1 Introduction ON NABLA DISCRETE FRACTIONAL CALCULUS OPERATOR FOR A MODIFIED BESSEL EQUATION. Resat YILMAZER a,, and Okkes OZTURK b

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1 ON NABLA DISCRETE FRACTIONAL CALCULUS OPERATOR FOR A MODIFIED BESSEL EQUATION by Resat YILMAZER a,, and Okkes OZTURK b a Department of Mathematics, Firat University, 23119, Elazig, Turkey b Department of Mathematics, Bitlis Eren University, 13000, Bitlis, Turkey Corresponding Author: rstyilmazer@gmail.com In thermal sciences, it is possible to encounter topics such as Bessel beams, Bessel functions or Bessel equations. In this work, we also present new discrete fractional solutions of the modified Bessel differential equation by means of the -discrete fractional calculus operator. We consider homogeneous and non-homogeneous modified Bessel differential equation. So, we acquire four new solutions of these equations in the discrete fractional forms via a newly developed method. Keywords: discrete fractional calculus, modified Bessel equation, nabla operator. 1 Introduction The fractional calculus studies that started about 300 years ago have been an interesting subject of presentday. This theory, which spans a wide field, has contributed many science fields and has been gained a lot of scientific publications to the literature [1 5]. Recently, there appeared a number of papers on the discrete fractional calculus, which has helped to build up some of the basic theory of this area. For example, Atici and Eloe introduced the discrete Laplace transform method for a family of finite fractional difference equations in [6]. Atici and Eloe [7] defined the initial value problems in the discrete fractional calculus. Atici and Eloe [8] studied properties of discrete fractional calculus with the nabla operator. They developed exponential laws and the product rule to the forward fractional calculus. Atici and Sengul [9] developed the Leibniz rule and summation by parts formula in discrete fractional calculus. Bastos and Torres [10] presented the more general discrete fractional operator and this operator was defined by the delta and nabla fractional sums. Holm [11] introduced fractional sums and difference operators. He applied this theory to solve the fractional initial value problems. Anastassiou derived the right discrete nabla fractional Taylor formula in [12]. Mohan [13] discussed the differrentiability properties of solutions of nabla fractional difference equations of order non-integer. Han et al. [14] studied existence and nonexistence of positive solutions to the discrete fractional boundary value problems. Solutions of the fractional modified Bessel equation were obtained asymptotic analysis was provided for these solutions in [15]. Robin [16] studied on Bessel equations and Bessel functions via an elementary factorization method. The solutions of the classical Bessel equation were presented by means of the fractional calculus theorems [17]. Before giving some results concerning the Bessel equation, we should 1

2 give its physical properties. The total energy of the particle is given by E = p2 2M = ħ2 k 2 2M = k2, where p is its initial or final momentum, and k the corresponding wavenumber [18]. Bessel beams, Bessel functions or Bessel equations can be seen in thermal sciences. For example, Doan et al. [19] presented a short paper about Bessel beam laser-scribing of thin film silicon solar cells. Semiclassical dynamics of dilute thermal atom clouds located in three-dimensional optical lattices generated by stationary optical Bessel beams were characterized [20]. A method to generate a Bessel beam using cross-phase modulation based on the thermal nonlinear optical effect was introduced [21]. The expansions of the modified Bessel functions were used to obtain the temperature field [22]. Jones [23] took advantage of the Bessel functions in his book on thermal sciences. The length of the temperature plume was calculated via an approximation of the modified Bessel function [24]. Swain et al. were studied on a straight triangular fin and a general porous pin fin profile. To formulate heat transfer equation in straight triangular fin modified Bessel s equation was used [25]. The aim of present study to solve the homogeneous and non-homogeneous modified Bessel equation by using the -discrete fractional calculus operator. In Section 2, some basic definitions of the discrete fractional calculus are presented main results are given in Section 3. Some conclusions and future perspectives are also introduced in the last section. 2 Preliminaries Here, we present some essential information about discrete fractional calculus theory. We use the some notations: N is the set of natural numbers including zero Z is the set of integers. N b = {b, b + 1, b + 2, } for b Z. Let f(t) and g (t) be a real-valued function defined on N + 0. These and other related results can be found in [6 14, 26 29]. 2.1 Definition The rising factorial power is given by Let α a real number. Then t α is defined to be t n = t (t + 1) (t + 2)... (t + n 1), n N, t 0 = 1. t α = where t R\ {..., 2,, 0} and 0 α = 0. Let us note that Γ (t + α), (1) Γ (t) ( t α) = αt α, (2) where u (t) = u (t) u (t 1). For n = 2, 3,... define n indeductively by n = n. 2.2 Definition The α th order fractional sum of f is given by α b f (t) = t (t δ (t)) α f (s), (3) Γ (α) where t N b, δ (t) = t 1 is backward jump operator of the time scale calculus. s=b 2

