Solution of Inhomogeneous Differential Equations with Polynomial Coefﬁcients in Terms of the Green’s Function, in Nonstandard Analysis

: Discussions are presented by Morita and Sato on the problem of obtaining the particular solution of an inhomogeneous differential equation with polynomial coefﬁcients in terms of the Green’s function. In a paper, the problem is treated in distribution theory, and in another paper, the formulation is given on the basis of nonstandard analysis, where fractional derivative of degree, which is a complex number added by an inﬁnitesimal number, is used. In the present paper, a simple recipe based on nonstandard analysis, which is closely related with distribution theory, is presented, where in place of Heaviside’s step function H ( t ) and Dirac’s delta function δ ( t ) in distribution theory, functions H (cid:101) ( t ) : = 1 Γ ( 1 + (cid:101) ) t (cid:101) H ( t ) and δ (cid:101) ( t ) : = ddt H (cid:101) ( t ) = 1 Γ ( (cid:101) ) t (cid:101) − 1 H ( t ) for a positive inﬁnitesimal number (cid:101) , are used. As an example, it is applied to Kummer’s differential equation.


Introduction
In the present paper, we treat the problem of obtaining the particular solutions of a differential equation with polynomial coefficients in terms of the Green's function.
In a preceding paper [1], this problem is studied in the framework of distribution theory, where the method is applied to Kummer's and the hypergeometric differential equation.In another paper [2], this problem is studied in the framework of nonstandard analysis, where a recipe of solution of the present problem is presented, and it is applied to a simple fractional and a first-order ordinary differential equation.
In the present paper, we present a compact recipe based on nonstandard analysis, which is obtained by revising the one given in [2].As an example, it is applied to Kummer's differential equation.
The presentation in this paper follows those in [1,2], in Introduction and in many descriptions in the following sections.
We consider a fractional differential equation, which takes the form: where n ∈ Z >−1 , t ∈ R, a l (t) for l ∈ Z [0,n] are polynomials of t, ρ l ∈ C for l ∈ Z [0,n] satisfy Re ρ 0 > Re ρ 1 ≥ • • • ≥ Re ρ n and Re ρ 0 > 0. We use Heaviside's step function H(t), which is equal to 1 if t > 0, and to 0 if t ≤ 0.Here R D Definition 1.Let t ∈ R, τ ∈ R, u 0 (t) be locally integrable on R >τ , u(t) = u 0 (t)H(t − τ), λ ∈ C + , n ∈ Z >−1 and ρ = n − λ.Then R D −λ t u(t) is the Riemann-Liouville fractional integral defined by and R D −λ t u(t) = 0 for t ≤ τ, where Γ(λ) is the gamma function, R D ρ t u(t) = R D n−λ t u(t) is the Riemann-Liouville fractional derivative defined by when n ≥ Re λ, and R D n t u(t) = d n dt n u 0 (t) Here Z, R and C are the sets of all integers, all real numbers and all complex numbers, respectively, and In accordance with Definition 1, when u 0 for ν ∈ C\Z <1 and τ ∈ R.Here R D t is used in place of usually used notation τ D R , in order to show that the variable is t.
Then, the index law: R D t g ν (t) does not always hold.An example is given in the book [4] (p. 108); see also [5] (p.48).
In [1,6], discussions are made of an ordinary differential equation, which is expressed by (1) for ρ l = n − l, in terms of distribution theory, and with the aid of the analytic continuation of Laplace transform, respectively.In those papers, solutions are given of differential equations with an inhomogeneous term f (t), which satisfies one of the following three conditions.

Green's Function in Distribution Theory
In a recent paper [5], the solution of Euler's differential equation in distribution theory is compared with the solution in nonstandard analysis.In distribution theory [1,[7][8][9], we use distribution H(t), which corresponds to function H(t), differential operator D and distribution δ(t) = D H(t), which is called Dirac's delta function.
As a consequence, when ρ = −ν ∈ C and n ∈ Z >0 , we have In place of (4), for ρ 1 ∈ C and ρ ∈ C, we now have Then, the index law: always holds.
In [1], the following theorem is given.
Theorem 1.Let f (t) satisfy Condition 1 (i) and G 0 (t, τ) be the one given in Lemma 1. Then u f (t) given by is a particular solution of Equation (1).
Proof.By using Equations and (9), we have By taking the derivative of the first and the last member in this equation with respect to t, we confirm that Equation ( 1) is satisfied by u(t) = u f (t).

