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Journal articles on the topic 'Fractional Difference'

Author: Grafiati

Published: 4 June 2025

Last updated: 15 July 2025

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1

Wang, Chao, Weiyu Xie, and Ravi P. Agarwal. "Quaternion Fractional Difference with Quaternionic Fractional Order and Applications to Fractional Difference Equation." Analele Universitatii "Ovidius" Constanta - Seria Matematica 32, no. 1 (2024): 271–304. https://doi.org/10.2478/auom-2024-0015.

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Abstract In this paper, we introduce the basic notions of the fractional summation, difference and q-difference with the quaternionic fractional order for the quaternion-valued functions and establish some of their basic properties. Based on this, the summation representations of solutions for the nonlinear quaternion-valued fractional difference equation and q-difference equation are obtained. In addition, several examples are provided to illustrate the feasibility of our obtained results in each section.

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2

G., Pushpalatha, and Ranjitha R. "Three Term Linear Fractional Nabla Difference Equation." International Journal of Trend in Scientific Research and Development 2, no. 3 (2018): 594–600. https://doi.org/10.31142/ijtsrd11059.

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In this present paper, a study on nabla difference equation and its third order linear fractional difference equation. A new generalized nabla difference equation is investigated from Three term linear fractional nabla difference equation. A relevant example is proved and justify the proposed notions. G. Pushpalatha | R. Ranjitha "Three-Term Linear Fractional Nabla Difference Equation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-3 , April 2018, URL: https://www.ijtsrd.com/papers/ijtsrd11059.pdf

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3

Pushpalatha, G., and R. Ranjitha. "Three-Term Linear Fractional Nabla Difference Equation." International Journal of Trend in Scientific Research and Development Volume-2, Issue-3 (2018): 594–600. http://dx.doi.org/10.31142/ijtsrd11059.

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4

Chasreechai, Saowaluck, and Thanin Sitthiwirattham. "On Separate Fractional Sum-Difference Equations with n-Point Fractional Sum-Difference Boundary Conditions via Arbitrary Different Fractional Orders." Mathematics 7, no. 5 (2019): 471. http://dx.doi.org/10.3390/math7050471.

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In this article, we study the existence and uniqueness results for a separate nonlinear Caputo fractional sum-difference equation with fractional difference boundary conditions by using the Banach contraction principle and the Schauder’s fixed point theorem. Our problem contains two nonlinear functions involving fractional difference and fractional sum. Moreover, our problem contains different orders in n + 1 fractional differences and m + 1 fractional sums. Finally, we present an illustrative example.

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5

Stanisławski, Rafał, and Krzysztof J. Latawiec. "Normalized finite fractional differences: Computational and accuracy breakthroughs." International Journal of Applied Mathematics and Computer Science 22, no. 4 (2012): 907–19. http://dx.doi.org/10.2478/v10006-012-0067-9.

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This paper presents a series of new results in finite and infinite-memory modeling of discrete-time fractional differences. The introduced normalized finite fractional difference is shown to properly approximate its fractional difference original, in particular in terms of the steady-state properties. A stability analysis is also presented and a recursive computation algorithm is offered for finite fractional differences. A thorough analysis of computational and accuracy aspects is culminated with the introduction of a perfect finite fractional difference and, in particular, a powerful adaptiv

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6

Cheng, Jin-Fa, and Yu-Ming Chu. "Fractional Difference Equations with Real Variable." Abstract and Applied Analysis 2012 (2012): 1–24. http://dx.doi.org/10.1155/2012/918529.

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We independently propose a new kind of the definition of fractional difference, fractional sum, and fractional difference equation, give some basic properties of fractional difference and fractional sum, and give some examples to demonstrate several methods of how to solve certain fractional difference equations.

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7

Mohan, J. Jagan, and G. V. S. R. Deekshitulu. "Fractional Order Difference Equations." International Journal of Differential Equations 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/780619.

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A difference equation is a relation between the differences of a function at one or more general values of the independent variable. These equations usually describe the evolution of certain phenomena over the course of time. The present paper deals with the existence and uniqueness of solutions of fractional difference equations.

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8

Abdeljawad, Thabet, and Bahaaeldin Abdalla. "Monotonicity results for delta and nabla caputo and Riemann fractional differences via dual identities." Filomat 31, no. 12 (2017): 3671–83. http://dx.doi.org/10.2298/fil1712671a.

