Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound

Christopher S. Goodrich; Benjamin Lyons; Mihaela T. Velcsov

doi:10.3934/cpaa.2020269

Previous Article
Isomorphism between one-dimensional and multidimensional finite difference operators
CPAA Home
This Issue
Next Article
Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional

January 2021, 20(1): 339-358. doi: 10.3934/cpaa.2020269

Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound

Christopher S. Goodrich ^1,, , Benjamin Lyons ^2, and Mihaela T. Velcsov ^3,

1.	School of Mathematics and Statistics, UNSW Sydney, NSW 2052 Australia
2.	Department of Mathematics, Creighton Preparatory School, Omaha, NE 68114 USA
3.	Department of Mathematics, University of Nebraska-Omaha, Omaha, NE 68182-0243 USA

^* Corresponding author

Received April 2020 Revised August 2020 Published November 2020

We investigate the relationship between the sign of the discrete fractional sequential difference

$ \big(\Delta_{1+a-\mu}^{\nu}\Delta_a^{\mu}f\big)(t) $

and the monotonicity of the function

$ t\mapsto f(t) $

. More precisely, we consider the special case in which this fractional difference can be negative and satisfies the lower bound

$ \begin{equation} \big(\Delta_{1+a-\mu}^{\nu}\Delta_a^{\mu}f\big)(t)\ge-\varepsilon f(a),\notag \end{equation} $

for some

$ \varepsilon>0 $

. We prove that even though the fractional difference can be negative, the monotonicity of the function

$ f $

, nonetheless, is still implied by the above inequality. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. Because of the challenges of a purely analytical approach, our analysis includes numerical simulation.

Keywords: Monotonicity, discrete fractional calculus, Gamma function, sequential difference, numerical approximation.

Mathematics Subject Classification: Primary: 26A48, 33F05, 39A12, 65D15, 65Q20; Secondary: 39A99, 39B62.

Citation: Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, 2021, 20 (1) : 339-358. doi: 10.3934/cpaa.2020269

References:

