Multi-Term Linear Fractional Nabla Difference Equations with Constant Coefficients
|
|
- Juliana Phillips
- 6 years ago
- Views:
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.
16 06 P. Eloe and Z. Ouyang [5] F. M. Atıcı and P. W. Eloe, Fractional q-calculus on a time scale, J. Nonlinear Math. Phys., 43: , [6] F. M. Atıcı and P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc., 37:98 989, [7] F. M. Atıcı and P. W. Eloe, Linear forward fractional difference equations, Commun. Appl. Anal., 9:3 42, 205. [8] F. M. Atıcı and S. Şengül, Modeling with fractional difference equations, J. Math. Anal. and Appl., 369: 9, 200. [9] B. Bonilla, M. Rivero and J.J. Trujillo, On systems of linear fracional differential equations with constant coefficients, Appl. Math. Comput., 87:68 78, [0] M. Cristina Caputo, Time scales: from nabla calculus to delta calculus and vice versa via duality, Int. J. Difference Equ, 5:25 40, 200. [] K. Diethelm. The Analysis of Fractional Differential Equations. An Applicationoriented Expostion Using Differential Operators of Caputo Type, Lecture Notes in Matheamtics, 2004, Springer-Verlag, Berlin, 200. [2] M. Fečkan, Note on fractional Gronwall inequalities, Elec. J. Qual. Theory Diff Eqns., 20444: 8, 204. [3] C. Goodrich, On discrete sequential fractional boundary value problems, J. Math. Anal. Appl., 385: 24, 202. [4] A. A. Kilbas, H. M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, [5] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Academic Press, New York, 993. [6] I. Podlubny, Fractional Differential Equations, Academic Press New York, 999. [7] G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 993. [8] H. M. Srivastava and Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput., 2:98 20, 2009.
Positive solutions for discrete fractional intiail value problem
Computational Methods for Differential Equations http://cmde.tabrizu.ac.ir Vol. 4, No. 4, 2016, pp. 285-297 Positive solutions for discrete fractional intiail value problem Tahereh Haghi Sahand University
More informationDual identities in fractional difference calculus within Riemann. Thabet Abdeljawad a b
Dual identities in fractional difference calculus within Riemann Thabet Abdeljawad a b a Department of Mathematics, Çankaya University, 06530, Ankara, Turkey b Department of Mathematics and Physical Sciences
More informationOscillatory Solutions of Nonlinear Fractional Difference Equations
International Journal of Difference Equations ISSN 0973-6069, Volume 3, Number, pp. 9 3 208 http://campus.mst.edu/ijde Oscillaty Solutions of Nonlinear Fractional Difference Equations G. E. Chatzarakis
More informationExistence of Minimizers for Fractional Variational Problems Containing Caputo Derivatives
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 8, Number 1, pp. 3 12 (2013) http://campus.mst.edu/adsa Existence of Minimizers for Fractional Variational Problems Containing Caputo
More informationANALYSIS OF NONLINEAR FRACTIONAL NABLA DIFFERENCE EQUATIONS
International Journal of Analysis and Applications ISSN 229-8639 Volume 7, Number (205), 79-95 http://www.etamaths.com ANALYSIS OF NONLINEAR FRACTIONAL NABLA DIFFERENCE EQUATIONS JAGAN MOHAN JONNALAGADDA
More informationOn Two-Point Riemann Liouville Type Nabla Fractional Boundary Value Problems
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 13, Number 2, pp. 141 166 (2018) http://campus.mst.edu/adsa On Two-Point Riemann Liouville Type Nabla Fractional Boundary Value Problems