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Recently, some authors have proved monotonicity results for delta and nabla fractional differences separately. In this article, we use dual identities relating delta and nabla fractional difference operators to prove shortly the monotonicity properties for the (left Riemann) nabla fractional differences using the corresponding delta type properties. Also, we proved some monotonicity properties for the Caputo fractional differences. Finally, we use the Q??operator dual identities to prove monotonicity results for the right fractional difference operators.

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9

D. Naga Purnima. "Fractional Difference Equations with Initial Time Difference." Communications on Applied Nonlinear Analysis 32, no. 8s (2025): 829–34. https://doi.org/10.52783/cana.v32.3825.

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10

Liu, Ru. "Fractional Difference Approximations for Time-Fractional Telegraph Equation." Journal of Applied Mathematics and Physics 06, no. 01 (2018): 301–9. http://dx.doi.org/10.4236/jamp.2018.61029.

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11

Abbas, Saïd, Mouffak Benchohra, Nadjet Laledj, and Yong Zhou. "Fractional ${q}$-difference equations on the half line." Archivum Mathematicum, no. 4 (2020): 207–23. http://dx.doi.org/10.5817/am2020-4-207.

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12

Wu, Guo–Cheng, and Dumitru Baleanu. "Stability analysis of impulsive fractional difference equations." Fractional Calculus and Applied Analysis 21, no. 2 (2018): 354–75. http://dx.doi.org/10.1515/fca-2018-0021.

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AbstractWe revisit motivation of the fractional difference equations and some recent applications to image encryption. Then stability of impulsive fractional difference equations is investigated in this paper. The fractional sum equation is considered and impulsive effects are introduced into discrete fractional calculus. A class of impulsive fractional difference equations are proposed. A discrete comparison principle is given and asymptotic stability of nonlinear fractional difference equation are discussed. Finally, an impulsive Mittag–Leffler stability is defined. The numerical result is p

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13

Chen, Minghua, and Weihua Deng. "Fourth Order Difference Approximations for Space Riemann-Liouville Derivatives Based on Weighted and Shifted Lubich Difference Operators." Communications in Computational Physics 16, no. 2 (2014): 516–40. http://dx.doi.org/10.4208/cicp.120713.280214a.

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AbstractHigh order discretization schemes play more important role in fractional operators than classical ones. This is because usually for classical derivatives the stencil for high order discretization schemes is wider than low order ones; but for fractional operators the stencils for high order schemes and low order ones are the same. Then using high order schemes to solve fractional equations leads to almost the same computational cost with first order schemes but the accuracy is greatly improved. Using the fractional linear multistep methods, Lubich obtains thev-th order (v <6) approxi

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14

Cheng, Jin-Fa, and Yu-Ming Chu. "On the Fractional Difference Equations of Order(2,q)." Abstract and Applied Analysis 2011 (2011): 1–16. http://dx.doi.org/10.1155/2011/497259.

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This paper presents a kind of new definition of fractional difference, fractional summation, and fractional difference equations and gives methods for explicitly solving fractional difference equations of order(2,q).

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15

Jagan Mohan, J., and G. V. S. R. Deekshitulu. "Difference inequalities of fractional order." Proyecciones (Antofagasta) 32, no. 3 (2013): 199–213. http://dx.doi.org/10.4067/s0716-09172013000300001.

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16

Jensen, Andreas Noack, and Morten Ørregaard Nielsen. "A FAST FRACTIONAL DIFFERENCE ALGORITHM." Journal of Time Series Analysis 35, no. 5 (2014): 428–36. http://dx.doi.org/10.1111/jtsa.12074.

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17

Atıcı, Ferhan M., and Sevgi Şengül. "Modeling with fractional difference equations." Journal of Mathematical Analysis and Applications 369, no. 1 (2010): 1–9. http://dx.doi.org/10.1016/j.jmaa.2010.02.009.

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18

Baliarsingh, P. "On a fractional difference operator." Alexandria Engineering Journal 55, no. 2 (2016): 1811–16. http://dx.doi.org/10.1016/j.aej.2016.03.037.

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19

Jonnalagadda, Jagan Mohan. "Impulsive nabla fractional difference equations." Fractional Differential Calculus, no. 2 (2022): 115–32. http://dx.doi.org/10.7153/fdc-2022-12-08.

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20

Abdeljawad, Thabet, and Arran Fernandez. "On a New Class of Fractional Difference-Sum Operators with Discrete Mittag-Leffler Kernels." Mathematics 7, no. 9 (2019): 772. http://dx.doi.org/10.3390/math7090772.