3 2.3 Theorem Let f(t) and g (t) : N + 0 R, α, β > 0 and h, v are scalars. The following equality holds: 1. α β f (t) = (α+β) f (t) = β α f (t). (4) 2. α [hf (t) + vg (t)] = h α f (t) + v α g (t). (5) 3. α f (t) = (α) f (t). (6) ( ) t + α 2 4. α f (t) = (1 α) f (t) f (0). (7) t Lemma For any α > 0, α th order fractional difference of the product fg is given by 2.5 Lemma α 0 (fg) (t) = t n=0 If the function f (t) is single-valued and analytic, then ( ) α [ α n 0 f (t n) ] [ n g (t)]. (8) n (f α (t)) β = f α+β (t) = (f β (t)) α (f α (t) 0; f β (t) 0; α, β R; t N). 3 Main Results Here, we will present the discrete fractional solutions of the modified Bessel differential equation by applying the -discrete fractional calculus operator by way of the following theorems: 3.1 Theorem Let ψ {ψ : 0 ψ α < ; α R} and f {f : 0 f α < ; α R}. Then the non-homogeneous modified Bessel equation has particular solutions of the forms: ψ 2 + ψ ( k 2 + ψ I = r ν+1/2 e kr {[ ( fr ν+1/2 e kr) q (ν+1/2) rν/2 e 2kr ] ψ II = r ν+1/2 e kr {[ ( fr ν+1/2 e kr) q (ν+1/2) rν/2 e 2kr ] ) 1/4 ν2 r 2 = f (r 0), (9) 3 r (ν+1/2) e 2kr } r (ν+1/2) e 2kr } +q (ν+1/2) +q (ν+1/2), (10), (11)

4 {[ ( ψ III = r ν+1/2 e kr fr ν+1/2 e kr) ] } q (ν/2) r (ν+1/2) e 2kr r ν/2 e 2kr, (12) +q ( ν+1/2) {[ ( ψ IV = r ν+1/2 e kr fr ν+1/2 e kr) ] } q (ν/2) r (ν+1/2) e 2kr r ν/2 e 2kr. (13) +q ( ν+1/2) Here, ψ 2 = d 2 ψ/dr 2, ψ 0 = ψ = ψ (r), f = f (r) (r R), and k, ν are given constants. Proof. Set hence, we have ψ = r τ φ φ = φ (r), (14) φ 2 r + φ 1 2τ + φ [(τ 2 τ + 14 ) ] ν2 r k 2 r = fr 1 τ. (15) And, we suppose τ as τ 2 τ ν2 = 0, that is τ = 1 2 ± ν. I) Let τ = ν + 1/2. From (14) and (15) we have ψ = r ν+1/2 φ, (16) Next, set φ 2 r + φ 1 (2ν + 1) φk 2 r = fr ν+1/2. (17) φ = e λr ϕ ϕ = ϕ (r), (18) then, equation (17) may be written in the form ϕ 2 r + ϕ 1 (2λr + 2ν + 1) + ϕ [( λ 2 k 2) r + 2νλ + λ ] = fr ν+1/2 e λr. (19) Choose λ such that λ 2 k 2 = 0, that is λ = ±k. I-i) If λ = k, we write φ = e kr ϕ, (20) ϕ 2 r + ϕ 1 (2kr + 2ν + 1) + ϕ [k (2ν + 1)] = fr ν+1/2 e kr, (21) from (18) and (19). Using the α to (21), we find the following equality: α (ϕ 2 r) + α [ϕ 1 (2kr + 2ν + 1)] + α {ϕ [k (2ν + 1)]} = α ( fr ν+1/2 e kr). (22) Using (1)-(8) we have α (ϕ 2 r) = ϕ 2+α r + αqϕ 1+α, (23) α [ϕ 1 (2kr + 2ν + 1)] = ϕ 1+α (2kr + 2ν + 1) + 2kαqϕ α, (24) 4