More informationA generalized Gronwall inequality and its application to fractional differential equations with Hadamard derivatives
A generalized Gronwall inequality and its application to fractional differential equations with Hadamard derivatives Deliang Qian Ziqing Gong Changpin Li Department of Mathematics, Shanghai University,
More informationIMPROVEMENTS OF COMPOSITION RULE FOR THE CANAVATI FRACTIONAL DERIVATIVES AND APPLICATIONS TO OPIAL-TYPE INEQUALITIES
Dynamic Systems and Applications ( 383-394 IMPROVEMENTS OF COMPOSITION RULE FOR THE CANAVATI FRACTIONAL DERIVATIVES AND APPLICATIONS TO OPIAL-TYPE INEQUALITIES M ANDRIĆ, J PEČARIĆ, AND I PERIĆ Faculty
More informationNonlocal Fractional Semilinear Delay Differential Equations in Separable Banach spaces
IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn:2319-765x. Volume 10, Issue 1 Ver. IV. (Feb. 2014), PP 49-55 Nonlocal Fractional Semilinear Delay Differential Equations in Separable Banach
More informationOn boundary value problems for fractional integro-differential equations in Banach spaces
Malaya J. Mat. 3425 54 553 On boundary value problems for fractional integro-differential equations in Banach spaces Sabri T. M. Thabet a, and Machindra B. Dhakne b a,b Department of Mathematics, Dr. Babasaheb
More informationarxiv: v2 [math.ca] 13 Oct 2015
arxiv:1510.03285v2 [math.ca] 13 Oct 2015 MONOTONICITY OF FUNCTIONS AND SIGN CHANGES OF THEIR CAPUTO DERIVATIVES Kai Diethelm 1,2 Abstract It is well known that a continuously differentiable function is
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS 1. Yong Zhou. Abstract
EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS 1 Yong Zhou Abstract In this paper, the initial value problem is discussed for a system of fractional differential
More informationNonlocal problems for the generalized Bagley-Torvik fractional differential equation
Nonlocal problems for the generalized Bagley-Torvik fractional differential equation Svatoslav Staněk Workshop on differential equations Malá Morávka, 28. 5. 212 () s 1 / 32 Overview 1) Introduction 2)
More informationNontrivial solutions for fractional q-difference boundary value problems
Electronic Journal of Qualitative Theory of Differential Equations 21, No. 7, 1-1; http://www.math.u-szeged.hu/ejqtde/ Nontrivial solutions for fractional q-difference boundary value problems Rui A. C.
More informationSOME RESULTS FOR BOUNDARY VALUE PROBLEM OF AN INTEGRO DIFFERENTIAL EQUATIONS WITH FRACTIONAL ORDER
Dynamic Systems and Applications 2 (2) 7-24 SOME RESULTS FOR BOUNDARY VALUE PROBLEM OF AN INTEGRO DIFFERENTIAL EQUATIONS WITH FRACTIONAL ORDER P. KARTHIKEYAN Department of Mathematics, KSR College of Arts
More informationPositive solutions for a class of fractional boundary value problems
Nonlinear Analysis: Modelling and Control, Vol. 21, No. 1, 1 17 ISSN 1392-5113 http://dx.doi.org/1.15388/na.216.1.1 Positive solutions for a class of fractional boundary value problems Jiafa Xu a, Zhongli
More informationDETERMINATION OF AN UNKNOWN SOURCE TERM IN A SPACE-TIME FRACTIONAL DIFFUSION EQUATION
Journal of Fractional Calculus and Applications, Vol. 6(1) Jan. 2015, pp. 83-90. ISSN: 2090-5858. http://fcag-egypt.com/journals/jfca/ DETERMINATION OF AN UNKNOWN SOURCE TERM IN A SPACE-TIME FRACTIONAL
More informationMahmoud M. El-Borai a, Abou-Zaid H. El-Banna b, Walid H. Ahmed c a Department of Mathematics, faculty of science, Alexandria university, Alexandria.
