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Article

# On Discrete Fractional Solutions of Non-Fuchsian Differential Equations

by 1,*,†, 1,† and
1
Department of Mathematics, Firat University, 23119 Elazig, Turkey
2
Department of Computer Engineering, Istanbul Gelisim University, 34315 Istanbul, Turkey
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2018, 6(12), 308; https://doi.org/10.3390/math6120308
Received: 15 October 2018 / Revised: 28 November 2018 / Accepted: 5 December 2018 / Published: 7 December 2018

## Abstract

In this article, we obtain new fractional solutions of the general class of non-Fuchsian differential equations by using discrete fractional nabla operator $∇ η ( 0 < η < 1 )$ . This operator is applied to homogeneous and nonhomogeneous linear ordinary differential equations. Thus, we obtain new solutions in fractional forms by a newly developed method.

## 1. Introduction

The history of fractional mathematics dates back to Leibniz (1695). This field of work is rapidly increasing and, nowadays, it has many applications in science and engineering [1,2,3,4]. Heat transfer, diffusion and Schrödinger equation are some fields where fractional analysis is used.
A similar theory was started for discrete fractional analysis and the definition and properties of fractional sums and differences theory were developed. Many articles related to this topic have appeared lately [5,6,7,8,9,10,11,12,13,14,15,16,17,18].
In 1956 [5], differences of fractional order was first introduced by Kuttner. Difference of fractional order has attracted more interest in recent years.
Diaz and Osler [6], defined the notion of fractional difference as follows
$Δ ς Φ ( t ) = ∑ k = 0 ∞ ( − 1 ) k ς k Φ ( t + ς − k )$
where $ς$ is any real number.
Granger and Joyeux [19] and Hosking [20], defined notion of the fractional difference as follows
$∇ ς Φ ( t ) = ( 1 − q ) ς Φ ( t ) ∑ k = 0 ∞ ( − 1 ) k Γ ( ς + 1 ) Γ ( k + 1 ) Γ ( ς − k + 1 ) q k Φ ( t ) = ∑ k = 0 ∞ ( − 1 ) ς k Φ ( t − k ) ,$
where $ς$ is any real number and $q Φ t = Φ t − 1$ is the shift operator. Gray and Zhang [21], Acar and Atici [10] studied on a new definition and characteristics of the fractional difference.

## 2. Preliminary and Properties

In this section, we first present sufficient fundamental definitions and formulas so that the article is self-contained.
The rising factorial power $t m ¯$ (t to the m rising, $m ∈ N$) is defined by
$t m ¯ = t t + 1 t + 2 … t + m − 1 , t 0 ¯ = 1 .$
Let $σ$ be any real number. Then “t to the $σ$ rising” is defined to be
$t σ ¯ = Γ t + σ Γ t , t ∈ R − … , − 2 , − 1 , 0 , 0 σ ¯ = 0 .$
Also, the ∇ operator of Equation (1) is given by
$∇ ( t σ ¯ ) = ∇ Γ ( t + σ ) Γ ( t ) = Γ ( t + σ ) Γ ( t ) − Γ ( t − 1 + σ ) Γ ( t − 1 ) = ( t − 1 + σ ) Γ ( t − 1 + σ ) Γ ( t ) − Γ ( t − 1 + σ ) Γ ( t − 1 ) = Γ ( t − 1 + σ ) Γ ( t − 1 ) t − 1 + σ t − 1 − 1 = σ Γ ( t − 1 + σ ) Γ ( t ) = σ t σ − 1 ¯$
where $∇ u t = u t − u t − 1 .$
Let $η ∈ R +$ such that $m − 1 ≤ η < m ,$ $m ∈ N .$ The $η$th-order fractional nabla sum of g is given by
$∇ b − η g t = 1 Γ η ∑ s = b t t − δ s η − 1 ¯ g s ,$
where $t ∈ N b = b + N 0 = b , b + 1 , b + 2 , … ,$ $b ∈ R ,$ $δ s = s − 1$ is backward jump operator. Also, we define the trivial sum by $∇ b − 0 g t = g t$ for $t ∈ N b .$
The $η$th-order Riemann-Liouville type nabla fractional difference of g is defined by
$∇ b η g t = ∇ m ∇ − m − η g t = ∇ m 1 Γ ( m − η ) ∑ s = b t ( t − δ ( s ) ) m − η − 1 ¯ g ( s ) ,$
where $g : N b + ⟶ R$ [10].
Theorem 1
([16]). Let f and $g : N 0 + ⟶ R ,$ $γ , ϕ > 0 .$ Then
$∇ − γ ∇ − ϕ f t = ∇ − γ + ϕ f t = ∇ − ϕ ∇ − γ f t ,$
$∇ γ h f t + k g t = h ∇ γ f t + k ∇ γ g t , h , k ∈ R$
$∇ ∇ − γ f t = ∇ − γ − 1 f t ,$
$∇ − γ ∇ f t = ∇ 1 − γ f t − t + γ − 2 t − 1 f 0 .$
Lemma 1
(Power Rule [10]). Let $v > 0$ and η be two real numbers so that $Γ η + 1 Γ η + v + 1$ is defined. Then,
$∇ b − v t − b + 1 η ¯ = Γ η + 1 Γ η + v + 1 t − b + 1 η + v ¯ , t ∈ N b .$
Lemma 2
(Leibniz Rule [10]). For any $η > 0 ,$ ηth-order fractional difference of the product $f g$ is given in this form
$∇ 0 η f g t = ∑ m = 0 t η m ∇ 0 η − m f t − m ∇ m g t ,$
where
$η m = Γ η + 1 Γ m + 1 Γ η − m + 1$
and $f ,$ g are defined on $N 0 ,$ and t is a positive integer.
Lemma 3 (Index Law).
Let $g t$ is single-valued and analytic. Then
$g γ η = g γ + η = g η γ g γ ≠ 0 ; g η ≠ 0 ; γ , η ∈ R ; t ∈ C .$