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We formulate a new class of fractional difference and sum operators, study their fundamental properties, and find their discrete Laplace transforms. The method depends on iterating the fractional sum operators corresponding to fractional differences with discrete Mittag–Leffler kernels. The iteration process depends on the binomial theorem. We note in particular the fact that the iterated fractional sums have a certain semigroup property, and hence, the new introduced iterated fractional difference-sum operators have this semigroup property as well.

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21

Abdeljawad, Thabet, Qasem M. Al-Mdallal, and Mohamed A. Hajji. "Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications." Discrete Dynamics in Nature and Society 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/4149320.

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Recently, Abdeljawad and Baleanu have formulated and studied the discrete versions of the fractional operators of order0<α≤1with exponential kernels initiated by Caputo-Fabrizio. In this paper, we extend the order of such fractional difference operators to arbitrary positive order. The extension is given to both left and right fractional differences and sums. Then, existence and uniqueness theorems for the Caputo (CFC) and Riemann (CFR) type initial difference value problems by using Banach contraction theorem are proved. Finally, a Lyapunov type inequality for the Riemann type fractional d

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22

Reunsumrit, Jiraporn, and Thanin Sitthiwirattham. "On the Nonlocal Fractional Delta-Nabla Sum Boundary Value Problem for Sequential Fractional Delta-Nabla Sum-Difference Equations." Mathematics 8, no. 4 (2020): 476. http://dx.doi.org/10.3390/math8040476.

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In this paper, we propose sequential fractional delta-nabla sum-difference equations with nonlocal fractional delta-nabla sum boundary conditions. The Banach contraction principle and the Schauder’s fixed point theorem are used to prove the existence and uniqueness results of the problem. The different orders in one fractional delta differences, one fractional nabla differences, two fractional delta sum, and two fractional nabla sum are considered. Finally, we present an illustrative example.

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23

J. Jagan Mohan and G.V.S.R. Deekshitulu. "Solutions of fractional difference equations using S-transforms." Malaya Journal of Matematik 1, no. 03 (2013): 7–13. http://dx.doi.org/10.26637/mjm103/002.

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In the present paper, we define the nabla discrete Sumudu transform (S-transform) and present some of its basic properties. We obtain the nabla discrete Sumudu transform of fractional sums and differences. We apply this transform to solve some fractional difference equations with initial value problems. Finally, using S-transforms, we prove that discrete Mittag-Leffler function is the eigen function of Caputo type fractional difference operator $\nabla^\alpha$.

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24

Stanisławski, Rafał. "New Laguerre Filter Approximators to the Grünwald-Letnikov Fractional Difference." Mathematical Problems in Engineering 2012 (2012): 1–21. http://dx.doi.org/10.1155/2012/732917.

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This paper presents a series of new results in modeling of the Grünwald-Letnikov discrete-time fractional difference by means of discrete-time Laguerre filers. The introduced Laguerre-based difference (LD) and combined fractional/Laguerre-based difference (CFLD) are shown to perfectly approximate its fractional difference original, for fractional order . This paper is culminated with the presentation of finite (combined) fractional/Laguerre-based difference (FFLD), whose excellent approximation performance is illustrated in simulation examples.

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25

Suwan, Iyad, Shahd Owies, Muayad Abussa, and Thabet Abdeljawad. "Monotonicity Analysis of Fractional Proportional Differences." Discrete Dynamics in Nature and Society 2020 (May 1, 2020): 1–11. http://dx.doi.org/10.1155/2020/4867927.

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In this work, the nabla discrete new Riemann–Liouville and Caputo fractional proportional differences of order 0<ε<1 on the time scale ℤ are formulated. The differences and summations of discrete fractional proportional are detected on ℤ, and the fractional proportional sums associated to ∇cRχε,ρz with order 0<ε<1 are defined. The relation between nabla Riemann–Liouville and Caputo fractional proportional differences is derived. The monotonicity results for the nabla Caputo fractional proportional difference are proved; specifically, if ∇c−1Rχε,ρz>0 then χz is ερ −increasing, an

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26

Mozyrska, Dorota. "Multiparameter Fractional Difference Linear Control Systems." Discrete Dynamics in Nature and Society 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/183782.