Preliminaries on Nonstandard Analysis
In the present paper, we use nonstandard analysis [10], where infinitesimal numbers are used.We denote the set of all infinitesimal real numbers by R 0 .We also use and N ∈ Z >0 , then < 1 N .We use R ns , which has subsets R and R 0 .If x ∈ R ns and x / ∈ R, x is expressed as x 1 + by x 1 ∈ R and ∈ R 0 , where x 1 may be 0 ∈ R. Equation x y for x ∈ R ns and y ∈ R ns , is used, when x − y ∈ R 0 .We denote the set of all infinitesimal complex numbers by C 0 , which is the set of complex numbers z which satisfy |Re z| + |Im z|∈ R 0 .We use C ns , which has subsets C and C 0 .If z ∈ C ns and z / ∈ C, z is expressed as z 1 + by z 1 ∈ C and ∈ C 0 , where z 1 may be 0 ∈ C. Remark 5.In nonstandard analysis [10], in addition to infinitesimal numbers, we use unlimited numbers, which are often called infinite numbers.In the present paper, we do not use them, but if we use them, we have to consider sets R ∞ and C ∞ such that if ω ∈ R ∞ , there exists ∈ R 0 satisfying ω = 1 , and if ω ∈ C ∞ , there exists ∈ C 0 satisfying ω = 1 , and then In place of (4), we now use for all ρ ∈ C and ν ∈ C, where ∈ R 0 >0 .
Remark 6.When ∈ R 0 or ∈ C 0 , we often ignore terms of O( ) compared with a term of O( 0 ).
In the present study in nonstandard analysis, ∈ R 0 >0 is used, and H(t) and δ(t) = D H(t), respectively, are replaced by which tends to H(t) in the limit → 0, and by , and we have

Summary of the Following Sections
In Section 2, a recipe of solution of Equation ( 1), in nonstandard analysis, is presented.We there consider the solution of the following equation for ũ(t): where ∈ R 0 >0 and pn, (t, Here, the inhomogeneous terms f (t) and f (t) are assumed to satisfy one of the following four conditions.
f β,0 (t) is locally integrable on R >0 , and c β, is a constant.
H(t). (iv) f (t) and f (t) are expressed as follows: respectively, where c l ∈ C are constants, does not always hold, and when Condition 2 (iii) is satisfied, R D t fβ (t) = 0.
In Sections 3 and 4, full expressions of the Green's functions and the solutions, are derived along the recipe given in Section 2, for Kummer's differential equation: where a, b and c are constants satifying a = 0 and b = 0. Section 5 is for Conclusion.In Section 6, a concluding remark is given.

Recipe of Solution of Differential Equation, in Nonstandard Analysis
In obtaining a particular solution of Equation (1) for f (t) satisfying Condition 2 (i), in place of the Green's function defined in Remark 4, we use it defined in the following definition.Definition 2. Let pn, (t, R D t ) be given by Equation (21).Then for ∈ R 0 >0 and τ ∈ R, the Green's function G (t, τ) for Equation (1) satisfies Lemma 5. Let G (t, τ) be defined as in Definition 2, and G 0 (t, τ) := R D t G (t, τ).Then G 0 (t, τ) is a complementary solution of Equation (1) on R >τ , and R D −1 t p n (t, R D t )G 0 (t, τ) = 1 at any value of t satisfying t > τ.
Proof.These are confirmed by applying R D t and R D −1+ t to Equation (25), by noting Lemma 3. Lemma 6.Let ũc (t) be a complementary solution of Equation (20) on R >0 , and u c (t Proof.This is confirmed by replacing ũ(t) and f (t) by ũc (t) and 0 in Equation (20), and then applying R D t to the equation.Theorem 2. Let Condition 2 (i) be satisfied, G (t, τ) and G 0 (t, τ) be given as in Lemma 5. Then ũ f (t) given by is the particular solution of Equation (20) for the term f (t), and u f (t) given by consists of the particular solution for the term f (t) and a complementary solution of Equation (1).
When Condition 2 (ii) is satisfied, we introduce the transformed differential equations for w(t) = R D −β t u(t) and w(t) = R D − t w(t) from Equations ( 1) and (20), respectively, by where are particular solutions of Equations (30) and (20), respectively.
Theorem 4 shows that if f (t) = R D β t δ (t), the particular solution of (20) is given by ũ f (t) = R D β t G β, (t, 0).As a consequence, we have Theorem 5. Let f (t) satisfy Condition 2 (iv), so that it is given by Equation (23).Then the particular solution of Equation (20) is given by