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:01 52 On Some Fractional-Integro Partial Differential Equations Mahmoud M. El-Borai a, Abou-Zaid H. El-Banna b, Walid H. Ahmed c
More informationSMOOTHNESS PROPERTIES OF SOLUTIONS OF CAPUTO- TYPE FRACTIONAL DIFFERENTIAL EQUATIONS. Kai Diethelm. Abstract
SMOOTHNESS PROPERTIES OF SOLUTIONS OF CAPUTO- TYPE FRACTIONAL DIFFERENTIAL EQUATIONS Kai Diethelm Abstract Dedicated to Prof. Michele Caputo on the occasion of his 8th birthday We consider ordinary fractional
More informationResearch Article A New Fractional Integral Inequality with Singularity and Its Application
Abstract and Applied Analysis Volume 212, Article ID 93798, 12 pages doi:1.1155/212/93798 Research Article A New Fractional Integral Inequality with Singularity and Its Application Qiong-Xiang Kong 1 and
More informationLong and Short Memory in Economics: Fractional-Order Difference and Differentiation
IRA-International Journal of Management and Social Sciences. 2016. Vol. 5. No. 2. P. 327-334. DOI: 10.21013/jmss.v5.n2.p10 Long and Short Memory in Economics: Fractional-Order Difference and Differentiation
More informationExistence and Uniqueness Results for Nonlinear Implicit Fractional Differential Equations with Boundary Conditions
Existence and Uniqueness Results for Nonlinear Implicit Fractional Differential Equations with Boundary Conditions Mouffak Benchohra a,b 1 and Jamal E. Lazreg a, a Laboratory of Mathematics, University
More informationExistence of solutions for multi-point boundary value problem of fractional q-difference equation
Electronic Journal of Qualitative Theory of Differential Euations 211, No. 92, 1-1; http://www.math.u-szeged.hu/ejtde/ Existence of solutions for multi-point boundary value problem of fractional -difference
More informationPolyexponentials. Khristo N. Boyadzhiev Ohio Northern University Departnment of Mathematics Ada, OH
Polyexponentials Khristo N. Boyadzhiev Ohio Northern University Departnment of Mathematics Ada, OH 45810 k-boyadzhiev@onu.edu 1. Introduction. The polylogarithmic function [15] (1.1) and the more general
More informationAbstract We paid attention to the methodology of two integral
Comparison of Homotopy Perturbation Sumudu Transform method and Homotopy Decomposition method for solving nonlinear Fractional Partial Differential Equations 1 Rodrigue Batogna Gnitchogna 2 Abdon Atangana
More informationEXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO HIGHER-ORDER NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION WITH INTEGRAL BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 212 (212), No. 234, pp. 1 11. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS
More informationApplied Mathematics Letters
Applied Mathematics Letters 24 (211) 219 223 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Laplace transform and fractional differential
More informationPicard s Iterative Method for Caputo Fractional Differential Equations with Numerical Results
mathematics Article Picard s Iterative Method for Caputo Fractional Differential Equations with Numerical Results Rainey Lyons *, Aghalaya S. Vatsala * and Ross A. Chiquet Department of Mathematics, University
More informationOn The Leibniz Rule And Fractional Derivative For Differentiable And Non-Differentiable Functions
On The Leibniz Rule And Fractional Derivative For Differentiable And Non-Differentiable Functions Xiong Wang Center of Chaos and Complex Network, Department of Electronic Engineering, City University of
More informationHomotopy Analysis Method for Nonlinear Differential Equations with Fractional Orders
Homotopy Analysis Method for Nonlinear Differential Equations with Fractional Orders Yin-Ping Liu and Zhi-Bin Li Department of Computer Science, East China Normal University, Shanghai, 200062, China Reprint
More informationarxiv: v2 [math.ca] 8 Nov 2014
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0894-0347(XX)0000-0 A NEW FRACTIONAL DERIVATIVE WITH CLASSICAL PROPERTIES arxiv:1410.6535v2 [math.ca] 8 Nov 2014 UDITA
More informationApplied Mathematics Letters. A reproducing kernel method for solving nonlocal fractional boundary value problems
Applied Mathematics Letters 25 (2012) 818 823 Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml A reproducing kernel method for
More informationNumerical solution of the Bagley Torvik equation. Kai Diethelm & Neville J. Ford