[1]	T. Abdeljawad and B. Abdalla, Monotonicity results for delta and nabla Caputo and Riemann fractional differences via dual identities, Filomat, 31 (2017), 3671-3683. doi: 10.2298/fil1712671a. Google Scholar
[2]	T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Adv. Differ. Equ., (2017), 9 pp. doi: 10.1186/s13662-017-1126-1. Google Scholar
[3]	G. A. Anastassiou, Nabla discrete fractional calculus and nabla inequalities, Math. Comput. Model., 51 (2010), 562-571. doi: 10.1016/j.mcm.2009.11.006. Google Scholar
[4]	F. M. Atici and P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Differ. Equ., 2 (2007), 165-176. Google Scholar
[5]	F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, P. Am. Math. Soc., 137 (2009), 981-989. doi: 10.1090/S0002-9939-08-09626-3. Google Scholar
[6]	F. M. Atici and P. W. Eloe, Two-point boundary value problems for finite fractional difference equations, J. Differ. Equ. Appl., 17 (2011), 445-456. doi: 10.1080/10236190903029241. Google Scholar
[7]	F. M. Atici and M. Uyanik, Analysis of discrete fractional operators, Appl. Anal. Discrete Math., 9 (2015), 139-149. doi: 10.2298/AADM150218007A. Google Scholar
[8]	B. C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, 1977. Google Scholar
[9]	R. Dahal and C. S. Goodrich, A monotonicity result for discrete fractional difference operators, Arch. Math., 102 (2014), 293-299. doi: 10.1007/s00013-014-0620-x. Google Scholar
[10]	R. Dahal and C. S. Goodrich, An almost sharp monotonicity result for discrete sequential fractional delta differences, J. Differ. Equ. Appl., 23 (2017), 1190-1203. doi: 10.1080/10236198.2017.1307351. Google Scholar
[11]	R. Dahal and C. S. Goodrich, Mixed order monotonicity results for sequential fractional nabla differences, J. Differ. Equ. Appl., 25 (2019), 837-854. doi: 10.1080/10236198.2018.1561883. Google Scholar
[12]	F. Du, B. Jia, L. Erbe and A. Peterson, Monotonicity and convexity for nabla fractional $(q, h)$-differences, J. Differ. Equ. Appl., 22 (2016), 1224-1243. doi: 10.1080/10236198.2016.1188089. Google Scholar
[13]	L. Erbe, C. S. Goodrich, B. Jia and A. Peterson, Survey of the qualitative properties of fractional difference operators: monotonicity, convexity, and asymptotic behavior of solutions, Adv. Differ Equ., (2016), 31 pp. doi: 10.1186/s13662-016-0760-3. Google Scholar
[14]	R. A. C. Ferreira, A discrete fractional Gronwall inequality, P. Am. Math. Soc., 140 (2012), 1605-1612. doi: 10.1090/S0002-9939-2012-11533-3. Google Scholar
[15]	C. S. Goodrich, On discrete sequential fractional boundary value problems, J. Math. Anal. Appl., 385 (2012), 111-124. doi: 10.1016/j.jmaa.2011.06.022. Google Scholar
[16]	C. S. Goodrich, A note on convexity, concavity, and growth conditions in discrete fractional calculus with delta difference, Math. Inequal. Appl., 19 (2016), 769-779. doi: 10.7153/mia-19-57. Google Scholar
[17]	C. S. Goodrich, A sharp convexity result for sequential fractional delta differences, J. Differ. Equ. Appl., 23 (2017), 1986-2003. doi: 10.1080/10236198.2017.1380635. Google Scholar
[18]	C. S. Goodrich, A uniformly sharp monotonicity result for discrete fractional sequential differences, Arch. Math., 110 (2018), 145-154. doi: 10.1007/s00013-017-1106-4. Google Scholar