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The Riemann-Liouville-, Caputo-, and Grünwald-Letnikov-type fractional order difference operators are discussed and used to state and solve the controllability and observability problems of linear fractional order discrete-time control systems with multiorder and multistep. It is shown that the obtained results do not depend on the type of fractional operators and steps. The comparison of systems is made under the number of steps needed, firstly to achieve a final point, and secondly to distinguish initial conditions for particular operator.

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27

Abdeljawad, Thabet. "On Delta and Nabla Caputo Fractional Differences and Dual Identities." Discrete Dynamics in Nature and Society 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/406910.

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We investigate two types of dual identities for Caputo fractional differences. The first type relates nabla and delta type fractional sums and differences. The second type represented by theQ-operator relates left and right fractional sums and differences. Two types of Caputo fractional differences are introduced; one of them (dual one) is defined so that it obeys the investigated dual identities. The relation between Riemann and Caputo fractional differences is investigated, and the delta and nabla discrete Mittag-Leffler functions are confirmed by solving Caputo type linear fractional differ

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28

Mohammed, Pshtiwan Othman, Thabet Abdeljawad, and Faraidun Kadir Hamasalh. "On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis." Mathematics 9, no. 11 (2021): 1303. http://dx.doi.org/10.3390/math9111303.

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Monotonicity analysis of delta fractional sums and differences of order υ∈(0,1] on the time scale hZ are presented in this study. For this analysis, two models of discrete fractional calculus, Riemann–Liouville and Caputo, are considered. There is a relationship between the delta Riemann–Liouville fractional h-difference and delta Caputo fractional h-differences, which we find in this study. Therefore, after we solve one, we can apply the same method to the other one due to their correlation. We show that y(z) is υ-increasing on Ma+υh,h, where the delta Riemann–Liouville fractional h-differenc

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29

Ghuge, Vijaymala, T. L. Holambe, Bhausaheb Sontakke, and Gajanan Shrimangale. "Solving Time-fractional Order Radon Diffusion Equation in Water by Finite Difference Method." Indian Journal Of Science And Technology 17, no. 19 (2024): 1994–2001. http://dx.doi.org/10.17485/ijst/v17i19.868.

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Objective: The aim of this research is to gain a comprehensive understanding of radon diffusion equation in water. Methods: A time fractional radon diffusion equation with Caputo sense is employed to find diffusion dynamics of radon in water medium. The fractional order explicit finite difference technique is used to find its numerical solution. A Python software is used to find numerical solution. Findings: The effect of fractional-order parameters on the distribution and concentration profiles of radon in water has been investigated. Furthermore, we study stability and convergence of the exp

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30

Liu, Kewei, and Wei Jiang. "Stability of Nonlinear Fractional Neutral Differential Difference Systems." Discrete Dynamics in Nature and Society 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/514631.

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We study the stability of a class of nonlinear fractional neutral differential difference systems equipped with the Caputo derivative. We extend Lyapunov-Krasovskii theorem for the nonlinear fractional neutral systems. Conditions of stability and instability are obtained for the nonlinear fractional neutral systems.

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31

Khader, M. M., and A. S. Hendy. "FRACTIONAL CHEBYSHEV FINITE DIFFERENCE METHOD FOR SOLVING THE FRACTIONAL BVPS." Journal of applied mathematics & informatics 31, no. 1_2 (2013): 299–309. http://dx.doi.org/10.14317/jami.2013.299.

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32

Edelman, M. "Caputo standard α-family of maps: Fractional difference vs. fractional". Chaos: An Interdisciplinary Journal of Nonlinear Science 24, № 2 (2014): 023137. http://dx.doi.org/10.1063/1.4885536.

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33

Hao, Zhaopeng, Zhongqiang Zhang, and Rui Du. "Fractional centered difference scheme for high-dimensional integral fractional Laplacian." Journal of Computational Physics 424 (January 2021): 109851. http://dx.doi.org/10.1016/j.jcp.2020.109851.

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34

MASSOPUST, PETER, and BRIGITTE FORSTER. "SOME REMARKS ABOUT THE CONNECTION BETWEEN FRACTIONAL DIVIDED DIFFERENCES, FRACTIONAL B-SPLINES, AND THE HERMITE–GENOCCHI FORMULA." International Journal of Wavelets, Multiresolution and Information Processing 06, no. 02 (2008): 279–90. http://dx.doi.org/10.1142/s0219691308002343.

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Fractional B-splines are a natural extension of classical B-splines. In this short paper, we show their relations to fractional divided differences and fractional difference operators, and present a generalized Hermite–Genocchi formula. This formula then allows the definition of a larger class of fractional B-splines.