## 3. Main Results

We start by considering the following differential equation
$1 + ℓ x d 2 y d x 2 + a + b x 1 + ℓ x d y d x + c + d x + ε x 2 1 + ℓ x y x = ψ$
where $ψ$ is a given function, $x ∈ C \ 0 , − ℓ ,$ and $a , b , c , d , ε$ and are parameters.
Let
$y x = x τ e κ x w x$
so that
$d y d x = x τ − 1 e κ x x d w d x + τ + κ x w x$
and
$d 2 y d x 2 = x τ − 2 e κ x x 2 d 2 w d x 2 + 2 τ + κ x x d w d x + κ 2 x 2 + 2 τ κ x + τ τ − 1 w x .$
By substituting (12)–(14) into the (11), we have
$x 2 x + ℓ d 2 w d x 2 + 2 τ + b ℓ + 2 τ + 2 κ ℓ + b x + 2 κ + a x 2 x d w d x + τ τ + b − 1 + ε ℓ + τ τ + 2 κ ℓ + b − 1 + κ b ℓ + ε x + κ 2 ℓ + 2 τ + b κ + τ a + d x 2 + κ 2 + κ a + c x 3 w x = x 3 − τ e − κ x ψ x , x ∈ C \ 0 , − ℓ .$
Finally, we find it to be suitable to restrict the different parameters involved in (11) and (15) by means of the following equalities;
$2 τ + b = 0 , τ τ + b − 1 + ε = 0 , κ 2 + κ a + c = 0 ,$
so that
$τ = − 1 2 b = − 1 ± 1 + 4 ε 2 ,$
and
$κ = − a ± a 2 − 4 c 2 .$
Under the parametric constraints given by (16), the Equation (15) will immediately decrease to a simpler form
$x + ℓ d 2 w d x 2 + 2 κ ℓ + 2 κ + a x d w d x + κ 2 ℓ + τ a + d w x = x 1 − τ e − κ x ψ x$
where $τ$ and $κ$ are given by (17) and (18), respectively.
Theorem 2.
Let $w , ψ ∈ w , ψ : 0 ≠ w η x , ψ η x < ∞ ,$ and $η ∈ R .$ Then the nonhomogeneous linear differential equation
$w 2 α x + β + w 1 γ x + ν α + δ + ν γ w x = ψ x , x ≠ − β α , α ≠ 0 , ν ∈ R$
has particular solutions in the below forms:
$w I x = ψ − q − 1 ν α x + β δ α − γ β − α 2 / α 2 e γ α x − 1 α x + β γ β − δ α / α 2 e − γ α x − 1 + q − 1 ν$
$w II x = α x + β − γ x + δ α − ν + 1 × ψ α x + β γ x + δ α + ν − 1 q − 1 ν α x + β − δ α + γ β α 2 + 1 e − γ α x − 1 × α x + β δ α − γ β α 2 − 2 e γ α x − 1 − q − 1 ν$
where $w n = d n w d x n$ $( n = 0 , 1 , 2 ) ,$ $w 0 = w = w ( x ) ,$ $α , β , γ , ν , δ$ are given constants.
Proof.
For $ψ x ≠ 0$,
(i) When we operate $∇ η$ to the both sides of (20), we have
$∇ η w 2 α x + β + ∇ η w 1 γ x + ν α + δ + ∇ η w ν γ = ∇ η ψ$
by using (9) and (10) we obtain
$∇ η w 2 α x + β = w 2 + η α x + β + q η α w 1 + η$
$∇ η w 1 γ x + ν α + δ = w 1 + η γ x + ν α + δ + q η γ w η$
where q is a shift operator which is defined by $w ( t − 1 ) = q w ( t )$. By substituting (24), (25) into the (23), we have
$w 2 + η α x + β + η q + ν α + γ x + δ w 1 + η + q η γ + ν γ w η = ψ η .$
We choose $η$ such that
