Next Article in Journal
Kempe-Locking Configurations
Next Article in Special Issue
Existence Results of a Coupled System of Caputo Fractional Hahn Difference Equations with Nonlocal Fractional Hahn Integral Boundary Value Conditions
Previous Article in Journal
n-Derivations and (n,m)-Derivations of Lattices
Previous Article in Special Issue
New Numerical Method for Solving Tenth Order Boundary Value Problems
Article

On Discrete Fractional Solutions of Non-Fuchsian Differential Equations

by 1,*,†, 1,† and 2,†
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.
Keywords: discrete fractional calculus; fractional nabla operator; non-Fuchsian equations discrete fractional calculus; fractional nabla operator; non-Fuchsian equations

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 + x d 2 y d x 2 + a + b x 1 + x d y d x + c + d x + ε x 2 1 + x y x = ψ
where ψ is a given function, x C \ 0 , , 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 + d 2 w d x 2 + 2 τ + b + 2 τ + 2 κ + b x + 2 κ + a x 2 x d w d x + τ τ + b 1 + ε + τ τ + 2 κ + b 1 + κ b + ε x + κ 2 + 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 , .
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 + d 2 w d x 2 + 2 κ + 2 κ + a x d w d x + κ 2 + 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 = , γ = 2 κ + a , δ = 2 κ ν , ν = κ 2 + 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 + ν a 1 e 2 κ + a x 1 x + ν + a e 2 κ + a x 1 + q 1 ν ,
w II x = x + 2 κ + a x 2 κ + 1 × x 1 τ e κ x ψ x x + 2 κ + a x + 2 κ 1 q 1 ν x + ν + a + 1 e 2 κ + a x 1 × x + ν a 2 e 2 κ + a x 1 q 1 ν
and that the corresponding homogeneous linear differential equation
x + d 2 w d x 2 + 2 κ + 2 κ + a x d w d x + κ 2 + τ a + d w x = 0
has solutions of the forms
w I x = h x + ν + a e 2 κ + a x 1 + q 1 ν ,
w II x = h x + 2 κ + a x 2 κ + 1 x + ν a 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 + ν a 1 e 2 κ + a x 1 x + ν + a e 2 κ + a x 1 + q 1 ν x C \ 0 , , v R
and
y II x = x τ e κ x x + 2 κ + a x 2 κ + 1 × x 1 τ e κ x ψ x x + 2 κ + a x + 2 κ 1 q 1 ν x + ν + a + 1 e 2 κ + a x 1 × x + ν a 2 e 2 κ + a x 1 q 1 ν
and that the corresponding homogeneous linear differential equation
1 + x d 2 y d x 2 + a + b x 1 + x d y d x + c + d x + ε x 2 1 + x y x = 0
has solutions given by
y I x = h x τ e κ x x + ν + a e 2 κ + a x 1 + q 1 ν ,
y II x = h x τ e κ x x + 2 κ + a x 2 κ + 1 x + ν a 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 = 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
= 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]
Back to TopTop