Multi-Term Linear Fractional Nabla Difference Equations with Constant Coefficients

Transcription

1 International Journal of Difference Equations ISSN , Volume 0, Number, pp Multi-Term Linear Fractional Nabla Difference Equations with Constant Coefficients Paul Eloe University of Dayton Department of Mathematics Dayton, 45469, USA peloe@udayton.edu Zi Ouyang University of Massachusetts Lowell Department of Physics and Applied Physics Lowell, 0854, USA zi ouyang@student.uml.edu Abstract We shall consider a linear fractional nabla backward fractional difference equation of Riemann Liouville type with constant coefficients. We apply a transform method to construct solutions. Sufficient conditions in terms of the coefficients are given so that the solutions are absolutely convergent. The method is known for two-term fractional difference equations; the method is new for fractional equations with three or more terms. As a corollary, we exhibit new summation representations of a discrete exponential function, a t, t 0,,.... AMS Subject Classifications: 39A2, 34A25, 26A33. Keywords: Discrete fractional calculus, transform methods, Riemann Liouville fractional difference. Introduction In this article we shall apply the transform method to obtain solutions of a linear fractional backward difference equation of the form 0yt + A yt + A 2 yt ft, t, 2...,. Received November 24, 204; Accepted March, 205 Communicated by Agnieszka Malinowska

2 92 P. Eloe and Z. Ouyang where < 2. The fractional difference operator, 0, is of Riemann Liouville type and is defined below. The solution algorithm is standard in the case A 0; see [2]. For the sake of exposition, we shall illustrate the solution algorithm with A 0 in the paper as well. The motivation for this work is to continue to develop the theory of linear fractional difference equations as an analogue of the theory of linear difference equations. Progress on what we shall call three term equations,., is limited. An equation of the form 2µ 0 yt + A µ 0yt + A 2 yt ft, t, 2..., is called a sequential fractional difference equation; important progress has been made in the study of sequential fractional equations see [5], [4], for fractional differential equations, and see [3], [], for fractional difference equations. For a more general equation, 2 0 yt + A 0 yt + A 2 yt ft, t, 2..., we prefer to assume 0 < < 2 2 as the only relation between and 2. For simplicity in the development that follows, we have taken. Let us recall the notation that will be employed throughout. The nabla operator commonly represents the backward difference operator and in this article yt yt yt, k yt k yt, k, 2,.... The operator a, a Riemann Liouville fractional difference operator, is defined as follows. If µ > 0, define the µth fractional sum by µ a yt t µ t ρs ys.2 Γµ where ρs s, and the raising factorial power function is defined by sa t α Γt + α. Γt Then if 0 n < n, define the th fractional difference a Riemann Liouville fractional difference by ayt n n a yt where n denotes the standard n th order backward difference. In Section 2, so that the article is self-contained, we shall give the definition of the discrete Laplace transform N -transform and provide the basic properties employed in this work. We shall provide some basic algebra properties related to the function t α. We shall also apply the transform method to find solutions of. in the case A 0. We obtain explicit sufficient conditions as a function of A 2 for the absolute convergence of these solutions. In Section 3, we shall apply the transform method to. and obtain

3 Multi-Term Equations 93 solutions; again, we shall obtain explicit sufficient conditions as a function of A and A 2 for convergence of the solutions. We believe these series representations are new and so, in Section 4, in an effort to recognize the series solutions, we apply the algorithm in the case when a solution is 2 t and verify independently that the series represents the known function. For further reading in this area, we refer the reader to the books on fractional differential equations [, 6, 7] and the articles on fractional difference equations [4 6, 8]. 2 Algebraic Properties and the Discrete Transform For the sake of exposition, we first introduce algebra and calculus properties related to the raising factorial power function. Lemma 2.. The following identities are valid: i t α αt α ; ii t α t + α β t α+β ; iii t µ iv 0 t + µ v k t n Γn + Γµ + Γµ + + tµ+ ; Γµ + Γµ + + t + µ+ ; t + k Γk +. vi µ a ft ta fa, if 0 < µ. vii µ a ft ta+ µfa + fa +, if 0 < µ. Proof. i and ii are easily observed by applying standard gamma function identities. iii is proved in [2] and is referred to as the power rule. iv is another form of the power rule and is stated for clarity since it is used in the specific applications in this article. v is obtained by induction on k and is in fact one of the identities found in Pascal s triangle since k+ t n Γn + t + k Γk + t + k t t + k t + tk+ Γk + 2. Thus, by induction on k, t + k + t t + k t vi and vii are obtained directly from the definition given by.2. t + k + Transform methods are commonly used to solve fractional differential and fractional difference equations. We have need to develop further transform properties to produce the applications given here. The following definition and the derivation of most of the following properties are found in [2]. Define the discrete Laplace transform N -transform by t.