$q η γ + ν γ = 0 , η = − q − 1 ν .$
Then we obtain
$w 2 − q − 1 ν α x + β + w 1 − q − 1 ν γ x + δ = ψ − q − 1 ν$
from (26).
Therefore, setting
$w 1 − q − 1 ν = u w = u − 1 + q − 1 ν$
we have
$u 1 + u γ x + δ α x + β = ψ − q − 1 ν α x + β − 1$
from (27). A particular solution of a first order ordinary differential Equation (29):
$u = ψ − q − 1 ν α x + β δ α − γ β − α 2 / α 2 e γ α x − 1 α x + β γ β − δ α / α 2 e − γ α x .$
Thus we obtain the solution (21) from (28) and (30).
(ii) Set
$w = α x + β σ W x$
The first and second derivations of (31) are acquired as follows:
$w 1 = σ α x + β σ − 1 α W + α x + β σ W 1$
$w 2 = σ σ − 1 α x + β σ − 2 α 2 W + 2 σ α x + β σ − 1 α W 1 + α x + β σ W 2 .$
Substitute (31)–(33) into (20), we have
$W 2 α x + β + W 1 2 σ α + γ x + ν α + δ + W α 2 σ σ − 1 + α σ γ x + ν α + δ α x + β + ν γ = ψ α x + β − σ .$
Here, we choose $σ$ such that
$α σ α σ − α + γ x + ν α + δ = 0$
that is $σ = 0 ,$ $σ = − γ x + δ α − ν + 1 .$
In the case $σ = 0 ,$ we have the same results as i.
Let $σ = − γ x + δ α − ν + 1 .$ From (31) and (34)
$w = α x + β − γ x + δ α − ν + 1 W$
and
$W 2 α x + β + W 1 α 2 − ν − δ − γ x + ν γ W = ψ α x + β γ x + δ α + ν − 1$
respectively.
Applying the operator $∇ η$ to both members of (36), we have
$W 2 + η α x + β + W 1 + η α 2 − ν + η q − δ − γ x + W η − γ η q + ν γ = ψ α x + β γ x + δ α + ν − 1 η .$
Choose $η$ such that
$− γ η q + ν γ = 0 , η = q − 1 ν$
we have then
$W 2 + q − 1 ν α x + β + W 1 + q − 1 ν 2 α − γ x + δ = ψ α x + β γ x + δ α + ν − 1 q − 1 ν$
from (37).
Therefore, setting
$W 1 + q − 1 ν = ϑ , W = ϑ − 1 − q − 1 ν$
we have
$ϑ 1 + ϑ 2 α α x + β − γ x + δ α x + β = ψ α x + β γ x + δ α + ν − 1 q − 1 ν α x + β − 1$
from (38). A particular solution of ordinary differential Equation (40) is given by
$ϑ = ψ α x + β γ x + δ α + ν − 1 q − 1 ν α x + β − δ α + γ β α 2 + 1 e − γ α x − 1 α x + β δ α − γ β α 2 − 2 e γ α x .$
Thus we obtain the solution (22) from (35), (39) and (41). □
Furthermore, we can prove for the homogen part such that the homogeneous linear ordinary differential equation
$w 2 α x + β + w 1 γ x + ν α + δ + ν γ w x = 0 , x ≠ − β α , α ≠ 0 , ν ∈ R$
has solutions of the forms
$w I x = h α x + β γ β − δ α / α 2 e − γ α x − 1 + q − 1 ν ,$
$w II x = h α x + β − γ x + δ α − ν + 1 α x + β δ α − γ β α 2 − 2 e γ α x − 1 − q − 1 ν$
where h is an arbitrary constant.
Now, in Theorem 1, we further set
