Example 7. We put , , , , and in Theorem 6. Then, Equation (53) becomes We have three solutions , and . In Reference [6], the last one is not mentioned. 5. Euler’s Equation Studied in Nonstandard Analysis
We first consider the solution of simple Euler’s equation:
We denote the solution of (
70) by
. For
, we adopt the solution:
Then, we note that they are related by for and .
In nonstandard analysis [
21], we consider
in place of (
70), where
is an infinitesimal number. When
, the solution of (
72) is
where
is Euler’s constant. From this, we have
When
, (
71) is expressed by
The leading term of
of this expression gives (
75), since
.
Lemma 2 shows that when the solution of Equation (
70) is
, we have
m solutions of (
77), which are given by
When
, in place of (
77), we consider
We now obtain the solutions:
where
is given by (
75).
Theorem 7. Let the condition in Theorem 1 be satisfied. Then, we have n solutions of Equation (7), which are classified into series. (i) If ,is a solution in the kth series, and if ,are also solutions in the kth series. (ii) If ,is a solution in the kth series, and if ,are also solutions in the kth series. Proof. Every solution given in the theorem is confirmed to be a solution of Equation (
13), which represents Equation (7). □
5.1. AC-Laplace Transform of Euler’s Equation
In Reference [
18,
19], the AC-Laplace transform, which is an analytic continuation of the Laplace transform, is introduced. The AC-Laplace transform of
given by (
71) is defined by
for
. When
,
is not defined. In
Section 5, we consider
expressed by (
76), in its place. Now, the AC-Laplace transform of this function is given by
.
In place of (
77), we have
Its solutions are given by
When
, in place of (
79), we consider
We now obtain the solutions:
Theorem 8. Let the condition in Theorem 1 be satisfied. Then, the Laplace transform of a solution of Equation (7) satisfiesand we have n solutions of Equation (89), which are classified into series. (i) If ,is a solution in the kth series, and if ,are also solutions in the kth series.(ii)
If ,is a solution in the kth series, and if ,are also solutions in the kth series. We obtain n solutions of Equation (7) by the inverse Laplace transform of the n solutions of Equation (89). Remark 13. In Reference [22], Ghil and Kim adopt that the inverse Laplace transform of for gives , which is justified by the present study, where we obtain solution and the Laplace transform for , where C is a constant, with the aid of nonstandard analysis.