On Discrete Fractional Solutions of Non-Fuchsian Differential Equations

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 [1–4]. 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 [5–18]. 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


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.
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 where ς is any real number.Granger and Joyeux [19] and Hosking [20], defined notion of the fractional difference as follows where ς is any real number and qΦ (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.

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 Let σ be any real number.Then "t to the σ rising" is defined to be Also, the ∇ operator of Equation ( 1) is given by where ∇u The ηth-order fractional nabla sum of g is given by where we define the trivial sum by ∇ −0 b g (t) = g (t) for t ∈ N b .The ηth-order Riemann-Liouville type nabla fractional difference of g is defined by Theorem 1 ([16]).Let f and g : ∇ 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, Lemma 2 (Leibniz Rule [10]).For any η > 0, ηth-order fractional difference of the product f g is given in this form where 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

Main Results
We start by considering the following differential equation where ψ is a given function, x ∈ C\ {0, − } , and a, b, c, d, ε and are parameters.Let y (x) = x τ e κx w (x) (12) so that and By substituting ( 12)-( 14) into the (11), we have Finally, we find it to be suitable to restrict the different parameters involved in (11) and ( 15) by means of the following equalities; and Under the parametric constraints given by ( 16), the Equation (15) will immediately decrease to a simpler form where τ and κ are given by ( 17) and ( 18), respectively.
Then we obtain from (26).Therefore, setting we have from (27).A particular solution of a first order ordinary differential Equation (29): Thus we obtain the solution ( 21) from ( 28) and ( 30). (ii The first and second derivations of (31) are acquired as follows: Substitute ( 31)-( 33) into ( 20), we have Here, we choose σ such that ασ (ασ In the case σ = 0, we have the same results as i. 31) and (34) respectively.
Applying the operator ∇ η to both members of (36), we have .
Furthermore, we can prove for the homogen part such that the homogeneous linear ordinary differential equation has solutions of the forms where h is an arbitrary constant.Now, in Theorem 1, we further set and let We thus find that the nonhomogeneous differential Equation ( 19) has a particular solution given by and that the corresponding homogeneous linear differential equation has solutions of the forms where h is an arbitrary constant.Therefore, the linear differential Equation (11), has a particular solution in the following forms and and that the corresponding homogeneous linear differential equation has solutions given by where h ∈ R, the parameters τ, κ and ν are given by ( 17), ( 18) and (45).(57) by using (21).The function obtained in (57) provide the Equation (56).

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.

Remark 1 .ε x 2 y
First of all, when = 0, the differential Equation (11) reduces to the following version of the Tricomi equation= ψ (x) .