ISSN 1360-1725 UMIST Numerical solution of the Bagley Torvik equation Kai Diethelm & Neville J. Ford Numerical Analysis Report No. 378 A report in association with Chester College Manchester Centre for
More informationOn four-point nonlocal boundary value problems of nonlinear impulsive equations of fractional order
On four-point nonlocal boundary value problems of nonlinear impulsive equations of fractional order Dehong Ji Tianjin University of Technology Department of Applied Mathematics Hongqi Nanlu Extension,
More informationFRACTIONAL INTEGRAL INEQUALITIES FOR DIFFERENTIABLE CONVEX MAPPINGS AND APPLICATIONS TO SPECIAL MEANS AND A MIDPOINT FORMULA
Journal of Applied Mathematics, Statistics and Informatics (JAMSI), 8 (), No. FRACTIONAL INTEGRAL INEQUALITIES FOR DIFFERENTIABLE CONVEX MAPPINGS AND APPLICATIONS TO SPECIAL MEANS AND A MIDPOINT FORMULA
More informationOscillation results for certain forced fractional difference equations with damping term
Li Advances in Difference Equations 06) 06:70 DOI 0.86/s66-06-0798- R E S E A R C H Open Access Oscillation results for certain forced fractional difference equations with damping term Wei Nian Li * *
More informationA computationally effective predictor-corrector method for simulating fractional order dynamical control system
ANZIAM J. 47 (EMA25) pp.168 184, 26 168 A computationally effective predictor-corrector method for simulating fractional order dynamical control system. Yang F. Liu (Received 14 October 25; revised 24
More informationOn integral representations of q-gamma and q beta functions
On integral representations of -gamma and beta functions arxiv:math/3232v [math.qa] 4 Feb 23 Alberto De Sole, Victor G. Kac Department of Mathematics, MIT 77 Massachusetts Avenue, Cambridge, MA 239, USA
More informationEXISTENCE THEOREM FOR A FRACTIONAL MULTI-POINT BOUNDARY VALUE PROBLEM
Fixed Point Theory, 5(, No., 3-58 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html EXISTENCE THEOREM FOR A FRACTIONAL MULTI-POINT BOUNDARY VALUE PROBLEM FULAI CHEN AND YONG ZHOU Department of Mathematics,
More informationFRACTIONAL BOUNDARY VALUE PROBLEMS ON THE HALF LINE. Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi
Opuscula Math. 37, no. 2 27), 265 28 http://dx.doi.org/.7494/opmath.27.37.2.265 Opuscula Mathematica FRACTIONAL BOUNDARY VALUE PROBLEMS ON THE HALF LINE Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi
More informationDIfferential equations of fractional order have been the
Multistage Telescoping Decomposition Method for Solving Fractional Differential Equations Abdelkader Bouhassoun Abstract The application of telescoping decomposition method, developed for ordinary differential
More informationarxiv: v1 [math.ap] 26 Mar 2013
Analytic solutions of fractional differential equations by operational methods arxiv:134.156v1 [math.ap] 26 Mar 213 Roberto Garra 1 & Federico Polito 2 (1) Dipartimento di Scienze di Base e Applicate per
More informationExistence of triple positive solutions for boundary value problem of nonlinear fractional differential equations
Computational Methods for Differential Equations http://cmde.tabrizu.ac.ir Vol. 5, No. 2, 217, pp. 158-169 Existence of triple positive solutions for boundary value problem of nonlinear fractional differential
More informationQ-INTEGRAL EQUATIONS OF FRACTIONAL ORDERS. 1. Introduction In this paper, we are concerned with the following functional equation
Electronic Journal of Differential Equations, Vol. 216 216, No. 17, pp. 1 14. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu Q-INTEGRAL EQUATIONS
More informationk -Fractional Integrals and Application
Int. J. Contem. Math. Sciences, Vol. 7,, no., 89-94 -Fractional Integrals and Alication S. Mubeen National College of Business Administration and Economics Gulberg-III, Lahore, Paistan smhanda@gmail.com
More informationA Comparison Result for the Fractional Difference Operator
International Journal of Difference Equations ISSN 0973-6069, Volume 6, Number 1, pp. 17 37 (2011) http://campus.mst.edu/ijde A Comparison Result for the Fractional Difference Operator Christopher S. Goodrich
More informationOn Local Asymptotic Stability of q-fractional Nonlinear Dynamical Systems
Available at http://pvamuedu/aam Appl Appl Math ISSN: 1932-9466 Vol 11, Issue 1 (June 2016), pp 174-183 Applications and Applied Mathematics: An International Journal (AAM) On Local Asymptotic Stability
More informationThis work has been submitted to ChesterRep the University of Chester s online research repository.