5 where q is a shift operator. Making use of the relations (23) and (24), rewriting (22) in the following form: ( ϕ 2+α r + ϕ 1+α (2kr + 2ν + αq + 1) + ϕ α [k (2αq + 2ν + 1)] = fr ν+1/2 e kr). (25) α Choose α such that α = q (ν + 1/2), we have then ϕ 2 q (ν+1/2)r + ϕ 1 q (ν+1/2) (2kr + ν + 1/2) = ( fr ν+1/2 e kr) q (ν+1/2), (26) from (25). Next, writing ϕ 1 q (ν+1/2) = u (r) [ ] ϕ = u+q (ν+1/2), (27) we obtain [ u 1 + u 2k + ν + 1/2 ] ( = fr ν+1/2 e kr) r q (ν+1/2) r, (28) from (26), (27). A particular solution of a first-order ordinary differential equation (28) is [ ( u = fr ν+1/2 e kr) ] q (ν+1/2) rν/2 e 2kr r (ν+1/2) e 2kr. (29) Thus, we obtain the solution (10) from (16), (20), (27) and (29). I-ii) If λ = k, we write φ = e kr ϕ, (30) ϕ 2 r + ϕ 1 ( 2kr + 2ν + 1) + ϕ [ k (2ν + 1)] = fr ν+1/2 e kr, (31) from (18) and (19). Using the α to (31), we have ϕ 2+α r + ϕ 1+α ( 2kr + 2ν + αq + 1) + ϕ α [ k (2αq + 2ν + 1)] = Choose α such that α = q (ν + 1/2), and replacing ( fr ν+1/2 e kr) α. (32) ϕ 1 q (ν+1/2) = ω (r) [ ] ϕ = ω+q (ν+1/2), (33) we obtain [ ω 1 + ω 2k + ν + 1/2 ] ( = fr ν+1/2 e kr) r q (ν+1/2) r, (34) from (32), (33). A particular solution of a first-order ordinary differential equation (34) is [ ( ω = fr ν+1/2 e kr) ] q (ν+1/2) rν/2 e 2kr r (ν+1/2) e 2kr. (35) Thus, we obtain the solution (11) from (16), (30), (33) and (35). II) Let τ = ν + 1/2. In the same way as the procedure in I, replacing ν by ν in I-i and in I-ii, we have other solutions (12) and (13) different from (10) and (11) respectively, if ν 0. 5

6 3.2 Theorem Let ψ {ψ : 0 ψ α < ; α R}. Then the homogeneous modified Bessel equation: ) ψ 2 + ψ ( k 2 1/4 ν2 + r 2 = 0 (r 0), (36) has solutions of the forms; ψ (I) = hr ν+1/2 e kr [ r (ν+1/2) e 2kr] +q (ν+1/2), (37) ψ (II) = hr ν+1/2 e kr [ r (ν+1/2) e 2kr] +q (ν+1/2), (38) ψ (III) = hr ν+1/2 e kr [ r ν/2 e 2kr] +q ( ν+1/2), (39) where h is an arbitrary constant and ν 0. ψ (IV ) = hr ν+1/2 e kr [ r ν/2 e 2kr] +q ( ν+1/2), (40) Proof. If f = 0 under the hypotheses of 3.1 Theorem, we have [ u 1 + u 2k + ν + 1/2 ] = 0, (41) r for λ = k and λ = k, instead of (28) and (34), respectively. [ ω 1 + ω 2k + ν + 1/2 ] = 0, (42) r Therefore, we obtain (37) for (41) and (38) for (42). And, for τ = ν + 1/2, replacing ν by ν in (41) and (42), we have (39) and (40). 3.3 Theorem Let ψ and f are just as in 3.1 Theorem. Then the non-homogeneous modified Bessel equation (9) is satisfied by the fractional differintegrated functions ψ = ψ I + ψ (I). Proof. It is clear by above theorems. 4 Conclusion In our study, we applied the -discrete fractional calculus operator to the homogeneous and nonhomogeneous modified Bessel differential equation. We obtained the discrete fractional solutions of these equations via this new operator method. For the first time, the present method was used to solve these equations in this article, and it will be applied to the similar equations in the future time. 6

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