$α = 1 , β = ℓ , γ = 2 κ + a , δ = 2 κ ℓ − ν , ν = κ 2 ℓ + τ a + d 2 κ + a$
and let
$ψ x → x 1 − τ e − κ x ψ x .$
We thus find that the nonhomogeneous differential Equation (19) has a particular solution given by
$w I x = x 1 − τ e − κ x ψ x − q − 1 ν x + ℓ − ν − a ℓ − 1 e 2 κ + a x − 1 x + ℓ ν + a ℓ e − 2 κ + a x − 1 + q − 1 ν ,$
$w II x = x + ℓ − 2 κ + a x − 2 κ ℓ + 1 × x 1 − τ e − κ x ψ x x + ℓ 2 κ + a x + 2 κ ℓ − 1 q − 1 ν x + ℓ ν + a ℓ + 1 e − 2 κ + a x − 1 × x + ℓ − ν − a ℓ − 2 e 2 κ + a x − 1 − q − 1 ν$
and that the corresponding homogeneous linear differential equation
$x + ℓ d 2 w d x 2 + 2 κ ℓ + 2 κ + a x d w d x + κ 2 ℓ + τ a + d w x = 0$
has solutions of the forms
$w I x = h x + ℓ ν + a ℓ e − 2 κ + a x − 1 + q − 1 ν ,$
$w II x = h x + ℓ − 2 κ + a x − 2 κ ℓ + 1 x + ℓ − ν − a ℓ − 2 e 2 κ + a x − 1 − q − 1 ν$
where h is an arbitrary constant.
Therefore, the linear differential Equation (11), has a particular solution in the following forms
$y I x = x τ e κ x w x = x τ e κ x x 1 − τ e − κ x ψ x − q − 1 ν x + ℓ − ν − a ℓ − 1 e 2 κ + a x − 1 x + ℓ ν + a ℓ e − 2 κ + a x − 1 + q − 1 ν x ∈ C \ 0 , − ℓ , v ∈ R$
and
$y II x = x τ e κ x x + ℓ − 2 κ + a x − 2 κ ℓ + 1 × x 1 − τ e − κ x ψ x x + ℓ 2 κ + a x + 2 κ ℓ − 1 q − 1 ν x + ℓ ν + a ℓ + 1 e − 2 κ + a x − 1 × x + ℓ − ν − a ℓ − 2 e 2 κ + a x − 1 − q − 1 ν$
and that the corresponding homogeneous linear differential equation
$1 + ℓ x d 2 y d x 2 + a + b x 1 + ℓ x d y d x + c + d x + ε x 2 1 + ℓ x y x = 0$
has solutions given by
$y I x = h x τ e κ x x + ℓ ν + a ℓ e − 2 κ + a x − 1 + q − 1 ν ,$
$y II x = h x τ e κ x x + ℓ − 2 κ + a x − 2 κ ℓ + 1 x + ℓ − ν − a ℓ − 2 e 2 κ + a x − 1 − q − 1 ν$
where $h ∈ R$, the parameters $τ ,$ $κ$ and $ν$ are given by (17), (18) and (45).
Remark 1.
First of all, when $ℓ = 0 ,$ the differential Equation (11) reduces to the following version of the Tricomi equation:
$d 2 y d x 2 + a + b x d y d x + c + d x + ε x 2 y = ψ x .$
By setting
$ℓ = 0 , a = b = 0 , c = k 2 , d = n , ε = 1 4 − m 2 ,$
in the Equation (11), we readily obtain the following Hydrogen atom equation:
$d 2 y d x 2 + k 2 + n x + 1 4 − m 2 x 2 y = ψ x .$
Example 1.
In the case $α = 1 , β = γ = v = 0 , δ = 2$ and $ψ ( x ) = x$, we have
$w ( x ) + 2 x w 1 = 1 ( x ≠ 0 )$
from (20). Solution of Equation (56) is obtained as
$w ( x ) = [ x 2 ] − 1 x − 2 = x 3 3 x − 2 = 1 6 x 2$
by using (21). The function obtained in (57) provide the Equation (56).