This work has been submitted to ChesterRep the University of Chester s online research repository http://chesterrep.openrepository.com Author(s): Kai Diethelm; Neville J Ford Title: Volterra integral equations
More informationFRACTIONAL FOURIER TRANSFORM AND FRACTIONAL DIFFUSION-WAVE EQUATIONS
FRACTIONAL FOURIER TRANSFORM AND FRACTIONAL DIFFUSION-WAVE EQUATIONS L. Boyadjiev*, B. Al-Saqabi** Department of Mathematics, Faculty of Science, Kuwait University *E-mail: boyadjievl@yahoo.com **E-mail:
More informationOscillation theorems for nonlinear fractional difference equations
Adiguzel Boundary Value Problems (2018) 2018:178 https://doi.org/10.1186/s13661-018-1098-4 R E S E A R C H Open Access Oscillation theorems for nonlinear fractional difference equations Hakan Adiguzel
More informationExistence of Solutions for Nonlocal Boundary Value Problems of Nonlinear Fractional Differential Equations
Advances in Dynamical Systems and Applications ISSN 973-5321, Volume 7, Number 1, pp. 31 4 (212) http://campus.mst.edu/adsa Existence of Solutions for Nonlocal Boundary Value Problems of Nonlinear Fractional
More informationNONLINEAR EIGENVALUE PROBLEMS FOR HIGHER ORDER LIDSTONE BOUNDARY VALUE PROBLEMS
NONLINEAR EIGENVALUE PROBLEMS FOR HIGHER ORDER LIDSTONE BOUNDARY VALUE PROBLEMS PAUL W. ELOE Abstract. In this paper, we consider the Lidstone boundary value problem y (2m) (t) = λa(t)f(y(t),..., y (2j)
More informationElena Gogovcheva, Lyubomir Boyadjiev 1 Dedicated to Professor H.M. Srivastava, on the occasion of his 65th Birth Anniversary Abstract
FRACTIONAL EXTENSIONS OF JACOBI POLYNOMIALS AND GAUSS HYPERGEOMETRIC FUNCTION Elena Gogovcheva, Lyubomir Boyadjiev 1 Dedicated to Professor H.M. Srivastava, on the occasion of his 65th Birth Anniversary
More information1 Introduction ON NABLA DISCRETE FRACTIONAL CALCULUS OPERATOR FOR A MODIFIED BESSEL EQUATION. Resat YILMAZER a,, and Okkes OZTURK b
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
More informationRESOLVENT OF LINEAR VOLTERRA EQUATIONS
Tohoku Math. J. 47 (1995), 263-269 STABILITY PROPERTIES AND INTEGRABILITY OF THE RESOLVENT OF LINEAR VOLTERRA EQUATIONS PAUL ELOE AND MUHAMMAD ISLAM* (Received January 5, 1994, revised April 22, 1994)
More informationExact Solution of Some Linear Fractional Differential Equations by Laplace Transform. 1 Introduction. 2 Preliminaries and notations
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.16(213) No.1,pp.3-11 Exact Solution of Some Linear Fractional Differential Equations by Laplace Transform Saeed
More informationA generalized Gronwall inequality and its application to a fractional differential equation
J. Math. Anal. Appl. 328 27) 75 8 www.elsevier.com/locate/jmaa A generalized Gronwall inequality and its application to a fractional differential equation Haiping Ye a,, Jianming Gao a, Yongsheng Ding
More informationNumerical solution for complex systems of fractional order
Shiraz University of Technology From the SelectedWorks of Habibolla Latifizadeh 213 Numerical solution for complex systems of fractional order Habibolla Latifizadeh, Shiraz University of Technology Available
More informationAnalytic solution of fractional integro-differential equations
Annals of the University of Craiova, Mathematics and Computer Science Series Volume 38(1), 211, Pages 1 1 ISSN: 1223-6934 Analytic solution of fractional integro-differential equations Fadi Awawdeh, E.A.