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35

Monisha., G., and S. Savitha. "Fractional Quadruple Laplace Transform and its Properties." International Journal of Trend in Scientific Research and Development 2, no. 5 (2018): 2404–9. https://doi.org/10.31142/ijtsrd18280.

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In this paper, we introduce definition for fractional quadruple Laplace transform of order a,0 a = 1, for fractional differentiable functions. Some main properties and inversion theorem of fractional quadruple Laplace transform are established. Further, the connection between fractional quadruple Laplace transform and fractional Sumudu transform are presented. Monisha. G | Savitha. S "Fractional Quadruple Laplace Transform and its Properties" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-5 , August 2018,

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36

Vijaymala, Ghuge, L. Holambe T, Sontakke Bhausaheb, and Shrimangale Gajanan. "Solving Time-fractional Order Radon Diffusion Equation in Water by Finite Difference Method." Indian Journal of Science and Technology 17, no. 19 (2024): 1994–2001. https://doi.org/10.17485/IJST/v17i19.868.

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Abstract <strong>Objective:</strong> The aim of this research is to gain a comprehensive understanding of radon diffusion equation in water. <strong>Methods:</strong> A time fractional radon diffusion equation with Caputo sense is employed to find diffusion dynamics of radon in water medium. The fractional order explicit finite difference technique is used to find its numerical solution. A Python software is used to find numerical solution.<strong> Findings:</strong> The effect of fractional-order parameters on the distribution and concentration profiles of radon in wa

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37

Raj, Kuldip, Kavita Saini та Anu Choudhary. "Orlicz lacunary sequence spaces of 𝑙-fractional difference operators". Journal of Applied Analysis 26, № 2 (2020): 173–83. http://dx.doi.org/10.1515/jaa-2020-2018.

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AbstractRecently, S. K. Mahato and P. D. Srivastava [A class of sequence spaces defined by 𝑙-fractional difference operator, preprint 2018, http://arxiv.org/abs/1806.10383] studied 𝑙-fractional difference sequence spaces. In this article, we intend to make a new approach to introduce and study some lambda 𝑙-fractional convergent, lambda 𝑙-fractional null and lambda 𝑙-fractional bounded sequences over 𝑛-normed spaces. Various algebraic and topological properties of these newly formed sequence spaces have been explored, and some inclusion relations concerning these spaces are also established. F

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38

Xiao-yan, Li, and Jiang Wei. "Solving Fractional Difference Equations Using the Laplace Transform Method." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/230850.

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We discuss the Laplace transform of the Caputo fractional difference and the fractional discrete Mittag-Leffer functions. On these bases, linear and nonlinear fractional initial value problems are solved by the Laplace transform method.

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39

Ferreira, Rui, and Delfim Torres. "Fractional h-difference equations arising from the calculus of variations." Applicable Analysis and Discrete Mathematics 5, no. 1 (2011): 110–21. http://dx.doi.org/10.2298/aadm110131002f.

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The recent theory of fractional h-difference equations introduced in [N.R.O. Bastos, R. A. C. Ferreira, D. F. M. Torres: Discrete-time fractional variational problems, Signal Process. 91 (2011), no. 3, 513{524], is enriched with useful tools for the explicit solution of discrete equations involving left and right fractional difference operators. New results for the right fractional h sum are proved. Illustrative examples show the effectiveness of the obtained results in solving fractional discrete Euler{Lagrange equations.

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40

Tarasov, Vasily E. "Fractional Diffusion Equations for Lattice and Continuum: Grünwald-Letnikov Differences and Derivatives Approach." International Journal of Statistical Mechanics 2014 (December 8, 2014): 1–7. http://dx.doi.org/10.1155/2014/873529.

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Fractional diffusion equations for three-dimensional lattice models based on fractional-order differences of the Grünwald-Letnikov type are suggested. These lattice fractional diffusion equations contain difference operators that describe long-range jumps from one lattice site to another. In continuum limit, the suggested lattice diffusion equations with noninteger order differences give the diffusion equations with the Grünwald-Letnikov fractional derivatives for continuum. We propose a consistent derivation of the fractional diffusion equation with the fractional derivatives of Grünwald-Letn

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41

Edelman, M. "Fractional Maps and Fractional Attractors. Part II: Fractional Difference Caputo α- Families of Maps". Interdisciplinary journal of Discontinuity, Nonlinearity, and Complexity 4, № 4 (2015): 391–402. http://dx.doi.org/10.5890/dnc.2015.11.003.