4 94 P. Eloe and Z. Ouyang N t0 ft s Lemma 2.2. For any R \ {..., 2,, 0}, i N t s Γ s, s <. ii N t α t s iii N t s s N t. α Γ, s < α. s + α iv N a ft + s N a+ ft. v AN ft Af0 s + AN 0ft. vi N a a ft s N a ft s. vii If 0 <, viii tt 0 s t ft. 2. N a+ a ft s s N a ft s s a fa. N a+2 2 ft s s 2 N a ft s s s a fa s a fa +. ix If < 2, N a+2 a ft s s N a ft s s s a fa x N 0 a 2 t+ s a a fa + s N a ft s s s a fa s a fa + fa. ss a 2. We point out that one can obtain viii using repeated applications of vii with. ix is then obtained by applying viii with f replaced by 2 a f and employing vi and vii of Lemma 2. to compute the initial values at t a and t a +. We shall also tabulate some inverse transforms which are obtained from Lemma 2.2 and geometric series expansions.

5 Multi-Term Equations 95 Lemma 2.3. For any R \ {..., 2,, 0}, i ii iii s + A n A n s n+ n A n Γ n + N t n+. ss + A n A n Γ n + + N t n+. ss + A n A n Γ n + N 0 t + n+. Proof. Statements ii and iii follow from Lemma 2.2 iii and iv respectively; in particular, and n A n Γ n + s N t n+ n A n Γ n + s N t n+ n A n Γ n + + N t n+ n A n Γ n + N 0 t + n+. An initial value problem for a two term linear equation of the form 0yt + Ayt 0, y0 a, 0 <, for t, 2,..., was solved in [3]. In that article, the authors expressed solutions as functions of Mittag Leffler type functions and obtained series solutions in the case, A <. The method is readily extended to an initial value problem for a two-term, linear, nonhomogeneous 0yt + Ayt k, y0 a, 0 <, t, 2, and is illustrated here. Apply the N -transform to each side of the equation 2.2 to obtain Then N 0 yt + AN yt N k. or s N 0 yt s a + AN yt k s s + AN 0 yt + A s a k s.

6 96 P. Eloe and Z. Ouyang Thus, N 0 yt Use Lemma 2.3 to calculate yt. k ss + A + + Aa ss + A. k N 0 yt ss + A + + Aa ss + A n A n k Γ n + + N t n+ n A n + + Aa Γ n + N 0 t + n+ n A n k Γ n + + N 0t n+ + + Aa n A n Γ n + N 0 t + n+. Note, we claim N t n+ N 0 t n+. We have employed Lemma 2.2 v with ft t n+ and with the convention, 0. Thus, Γ0 yt k n A n Γ n + + tn+ ++Aa n A n Γ n + t+n+ 2.3 provides a solution of 2.2. As noted in a number of references, [7] or [2], the asymptotic property Γk + α lim k k α Γk, α R, 2.4 and the ratio test imply that y, given by 2.3, converges absolutely if A <. In order that the paper is self-contained, we outline how 2.4 is employed with the ratio test. Consider the term n A n Γ n + + tn+ and consider the limit of the ratio lim n t n+2 / Γ n t n+ Γ n + +.

7 Multi-Term Equations 97 Write t n+2 Γ n Γ n + + Γ t + n + 2 Γ n t n+ n + + t + n + Γ n + + n + + t + n + Γt + n + Γ t + n + 2 t + n + n + + Γ n + + n + + Γt + n + n + + Γ n Apply 2.4 to and with respectively. Then Γ t + n + 2 n + + Γt + n + n + + Γ n + + Γ n k t + n +, α, and k n + +, α, lim n t n+2 Γ n Γ n + +. t n+ So, one can apply the ratio test and the convergence reduces to the condition A <. The method is valid regardless of the order of. For example, consider an initial value problem Thus, 0yt + Ayt k, y0 a 0, y a < 2, t 2, 3, N 2 0 yt + AN 2 yt N 2 0 yt + AN 2 yt + A y + y0 A a a 0 s s N 2 0 yt + AN 0 yt Aa A a 0 s N2 k k. s s Apply Condition ix of Lemma 2.2 to obtain s + AN 0 yt a 0s s a a 0 Aa A s a 0 k s s.