[19]	C. S. Goodrich, Sharp monotonicity results for fractional nabla sequential differences, J. Differ. Equ. Appl., 25 (2019), 801-814. doi: 10.1080/10236198.2018.1542431. Google Scholar
[20]	C. S. Goodrich and C. Lizama, A transference principle for nonlocal operators using a convolutional approach: fractional monotonicity and convexity, Israel J. Math., 236 (2020), 533-589. doi: 10.1007/s11856-020-1991-2. Google Scholar
[21]	C. S. Goodrich and C. Lizama, Positivity, monotonicity, and convexity for convolution operators, Discrete Contin. Dyn. Syst., 40 (2020), 4961-4983. doi: 10.3934/dcds.2020207. Google Scholar
[22]	C. S. Goodrich and B. Lyons, Positivity and monotonicity results for triple sequential fractional differences via convolution, Analysis (Berlin), 40 (2020), 89-103. doi: 10.1515/anly-2019-0050. Google Scholar
[23]	C. S. Goodrich and M. Muellner, An analysis of the sharpness of monotonicity results via homotopy for sequential fractional operators, Appl. Math. Lett., 98 (2019), 446-452. doi: 10.1016/j.aml.2019.07.003. Google Scholar
[24]	C. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer, New York, 2015. doi: 10.1007/978-3-319-25562-0. Google Scholar
[25]	M. Holm, Sum and difference compositions in discrete fractional calculus, Cubo, 13 (2011), 153-184. doi: 10.4067/s0719-06462011000300009. Google Scholar
[26]	B. Jia, L. Erbe, C. S. Goodrich and A. Peterson, Monotonicity results for delta fractional differences revisited, Math. Slovaca, 67 (2017), 895-906. doi: 10.1515/ms-2017-0018. Google Scholar
[27]	B. Jia, L. Erbe and A. Peterson, Two monotonicity results for nabla and delta fractional differences, Arch. Math., 104 (2015), 589-597. doi: 10.1007/s00013-015-0765-2. Google Scholar
[28]	B. Jia, L. Erbe and A. Peterson, Monotonicity and convexity for nabla fractional $q$-differences, Dynam. Systems Appl., 25 (2016), 47-60. Google Scholar
[29]	B. Jia, L. Erbe and A. Peterson, Convexity for nabla and delta fractional differences, J. Differ. Equ. Appl., 21 (2015), 360-373. doi: 10.1080/10236198.2015.1011630. Google Scholar
[30]	B. Jia, L. Erbe and A. Peterson, Some relations between the Caputo fractional difference operators and integer-order differences, Electron. J. Differ. Equ., (2015), 7 pp. Google Scholar
[31]	J. M. Jonnalagadda, An ordering on Green's function and a Lyapunov-type inequality for a family of nabla fractional boundary value problems, Fract. Differ. Calc., 9 (2019), 109-124. doi: 10.7153/fdc-2019-09-08. Google Scholar
[32]	A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland, New York, 2006. Google Scholar
[33]	C. Lizama, The Poisson distribution, abstract fractional difference equations, and stability, P. Am. Math. Soc., 145 (2017), 3809-3827. doi: 10.1090/proc/12895. Google Scholar
[34]	C. Lizama and M. Murillo-Arcila, Well posedness for semidiscrete fractional Cauchy problems with finite delay, J. Comput. Appl. Math., 339 (2018), 356-366. doi: 10.1016/j.cam.2017.07.027. Google Scholar
[35]	R. Wong and R. Beals, Special Functions: A Graduate Text, Cambridge University Press, New York, 2010. doi: 10.1017/CBO9780511762543. Google Scholar