More informationFractional differential equations with integral boundary conditions
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 8 (215), 39 314 Research Article Fractional differential equations with integral boundary conditions Xuhuan Wang a,, Liping Wang a, Qinghong Zeng
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS TO FUNCTIONAL INTEGRO-DIFFERENTIAL FRACTIONAL EQUATIONS
Electronic Journal of Differential Equations, Vol. 212 212, No. 13, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS
More informationA study on nabla discrete fractional operator in mass - spring - damper system
NTMSCI 4, No. 4, 137-144 (2016) 137 New Trends in Mathematical Sciences http://dx.doi.org/10.20852/ntmsci.2016422559 A study on nabla discrete fractional operator in mass - spring - damper system Okkes
More informationExistence of Ulam Stability for Iterative Fractional Differential Equations Based on Fractional Entropy
Entropy 215, 17, 3172-3181; doi:1.339/e1753172 OPEN ACCESS entropy ISSN 199-43 www.mdpi.com/journal/entropy Article Existence of Ulam Stability for Iterative Fractional Differential Equations Based on
More informationOn the Finite Caputo and Finite Riesz Derivatives
EJTP 3, No. 1 (006) 81 95 Electronic Journal of Theoretical Physics On the Finite Caputo and Finite Riesz Derivatives A. M. A. El-Sayed 1 and M. Gaber 1 Faculty of Science University of Alexandria, Egypt
More informationComputers and Mathematics with Applications. The controllability of fractional control systems with control delay
Computers and Mathematics with Applications 64 (212) 3153 3159 Contents lists available at SciVerse ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
More informationEXACT TRAVELING WAVE SOLUTIONS FOR NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS USING THE IMPROVED (G /G) EXPANSION METHOD
Jan 4. Vol. 4 No. 7-4 EAAS & ARF. All rights reserved ISSN5-869 EXACT TRAVELIN WAVE SOLUTIONS FOR NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS USIN THE IMPROVED ( /) EXPANSION METHOD Elsayed M.
More informationInternational Journal of Mathematics Trends and Technology (IJMTT) Volume 48 Number 4 August 2017
Solving Fuzzy Fractional Differential Equation with Fuzzy Laplace Transform Involving Sine function Dr.S.Rubanraj 1, J.sangeetha 2 1 Associate professor, Department of Mathematics, St. Joseph s College
More informationREFLECTION SYMMETRIC FORMULATION OF GENERALIZED FRACTIONAL VARIATIONAL CALCULUS
RESEARCH PAPER REFLECTION SYMMETRIC FORMULATION OF GENERALIZED FRACTIONAL VARIATIONAL CALCULUS Ma lgorzata Klimek 1, Maria Lupa 2 Abstract We define generalized fractional derivatives GFDs symmetric and
More informationComputational Non-Polynomial Spline Function for Solving Fractional Bagely-Torvik Equation
Math. Sci. Lett. 6, No. 1, 83-87 (2017) 83 Mathematical Sciences Letters An International Journal http://dx.doi.org/10.18576/msl/060113 Computational Non-Polynomial Spline Function for Solving Fractional
More informationA collocation method for solving the fractional calculus of variation problems
Bol. Soc. Paran. Mat. (3s.) v. 35 1 (2017): 163 172. c SPM ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v35i1.26333 A collocation method for solving the fractional
More informationInternational Journal of Engineering Research and Generic Science (IJERGS) Available Online at
International Journal of Engineering Research and Generic Science (IJERGS) Available Online at www.ijergs.in Volume - 4, Issue - 6, November - December - 2018, Page No. 19-25 ISSN: 2455-1597 Fractional
More informationFractional Order Riemann-Liouville Integral Equations with Multiple Time Delays
Applied Mathematics E-Notes, 12(212), 79-87 c ISSN 167-251 Available free at mirror sites of http://www.math.nthu.edu.tw/amen/ Fractional Order Riemann-Liouville Integral Equations with Multiple Time Delays
More informationBoundary value problems for fractional differential equations with three-point fractional integral boundary conditions
Sudsutad and Tariboon Advances in Difference Equations 212, 212:93 http://www.advancesindifferenceequations.com/content/212/1/93 R E S E A R C H Open Access Boundary value problems for fractional differential
More informationarxiv: v1 [math.na] 8 Jan 2019
arxiv:190102503v1 [mathna] 8 Jan 2019 A Numerical Approach for Solving of Fractional Emden-Fowler Type Equations Josef Rebenda Zdeněk Šmarda c 2018 AIP Publishing This article may be downloaded for personal
More informationFinite Difference Method for the Time-Fractional Thermistor Problem
International Journal of Difference Equations ISSN 0973-6069, Volume 8, Number, pp. 77 97 203) http://campus.mst.edu/ijde Finite Difference Method for the Time-Fractional Thermistor Problem M. R. Sidi
More informationON THE C-LAGUERRE FUNCTIONS
ON THE C-LAGUERRE FUNCTIONS M. Ishteva, L. Boyadjiev 2 (Submitted by... on... ) MATHEMATIQUES Fonctions Specialles This announcement refers to a fractional extension of the classical Laguerre polynomials.