Solution of Kummer's Differential Equation, I
We construct the transformed differential equation of Equation (24), which corresponds to Equation (20).For this purpose, we use the following lemma.
Proof.When m = 0 and ρ = −λ, this is confirmed with the aid of Formula (2), as follows: We prove (38) by mathematical induction.In fact, when (38) holds for a value n ∈ Z >−1 of m, we confirm it to hold even for m = n + 1, by applying d dt to (38).
In the present paper, these equations are proved in Lemmas 11 and 12 given below.Lemma 11.Let K 1 (t) be given by (41).Then G K, (t, 0) and G K,0 (t, 0), given by are a particular solution of Equation (40) for τ = 0, and a complementary solution of Equation (24), respectively.
A proof of the statement for G K, (t, 0) is given in Section 3.1, and the statement for G K,0 (t, 0) is due to Lemma 5. Lemma 12. Let K 2 (t) be given by (42).Then ũc (t) and u c (t), given by are complementary solutions of Equations (39) and (24), respectively.
A proof of the statement for ũc (t) is given in Section 3.1, and the statement for u c (t) is due to Lemma 6.
Then Theorem 2 shows that we have the solutions ũ f (t) and u f (t) of Equations (39) and (24), respectively, which are given by See Lemma 12 for the complementary solutions ũc (t) and u c (t).This result is derived with the aid of the complementary solutions given by Equations (41) and (42), and hence by assuming c / ∈ Z <1 .

Solution of Kummer's Differential Equation, II
We construct the transformed differential equations of Equation ( 24), which appear in Theorems 3-5.Corresponding to Equations ( 29) and (30), we have the following equations for w(t) = R D −β t u(t) and w(t) = R D − t w(t) from Equation (24) satisfying Condition 2 (ii), as follows: Remark 10.In this section, we consider Equations (56) and (57) in place of Equations (24) and (39), respectively, and hence the equations in this section are obtained from the corresponding equations in Section 3, by replacing c by c − β, a by a − β, f by f β , f by fβ , u by w, and ũ by w.They will be given without derivation.
Lemma 14. Lemma 10 and Remark 10 show that if c − β / ∈ Z <1 , there exist two complementary solutions of Equation (56), which are given by In accordance with Definition 3, we define the Green's function for τ ∈ R. The solutions of Equations ( 57), ( 56), (39) and ( 24) are then given with the aid of Theorems 3, 4 and 5, and Lemma 14.In Section 4, formulas are derived with the aid of two complementary solutions given by ( 58) and (59), and hence they hold when c − β / ∈ Z <1 .
With the aid of Remark 11, we have the following lemma for G K,β, (t, τ) for τ > 0.
Corollary 1.Let β = n ∈ Z >−1 , and ũ f (t) be the solution of (39), given by Equation (62).Then u f (t) = R D t ũ f (t) and ũ f (t) are expressed by Equations ( 39) and (24) from them.In Corollary 1, a nonstandard solution, which involves infinitestimal terms, is presented.In [11], an ordinary differential equation is expressed in terms of blocks of classified terms.When the equation is expressed by two blocks of classified terms, the complementary solutions are obtained by using Frobenius' method.In Section 3.1, the Green's function and a complementary solution for Equation (39) are presented by using Frobenius' method.
One of reviewers of this paper asked the author to cite papers [12][13][14], which discuss the solutions of fractional differential equations.When the solutions of the differential equations, which are obtained with the aid of distribution theory, are of interest, the solution by using nonstandard analysis will be useful.

Concluding Remark
In the book of [9], Dirac's delta function δ(t) is introduced as a limit of zero width, of a function which has a single peak at t = 0 and unit area, and is defined as a functional.In the present paper, we study problems in nonstandard analysis, by using a function δ (t) which has an infinitesimal width and unit area.
In a preceding paper [1], the problem of obtaining the particular solution of an inhomogeneous ordinary differential equation, is discussed in terms of distribution theory.In another paper [2], we discussed solution of a fractional and a simple ordinary differential equation, in terms of nonstandard analysis by using two functions δ 1 (t) and δ (t) expressed by two infinitesimal numbers 1 and .In the present paper, we proposed a revised recipe in terms of nonstandard analysis, by using the function δ (t) in place of distribution δ(t) in distribution theory.In the present paper, the recipe is applied only to Kummer's differential equation.The application of the present recipe to other differential equations studied in [1,2], will be given in a separate paper in preparation.
The author desires to have a day when we discuss the merit of using two functions δ 1 (t) and δ (t).