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.
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 [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
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
for, where is the gamma function.
In [4], we take up Kummer’s DE as an example, which is
where are constants. If, one of the solutions given in [7,8] is
where for andand. The other solution is
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;
where are constants.
Lemma 1 For,
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
We then define the Riemann-Liouville fD, , of order, by
if it exists, where, and for.
For, we have
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
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
Let, and let be such a regular distribution that is continuous and differentiable on
, for every. Then is defined by
Let, , and let
be continuous and differentiable on for every. Then
When is a regular distribution, is defined for all.
Lemma 3 For, the index law:
is valid for every.
Dirac’s delta function is the distribution defined by.
Let for be defined by
Lemma 4 If,
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
Then and are expressed as
where
Because of (2.11), we have
Lemma 5 Let. Then
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
2.3. From <img src="5-7401796\75085d4e-0bc4-461e-bcc4-ae30df0a8bc0.jpg" /> to <img src="5-7401796\45955314-3da5-4c2e-b702-a4d13ae872cf.jpg" /> and Vice Versa
Lemma 7 Let, satisfy. Then
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
Lemma 8 Let, satisfy. Then
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
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
if satisfies.
Proof By (2.13) and (2.16), we have
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:
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
where
3.2. Solution Via Operational Calculus
By using (2.14) and (2.19), we express (3.2) as
where
In order to solve the Equation (3.4) for
we solve the following equation for function of real variable:
Lemma 11 The complementary solution (C-solution) of equation (3.7) is given by, where is an arbitrary constant and
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
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
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
By (3.5) and (3.6), , and are
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:
where
B(x) is now expressed as.
By using (3.8), we obtain
where for and are the binomial coefficients.
The C-solution of (3.2) is given by
If, we obtain a C-solution of (4.1), by using Lemma 9:
Remark 1 In Introduction, Kummer’s DE is given by (1.4). It is equal to (4.1) for, , and. In this case,
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]:
where
Lemma 13 When, defined by (4.11) is expressed as
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
Theorem 1 Let, , and. Then we have a P-solution of (4.1), given by
where
Proof Applying Lemma 9 to (4.13), we obtain
By using (4.12) in (4.16), we obtain (4.14) with (4.15).
We note that is expressed as
4.4. Complementary Solution of (4.1)
By (4.3) and (4.5),. When
and, the P-solution of (4.7) is given by
By using (4.14) for, if, we obtain a C-solution of (4.1):
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
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
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 <img src="5-7401796\63a0a436-070e-44df-8547-740d57dd918b.jpg" />
In this section, we consider the case of, ,
, andThen the Equation (3.1) to be solved is
Now (3.5) and (3.6) are expressed as
where.
5.1. Complementary Solution of (3.7)
By using (5.2), is expressed as
where
By (3.8), the C-solution of (3.7) is given by
5.2. Complementary Solution of (3.2) and (5.1)
The C-solution of (3.2) is given by
If, by applying Lemma 9 to this, we obtain the C-solution of (5.1):
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:
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
where
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
REFERENCESAppendix A: Polynomial Form of P-Solution of (4.1)
Let and. Then (4.15) gives
where
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):
Theorem 4 Let, , and for. Then we have the polynomial form of P-solution of (4.1):
Appendix B: Polynomial Form of P-Solution of Hermite DE
We now consider the inhomogeneous Hermite DE given by
for and. We put and. Then the equation for is given by
This is Laplace’s DE (4.1) with parameters
and the inhomogeneous term.
Theorem 5 Let, , and,. Then we have the polynomial form of Psolution of (B.2):
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):
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):
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):
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
where
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):
ReferencesK. 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.
and 
for 
are constants. His discussion is based on Mikusiński’s operational calculus [
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
.
for
, to denote the least integer that is not less than
.
only for such a locally integrable function 
on 
that
is finite. Practically, we adopt Condition B in
and 
are expressed as a linear combination of 
for
.
is Heaviside’s step function, and when 
is defined on
, 
is assumed to be equal to 
when 
and to 
when
. 
is defined by
, where 
is the gamma function.
are constants. If
, one of the solutions given in [7,8] is
for 
and
and
. The other solution is
, we obtain both of the solutions. But when
, (1.6) does not satisfy Condition IB and we could not get it.
, 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
.
and 
are expressed as a linear combination of 
for
, where 
is a set of 
for some
.
. For (1.4), we obtain both solutions (1.5) and (1.6) if
.