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42

Shiv, Narain*1 &. Meenu Goel2. "AN IMPLICIT NUMERICAL SCHEME FOR FRACTIONAL ADVENTION DIFFUSION EQUATION." GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES 6, no. 5 (2019): 94–99. https://doi.org/10.5281/zenodo.2694021.

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In this paper, a finite difference scheme is presented for time fractional advection diffusion equation (TFADE). This equation is derived from classical advection diffusion equation with variable coefficients on replacing classical integer order derivatives by their fractional counterpart. An advection diffusion equation describes physical phenomenon where particle, energy or other physical quantities are transferred inside a physical system due to combined effect of advection and diffusion. To address anomalous diffusion like sub diffusion or super diffusion, classical integer derivatives are

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43

Tarasov, Vasily E., and Valentina V. Tarasova. "Long and Short Memory in Economics: Fractional-Order Difference and Differentiation." IRA-International Journal of Management & Social Sciences (ISSN 2455-2267) 5, no. 2 (2016): 327. http://dx.doi.org/10.21013/jmss.v5.n2.p10.

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<div><p><em>Long and short memory in economic processes is usually described by the so-called discrete fractional differencing and fractional integration. We prove that the discrete fractional differencing and integration are the </em><em>Grunwald-Letnikov fractional differences of non-integer order d. Equations of ARIMA(p,d,q) and ARFIMA(p,d,q) models are the fractional-order difference equations with the Grunwald-Letnikov differences of order d. We prove that the long and short memory with power law should be described by the exact fractional-order differences,

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44

Chatzarakis, George E., Palaniyappan Gokulraj, and Thirunavukarasu Kalaimani. "Oscillation Tests for Fractional Difference Equations." Tatra Mountains Mathematical Publications 71, no. 1 (2018): 53–64. http://dx.doi.org/10.2478/tmmp-2018-0005.

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Abstract In this paper, we study the oscillatory behavior of solutions of the fractional difference equation of the form $$\Delta \left( {r\left( t \right)g\left( {{\Delta ^\alpha }x(t)} \right)} \right) + p(t)f\left( {\sum\limits_{s = {t_0}}^{t - 1 + \alpha } {{{(t - s - 1)}^{( - \alpha )}}x(s)} } \right) = 0, & t \in {_{{t_0} + 1 - \alpha }},$$ where Δα denotes the Riemann-Liouville fractional difference operator of order α, 0 < α ≤ 1, ℕt0+1−α={t0+1−αt0+2−α…}, t0 > 0 and γ > 0 is a quotient of odd positive integers. We establish some oscillatory criteria for the above equation,

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45

Meganathan, M., and G. B. A. Xavier. "IMPULSIVE FRACTIONAL DIFFERENCE EQUATION ON hZ." Advances in Mathematics: Scientific Journal 9, no. 3 (2020): 1215–24. http://dx.doi.org/10.37418/amsj.9.3.65.

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46

Gopal, N. S., and Jagan Mohan Jonnalagadda. "Linear Hilfer nabla fractional difference equations." International Journal of Dynamical Systems and Differential Equations 11, no. 3/4 (2021): 322. http://dx.doi.org/10.1504/ijdsde.2021.10040327.

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Jonnalagadda, Jagan Mohan, and N. S. Gopal. "Linear Hilfer nabla fractional difference equations." International Journal of Dynamical Systems and Differential Equations 11, no. 3/4 (2021): 322. http://dx.doi.org/10.1504/ijdsde.2021.117361.

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Agiakloglou, Christos, and Paul Newbold. "LAGRANGE MULTIPLIER TESTS FOR FRACTIONAL DIFFERENCE." Journal of Time Series Analysis 15, no. 3 (1994): 253–62. http://dx.doi.org/10.1111/j.1467-9892.1994.tb00190.x.

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Diekema, Enno. "The Fractional Orthogonal Difference with Applications." Mathematics 3, no. 2 (2015): 487–509. http://dx.doi.org/10.3390/math3020487.

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Feckan, Michal, and Michal Pospíšil. "Note on fractional difference Gronwall inequalities." Electronic Journal of Qualitative Theory of Differential Equations, no. 44 (2014): 1–18. http://dx.doi.org/10.14232/ejqtde.2014.1.44.

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