## 4. Conclusions

In this article, we use the discrete fractional operator for the homogeneous and non-homogeneous non-Fuchsian differential equations. This solution of the equation has not been obtained before by using ∇ operator. We can obtain particular solutions of the same type linear singular ordinary and partial differential equations by using the discrete fractional nabla operator in future works.

## Author Contributions

These authors contributed equally to this work.

## Funding

This research received no external funding.

## Acknowledgments

The authors gratefully thank the anonymous the editor and referees for valuable suggestions which improved the manuscript.

## Conflicts of Interest

The authors declare no conflict of interest.

## References

1. Miller, K.S.; Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations; John Wiley and Sons, Inc.: New York, NY, USA, 1993. [Google Scholar]
2. Oldham, K.; Spanier, J. The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order; Dover Publications, Inc.: Mineola, NY, USA, 2002. [Google Scholar]
3. Podlubny, I. Fractional Differential Equations; Academic Press: New York, NY, USA, 1999. [Google Scholar]
4. Baleanu, D.; Guven, Z.B.; Machado, J.A.T. New Trends in Nanotechnology and Fractional Calculus Applications; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
5. Kuttner, B. On differences of fractional order. Proc. Lond. Math. Soc. 1957, 3, 453–466. [Google Scholar] [CrossRef]
6. Diaz, J.B.; Osler, T.J. Differences of Fractional Order. Am. Math. Soc. 1974, 28, 185–202. [Google Scholar] [CrossRef]
7. Atici, F.M.; Eloe, P.W. A transform method in discrete fractional calculus. Int. J. Differ. Equ. 2007, 2, 165–176. [Google Scholar]
8. Atici, F.M.; Eloe, P.W. Discrete fractional calculus with the nabla operator. Electr. J. Qual. Theory Differ. Equ. 2009, 3, 1–12. [Google Scholar]
9. Lin, S.D.; Tu, S.T.; Srivastava, H.M. A Unified Presentation of Certain Families of Non-Fuchsian Differential Equations via Fractional Calculus Operators. Comput. Math. Appl. 2003, 45, 1861–1870. [Google Scholar] [CrossRef]
10. Acar, N.; Atici, F.M. Exponential functions of discrete fractional calculus. Appl. Anal. Discrete Math. 2013, 7, 343–353. [Google Scholar] [CrossRef]
11. Anastassiou, G.A. Right nabla discrete fractional calculus. Int. J. Differ. Equ. 2011, 6, 91–104. [Google Scholar]
12. Holm, M. Sum and Difference Compositions in Discrete Fractional Calculus. COBO Math. J. 2011, 13, 153–184. [Google Scholar] [CrossRef]
13. Yilmazer, R.; Ozturk, O. On Nabla Discrete Fractional Calculus Operator for a Modified Bessel Equation. Therm. Sci. 2018, 22, S203–S209. [Google Scholar] [CrossRef]
14. Yilmazer, R.; Inc, M.; Tchier, F.; Baleanu, D. Particular Solutions of the Confluent Hypergeometric Differential Equation by Using the Nabla Fractional Calculus Operator. Entropy 2016, 18, 49. [Google Scholar] [CrossRef]
15. Atici, F.M.; Sengül, S. Modeling with fractional difference equations. J. Math. Anal. Appl. 2010, 369, 1–9. [Google Scholar] [CrossRef]
16. Mohan, J.J. Solutions of perturbed nonlinear nabla fractional difference equations. Novi Sad. J. Math. 2013, 43, 125–138. [Google Scholar]
17. Mohan, J.J. Analysis of nonlinear fractional nabla difference equations. Int. J. Anal. Appl. 2015, 7, 79–95. [Google Scholar]
18. Yilmazer, R. N-fractional calculus operator Nμ method to a modified hydrogen atom equation. Math. Commun. 2010, 15, 489–501. [Google Scholar]
19. Granger, C.W.J.; Joyeux, R. An introduction to long-memory time series models and fractional differencing. J. Time Ser. Anal. 1980, 1, 15–29. [Google Scholar] [CrossRef]
20. Hosking, J.R.M. Fractional differencing. Biometrika 1981, 68, 165–176. [Google Scholar] [CrossRef]
21. Gray, H.L.; Zhang, N. On a New Definition of the Fractional Difference. Math. Comput. 1988, 50, 513–529. [Google Scholar] [CrossRef]