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 [
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
.
as follows;
are constants.
,
, we have
. 
be locally integrable on
. We then define the Riemann-Liouville fractional integral, 
, of order 
by
, of order
, by
, and 
for
.
, we have
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 [
to represent
, and hence we accept the following lemma.
,
.
defined by (2.1), we note that
.
. 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 of [
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
, 
, and let
for every
. Then
is a regular distribution, 
is defined for all
.
, the index law:
.
is the distribution defined by
.
for 
be defined by
,
in (2.7) and using (2.11) and (2.5), we obtain
to this and using (2.9) and (2.5), we obtain (2.12). 
expressed by (2.1), we define 
by
and 
are expressed as
. Then
is taken regarding 
as a variable.
is given in Appendix B of [
, 
, by Lemmas 4 and 1,
be expressed as a linear combination of 
for
. Then
, 
satisfy
. Then
and 
by
, and
, respectively, by using (2.2) and (2.17). 
, 
satisfy
. Then
is expressed as a linear combination of 
for
, where 
is a set of
, for some
.
is expressed as (2.13) with 
replaced by
.
satisfy Condition B. Then the corresponding 
is obtained from
, by
replaced by
.
and 
be given by (2.13) and (2.1), respectively. Then 
and 
are related by
satisfies
.
and noting that the first term on the righthand side is then equal to (2.23), we obtain (2.27). 
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 
be the particular solution (P-solution) of (3.7), when the inhomogeneous term is 
for
. Then
is any constant.
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.
of (3.7), 
is obtained by substituting 
by
. Then we confirm that (3.4) is satisfied by that 
operated to
.
is expanded into Neumann series [
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.
by (2.14), and
, 
and 
by (3.5) and (3.6).
by (3.8). The C-solution of (3.2) is given by
, the C-solution of (3.1) is obtained from this with the aid of Lemma 9.
or
, we obtain 
given by (3.9).
and
, the solution of (3.2) is given by
are constants. The C-solution of (3.1) is then obtained from this with the aid of Lemma 9.
, the P-solution of (3.2)
and 
are constants. The P-solution of (3.1) with inhomogeneous term
. 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.
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 of [
, and
. Then (3.1) reduces to
, 
and 
are
.
of (3.7) by using (3.8), we express 
as follows:
.
for 
and 
are the binomial coefficients.
, we obtain a C-solution of (4.1), by using Lemma 9:
, 
, 
and
. In this case,
agrees with (1.6), which is one of the C-solutions of Kummer’s DE given in [7,8].
for
.
, the P-solution of (3.7) is given by (3.9). By using (4.2) and (4.6), the following result is obtained in [
, 
defined by (4.11) is expressed as
for
, the P-solution of (3.2) is given by
, 
, and
. Then we have a P-solution 
of (4.1), given by
is expressed as
. When
and
, the P-solution of (4.7) is given by
, if
, we obtain a C-solution of (4.1):
, it can be expressed as
, 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 
and
respectively.
and 
are given in (4.9), and
, the set of (4.8) and (4.20) agrees with the set of (4.5) and (4.6).
, 
,
, and
Then the Equation (3.1) to be solved is
.
is expressed as
of (3.7) is given by
, by applying Lemma 9 to this, we obtain the C-solution of (5.1):
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
:
is defined by (4.11) and is given by
.
for
, the P-solution of (3.2) is
. By applying Lemma 9 to this, we obtain the following theorem.
, 
and 
. Then we have a P-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).
. Then the C-solution of (5.1) is given by
and
. Then (4.15) gives
, 
, and
. Then we have the polynomial form of P-solution of (4.1):
, 
, 
and 
for
. Then we have the polynomial form of P-solution of (4.1):
and
. We put 
and
. Then the equation for 
is given by
.
, 
, and
,
. Then we have the polynomial form of Psolution of (B.2):
, 
, and
. By Theorem 3, we obtain this result. 
, 
, and
,
. Then we have the polynomial form of Psolution of (B.2):
, 
, and
. By Theorem 4, we obtain this result. 
, 
, and
,
. Then we have the polynomial form of Psolution of (B.2):
, 
, and
. By Theorem 4, we obtain this result. 
, 
, and
,
. Then we have the polynomial form of Psolution of (B.2):
, 
, and
. By Theorem 3, we obtain this result. 
and
. Then (5.11) gives
, 
, 
and 
for
. Then we have the polynomial form of P-solution of (5.1):