More informationANALYSIS AND APPLICATION OF DIFFUSION EQUATIONS INVOLVING A NEW FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL
Electronic Journal of Differential Equations, Vol. 217 (217), No. 289, pp. 1 6. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ANALYSIS AND APPLICATION OF DIFFUSION EQUATIONS
More informationNUMERICAL SOLUTION OF FRACTIONAL ORDER DIFFERENTIAL EQUATIONS USING HAAR WAVELET OPERATIONAL MATRIX
Palestine Journal of Mathematics Vol. 6(2) (217), 515 523 Palestine Polytechnic University-PPU 217 NUMERICAL SOLUTION OF FRACTIONAL ORDER DIFFERENTIAL EQUATIONS USING HAAR WAVELET OPERATIONAL MATRIX Raghvendra
More informationBritish Journal of Applied Science & Technology 10(2): 1-11, 2015, Article no.bjast ISSN:
British Journal of Applied Science & Technology 10(2): 1-11, 2015, Article no.bjast.18590 ISSN: 2231-0843 SCIENCEDOMAIN international www.sciencedomain.org Solutions of Sequential Conformable Fractional
More informationHigher monotonicity properties of q gamma and q-psi functions
Advances in Dynamical Systems and Applications ISSN 973-5321, Volume x, Number x, pp. 1 13 (2xx) http://campus.mst.edu/adsa Higher monotonicity properties of q gamma and q-psi functions Mourad E. H. Ismail
More informationTema Tendências em Matemática Aplicada e Computacional, 18, N. 2 (2017),
Tema Tendências em Matemática Aplicada e Computacional, 18, N 2 2017), 225-232 2017 Sociedade Brasileira de Matemática Aplicada e Computacional wwwscielobr/tema doi: 105540/tema2017018020225 New Extension
More informationOSCILLATORY PROPERTIES OF A CLASS OF CONFORMABLE FRACTIONAL GENERALIZED LIENARD EQUATIONS
IMPACT: International Journal of Research in Humanities, Arts and Literature (IMPACT: IJRHAL) ISSN (P): 2347-4564; ISSN (E): 2321-8878 Vol 6, Issue 11, Nov 2018, 201-214 Impact Journals OSCILLATORY PROPERTIES
More informationBOUNDARY VALUE PROBLEMS FOR FRACTIONAL INTEGRODIFFERENTIAL EQUATIONS INVOLVING GRONWALL INEQUALITY IN BANACH SPACE
J. Appl. Math. & Informatics Vol. 34(216, No. 3-4, pp. 193-26 http://dx.doi.org/1.14317/jami.216.193 BOUNDARY VALUE PROBLEMS FOR FRACTIONAL INTEGRODIFFERENTIAL EQUATIONS INVOLVING GRONWALL INEQUALITY IN
More informationHERMITE HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRALS
HERMITE HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRALS MARIAN MATŁOKA Abstract: In the present note, we have established an integral identity some Hermite-Hadamard type integral ineualities for the
More informationPositive solutions for nonlocal boundary value problems of fractional differential equation
Positive solutions for nonlocal boundary value problems of fractional differential equation YITAO YANG Tianjin University of Technology Department of Applied Mathematics No. 39 BinShuiWest Road, Xiqing
More informationNotes on uniform convergence
Notes on uniform convergence Erik Wahlén erik.wahlen@math.lu.se January 17, 2012 1 Numerical sequences We begin by recalling some properties of numerical sequences. By a numerical sequence we simply mean
More informationarxiv: v1 [math.ca] 28 Feb 2014
Communications in Nonlinear Science and Numerical Simulation. Vol.18. No.11. (213) 2945-2948. arxiv:142.7161v1 [math.ca] 28 Feb 214 No Violation of the Leibniz Rule. No Fractional Derivative. Vasily E.