8 98 P. Eloe and Z. Ouyang Apply Lemma 2.3, and n A n N 0 yt k Γ n + + N t n+ k n A n Γ n + N t n+ + a 0 s + Aa 0 s n A n Γ n + N t n+ 2 n A n Γ n + N t n+ + + Aa a 0 n A n Γ n + N t n+. Apply Lemma 2.2 v and the convention, 0 in the first, second and last terms, Γ0 combine the second and last terms, and in the third and fourth terms, apply Lemma 2.2 iv to obtain n A n N 0 yt k Γ n + + N 0 t n+ Thus, yt k + a a 0 + Aa 0 + n A n Γ n + N 0 t + n+ 2 n A n Γ n + N 0 t + n+ + Aa a 0 k n A n Γ n + + tn+ n A n Γ n + t + n+ 2 + Aa 0 + Aa a 0 k n A n Γ n + N 0t n+. n A n Γ n + tn+ n A n Γ t + n+ n + provides a solution of 2.5. Again, 2.4 and the ratio test imply that y converges absolutely if A <, for t 0,,....

9 Multi-Term Equations 99 3 A Three Term Equation In this section, we begin by describing an algorithm to construct a solution of an initial value problem for a three term linear fractional difference equation of the form 0yt + A yt + A 2 yt 0, y0 a 0, y a, for t, 2,..., 3. where < 2. Apply the operator N 2 to 3. to obtain Apply Lemma 2.2 ix and note N 2 0 yt + A N 2 yt + A2 N 2 yt N 2 0 yt s N 0 ft s s s a 0 a a 0 ; 3.3 apply Lemma 2.2 vii and note Similarly, in particular, N 2 yt N2 yt + y y N yt y 3.4 sn yt s a 0 a a 0 sn 0 yt s s a 0 a. N 2 yt N 2 yt + a + s a 0 a + s a 0 ; N 2 yt N 0 yt a s a Substitute 3.3, 3.4, and 3.5 into 3.2 and obtain N 0 yt s + A s + A 2 a 0 ss + A s + A A + A 2 a + a s + A s + A 2 Write Note that s + A s + A 2 s + A s + n s n+ A n 2 mn A 2 s +A s + A s n+. n+ m m+n A + A s s m n n

10 00 P. Eloe and Z. Ouyang and so, Since s + A s + A 2 s m n N mn m0 m m n m m n t m+n+ Γ m + n + m0 A m n A n 2s m n 3.7 A m n A n 2s m n. N 0 t m+n+ Γ m + n + employ Lemma 2.2 v to reexpress N as N 0, and we have m t N s m A m n A n m+n+ 2 + A s + A 2 n Γ m + n m0 m t N 0 m A m n A n m+n+ 2 n Γ m + n + ; it follows from Lemma 2.2 iv that s s + A s + A 2 N 0 Moreover, from 3.7, and so, m0 s s + A s + A 2 s s s + A s + A 2 N 0 m0 m0 m A m n A n 2 m0 m n t + m+n+ Γ. 3.9 m + n + m m A m n A n n 2s m n + m A m n A n 2 m t + m+n+ 2 n Γ m + n Substitute 3.8, 3.9, 3.0 directly into 3.6, to obtain m t + yt A 2 a 0 m A m n A n m+n+ 2 n Γ m + n + 3. m0 m t A a 0 m A m n A n m+n+ 2 2 n Γ m + n + m0 m t + K m A m n A n m+n+ 2 n Γ m + n +,,

11 Multi-Term Equations 0 where K + A + A 2 a + a 0. To simplify this representation, note that t + m+n+ Γ m + n + + t m+n+ Γ m + n + t + m+n+ 2 Γ m + n + since t + m+n+ Γ m + n + Γ t + + m + n + Γt + Γ m + n + Thus, 3. can be expressed as where yt K + K 2 m0 m0 t Γ t + m + n + Γt + Γ m + n + + m + n + Γ t + m + n + Γt + Γ m + n +. m t + m A m n A n m+n+ 2 2 m A m n A n 2 n m n Γ m + n t m+n+ Γ m + n +, K a 0 +A +A 2, K 2 K +a 0 A 2 +A +A 2 a + +A 2 a Note that m0 m A m n A n 2 m t + m+n+ 2 n Γ m + n and m0 m A m n A n 2 m t m+n+ n Γ m + n are two linear independent solutions of 3.. We provide the details to obtain conditions for absolute convergence in 3.5 for fixed t; the details for 3.4 are analogous. First note that for each t, t m+n+ Γ m + n +

12 02 P. Eloe and Z. Ouyang is an increasing function in n. To see this, consider the ratio t m+n++ / Γ m + n + + t m+n+ Γ m + n +. Then Γ t + m + n + + ΓtΓ m + n + + ΓtΓ m + n + Γt + m + n + and the inequality is strict if t >. Thus, compare 3.5 to t + m + n + m + n +, t m+ Γm + m0 A m n A n 2 m t m+ n Γm + A + A 2 m m0 and apply 2.4 and the ratio test. Thus, each of 3.4 and 3.5 converge absolutely if A + A 2 <. 4 Representations of Known Functions We illustrate this method with an initial value problem for a classical second order finite difference equation. The unique solution of the initial value problem is yt 2 t ; thus we obtain a series representation of 2 t as a linear combination of the forms 3.4 and 3.5. We believe this representation is new. Consider the initial value problem 0 2 yt yt 2yt 0, t 2, 3,..., y0, y Then, apply 3.6 with A 0, A 2 5, a 0, a 2, 2, to obtain from 3.3, N 0 yt 9 s ss 2 s s s ; 0 5 K 9 0 5, K

13 Multi-Term Equations 03 and yt n m t + m+n 0 m n n Γm + n + m n m t m+n+ 0 m n n Γm + n + 2. m0 4.2 The unique solution of 4. is 2 t and the series given in 4.2 converges absolutely for each t 0,,.... Thus, 4.2 provides a series representation of 2 t, t 0,,.... However, we claim the representation is new and so we further validate the representation. Write 4.2 as where At Bt m0 m0 yt 7 0 At + 5 Bt 5 n m t + m+n 0 m n n Γm + n + 5 n m t m+n+ 0 m n n Γm + n + 2. To show yt + 2yt, or 7 0 At Bt At + 5 Bt it is sufficient to show Bt + At + Bt and 7 0 At At + 5 Bt. Treating Bt + is straightforward since, Bt + 5 n m t + m+n+ 0 m n n Γm + n + 2 m0 5 n m Γt + + m + n + 0 m n n Γt + Γm + n + 2 m0 5 n m Γt + m + n + 0 m n t + m + n + n Γt + Γm + n + 2 m0 5 n m Γt + m + n + 0 m n t n Γt + Γm + n + 2 m0 + 5 n m Γt + m + n + 0 m n m + n + n Γt + Γm + n + 2 m0 Bt + At.

14 04 P. Eloe and Z. Ouyang We have yet to obtain a straightforward approach to treat At+. We begin by showing directly that Bt satisfies Apply the power rule and Thus, 0 2 yt yt 2yt 0, t 2, 3,.... Bt Bt 2 Bt m0 m0 m 5 n m t m+n+ 0 m n n Γm + n + 2, 5 n m t m+n 0 m n n Γm + n +, 5 n m t m+n 0 m n n Γm + n. m+ 2 Bt 5 n m + t m+n 0 m+ n n Γm + n + m0 0 5 n m m t m+n 0 m n + n n Γm + n + m0 + t 2m+ 5 m+ Γ2m n m t m+n 0 m n n Γm + n + m0 m+ + 5 n m t m+n 0 m+ n n Γm + n + m0 0 Bt n m t m+n+ 0 m n n Γm + n + 2 m0 0 Bt + 5 Bt. A similar calculation shows that At satisfies We close by arguing that 0 2 yt yt 2yt 0, t 2, 3, At At + 5 Bt.

15 Multi-Term Equations 05 Simplify to obtain At + At + 2At + 0 At + 2At + At At + At 2At At At At 6 5 At At At. Now it is sufficient to show 7 0 At At 5 Bt. Employ At Bt + Bt and At Bt Bt to obtain 7 0 At At 7 Bt + Bt Bt Bt Bt + Bt + Bt Bt Bt + Bt Bt 5 Bt. Remark 4.. A duality between delta and nabla finite differences has been established for both classical finite differences and finite differences on time scales see [0]. To our knowledge, the calculations performed here for the multi-term nabla fractional equations have not been carried out for analogous multi-term delta fractional equations. Even in the two-term delta fractional case, one considers two independent time domains see [7], for example, one for the function space yt and one for the function space 0yt; for the multi-term delta fractional equation it is feasible that one introduces multiple time domains. The nabla fractional equation avoids the issue of multiple time domains; we have not yet considered the multi-term delta fractional equation. References [] N. Acar and F. M. Atıcı, Exponential functions of discrete calculus, Appl. Anal. Discrete Math. 7: , 203. [2] F. M. Atıcı and P. W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ. 2009, Special Edition I No. 3, 2, [3] F. M. Atıcı and P. W. Eloe, Linear systems of fractional nabla difference equations, Rocky Mountain J. Math., 4: , 20. [4] F. M. Atıcı and P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Difference Equ., 22:65 76, 2007.

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