More informationAbdulmalik Al Twaty and Paul W. Eloe
Opuscula Math. 33, no. 4 (23, 63 63 http://dx.doi.org/.7494/opmath.23.33.4.63 Opuscula Mathematica CONCAVITY OF SOLUTIONS OF A 2n-TH ORDER PROBLEM WITH SYMMETRY Abdulmalik Al Twaty and Paul W. Eloe Communicated
More informationOn a perturbed functional integral equation of Urysohn type. Mohamed Abdalla Darwish
On a perturbed functional integral equation of Urysohn type Mohamed Abdalla Darwish Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, Jeddah, Saudi Arabia Department of
More informationResearch Article New Method for Solving Linear Fractional Differential Equations
International Differential Equations Volume 2011, Article ID 814132, 8 pages doi:10.1155/2011/814132 Research Article New Method for Solving Linear Fractional Differential Equations S. Z. Rida and A. A.
More informationANALYTIC SOLUTIONS AND NUMERICAL SIMULATIONS OF MASS-SPRING AND DAMPER-SPRING SYSTEMS DESCRIBED BY FRACTIONAL DIFFERENTIAL EQUATIONS
ANALYTIC SOLUTIONS AND NUMERICAL SIMULATIONS OF MASS-SPRING AND DAMPER-SPRING SYSTEMS DESCRIBED BY FRACTIONAL DIFFERENTIAL EQUATIONS J.F. GÓMEZ-AGUILAR Departamento de Materiales Solares, Instituto de
More informationHigher Monotonicity Properties of q-gamma and q-psi Functions
Advances in Dynamical Systems and Applications ISSN 973-5321, Volume 8, Number 2, pp. 247 259 (213) http://campus.mst.edu/adsa Higher Monotonicity Properties of q-gamma and q-psi Functions Mourad E. H.
More informationIterative scheme to a coupled system of highly nonlinear fractional order differential equations
Computational Methods for Differential Equations http://cmde.tabrizu.ac.ir Vol. 3, No. 3, 215, pp. 163-176 Iterative scheme to a coupled system of highly nonlinear fractional order differential equations
More informationCritical exponents for a nonlinear reaction-diffusion system with fractional derivatives
Global Journal of Pure Applied Mathematics. ISSN 0973-768 Volume Number 6 (06 pp. 5343 535 Research India Publications http://www.ripublication.com/gjpam.htm Critical exponents f a nonlinear reaction-diffusion
More informationNONLINEAR DIFFERENTIAL INEQUALITY. 1. Introduction. In this paper the following nonlinear differential inequality
M athematical Inequalities & Applications [2407] First Galley Proofs NONLINEAR DIFFERENTIAL INEQUALITY N. S. HOANG AND A. G. RAMM Abstract. A nonlinear differential inequality is formulated in the paper.
More informationAN INTRODUCTION TO THE FRACTIONAL CALCULUS AND FRACTIONAL DIFFERENTIAL EQUATIONS
AN INTRODUCTION TO THE FRACTIONAL CALCULUS AND FRACTIONAL DIFFERENTIAL EQUATIONS KENNETH S. MILLER Mathematical Consultant Formerly Professor of Mathematics New York University BERTRAM ROSS University
More informationDifferential equations with fractional derivative and universal map with memory
IOP PUBLISHING JOURNAL OF PHYSIS A: MATHEMATIAL AND THEORETIAL J. Phys. A: Math. Theor. 42 (29) 46512 (13pp) doi:1.188/1751-8113/42/46/46512 Differential equations with fractional derivative and universal
More informationSome New Results on the New Conformable Fractional Calculus with Application Using D Alambert Approach
Progr. Fract. Differ. Appl. 2, No. 2, 115-122 (2016) 115 Progress in Fractional Differentiation and Applications An International Journal http://dx.doi.org/10.18576/pfda/020204 Some New Results on the
More informationFRACTIONAL DIFFERENTIAL EQUATIONS
FRACTIONAL DIFFERENTIAL EQUATIONS An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications by Igor Podlubny Technical University
More informationThe geometric and physical interpretation of fractional order derivatives of polynomial functions
The geometric and physical interpretation of fractional order derivatives of polynomial functions M.H. Tavassoli, A. Tavassoli, M.R. Ostad Rahimi Abstract. In this paper, after a brief mention of the definitions
More information