show all references

References:

[1]	T. Abdeljawad and B. Abdalla, Monotonicity results for delta and nabla Caputo and Riemann fractional differences via dual identities, Filomat, 31 (2017), 3671-3683. doi: 10.2298/fil1712671a. Google Scholar
[2]	T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Adv. Differ. Equ., (2017), 9 pp. doi: 10.1186/s13662-017-1126-1. Google Scholar
[3]	G. A. Anastassiou, Nabla discrete fractional calculus and nabla inequalities, Math. Comput. Model., 51 (2010), 562-571. doi: 10.1016/j.mcm.2009.11.006. Google Scholar
[4]	F. M. Atici and P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Differ. Equ., 2 (2007), 165-176. Google Scholar
[5]	F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, P. Am. Math. Soc., 137 (2009), 981-989. doi: 10.1090/S0002-9939-08-09626-3. Google Scholar
[6]	F. M. Atici and P. W. Eloe, Two-point boundary value problems for finite fractional difference equations, J. Differ. Equ. Appl., 17 (2011), 445-456. doi: 10.1080/10236190903029241. Google Scholar
[7]	F. M. Atici and M. Uyanik, Analysis of discrete fractional operators, Appl. Anal. Discrete Math., 9 (2015), 139-149. doi: 10.2298/AADM150218007A. Google Scholar
[8]	B. C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, 1977. Google Scholar
[9]	R. Dahal and C. S. Goodrich, A monotonicity result for discrete fractional difference operators, Arch. Math., 102 (2014), 293-299. doi: 10.1007/s00013-014-0620-x. Google Scholar
[10]	R. Dahal and C. S. Goodrich, An almost sharp monotonicity result for discrete sequential fractional delta differences, J. Differ. Equ. Appl., 23 (2017), 1190-1203. doi: 10.1080/10236198.2017.1307351. Google Scholar
[11]	R. Dahal and C. S. Goodrich, Mixed order monotonicity results for sequential fractional nabla differences, J. Differ. Equ. Appl., 25 (2019), 837-854. doi: 10.1080/10236198.2018.1561883. Google Scholar
[12]	F. Du, B. Jia, L. Erbe and A. Peterson, Monotonicity and convexity for nabla fractional $(q, h)$-differences, J. Differ. Equ. Appl., 22 (2016), 1224-1243. doi: 10.1080/10236198.2016.1188089. Google Scholar
[13]	L. Erbe, C. S. Goodrich, B. Jia and A. Peterson, Survey of the qualitative properties of fractional difference operators: monotonicity, convexity, and asymptotic behavior of solutions, Adv. Differ Equ., (2016), 31 pp. doi: 10.1186/s13662-016-0760-3. Google Scholar
[14]	R. A. C. Ferreira, A discrete fractional Gronwall inequality, P. Am. Math. Soc., 140 (2012), 1605-1612. doi: 10.1090/S0002-9939-2012-11533-3. Google Scholar
[15]	C. S. Goodrich, On discrete sequential fractional boundary value problems, J. Math. Anal. Appl., 385 (2012), 111-124. doi: 10.1016/j.jmaa.2011.06.022. Google Scholar
[16]	C. S. Goodrich, A note on convexity, concavity, and growth conditions in discrete fractional calculus with delta difference, Math. Inequal. Appl., 19 (2016), 769-779. doi: 10.7153/mia-19-57. Google Scholar
[17]	C. S. Goodrich, A sharp convexity result for sequential fractional delta differences, J. Differ. Equ. Appl., 23 (2017), 1986-2003. doi: 10.1080/10236198.2017.1380635. Google Scholar
[18]	C. S. Goodrich, A uniformly sharp monotonicity result for discrete fractional sequential differences, Arch. Math., 110 (2018), 145-154. doi: 10.1007/s00013-017-1106-4. Google Scholar
[19]	C. S. Goodrich, Sharp monotonicity results for fractional nabla sequential differences, J. Differ. Equ. Appl., 25 (2019), 801-814. doi: 10.1080/10236198.2018.1542431. Google Scholar
[20]	C. S. Goodrich and C. Lizama, A transference principle for nonlocal operators using a convolutional approach: fractional monotonicity and convexity, Israel J. Math., 236 (2020), 533-589. doi: 10.1007/s11856-020-1991-2. Google Scholar
[21]	C. S. Goodrich and C. Lizama, Positivity, monotonicity, and convexity for convolution operators, Discrete Contin. Dyn. Syst., 40 (2020), 4961-4983. doi: 10.3934/dcds.2020207. Google Scholar
[22]	C. S. Goodrich and B. Lyons, Positivity and monotonicity results for triple sequential fractional differences via convolution, Analysis (Berlin), 40 (2020), 89-103. doi: 10.1515/anly-2019-0050. Google Scholar
[23]	C. S. Goodrich and M. Muellner, An analysis of the sharpness of monotonicity results via homotopy for sequential fractional operators, Appl. Math. Lett., 98 (2019), 446-452. doi: 10.1016/j.aml.2019.07.003. Google Scholar
[24]	C. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer, New York, 2015. doi: 10.1007/978-3-319-25562-0. Google Scholar
[25]	M. Holm, Sum and difference compositions in discrete fractional calculus, Cubo, 13 (2011), 153-184. doi: 10.4067/s0719-06462011000300009. Google Scholar
[26]	B. Jia, L. Erbe, C. S. Goodrich and A. Peterson, Monotonicity results for delta fractional differences revisited, Math. Slovaca, 67 (2017), 895-906. doi: 10.1515/ms-2017-0018. Google Scholar
[27]	B. Jia, L. Erbe and A. Peterson, Two monotonicity results for nabla and delta fractional differences, Arch. Math., 104 (2015), 589-597. doi: 10.1007/s00013-015-0765-2. Google Scholar
[28]	B. Jia, L. Erbe and A. Peterson, Monotonicity and convexity for nabla fractional $q$-differences, Dynam. Systems Appl., 25 (2016), 47-60. Google Scholar
[29]	B. Jia, L. Erbe and A. Peterson, Convexity for nabla and delta fractional differences, J. Differ. Equ. Appl., 21 (2015), 360-373. doi: 10.1080/10236198.2015.1011630. Google Scholar
[30]	B. Jia, L. Erbe and A. Peterson, Some relations between the Caputo fractional difference operators and integer-order differences, Electron. J. Differ. Equ., (2015), 7 pp. Google Scholar
[31]	J. M. Jonnalagadda, An ordering on Green's function and a Lyapunov-type inequality for a family of nabla fractional boundary value problems, Fract. Differ. Calc., 9 (2019), 109-124. doi: 10.7153/fdc-2019-09-08. Google Scholar
[32]	A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland, New York, 2006. Google Scholar
[33]	C. Lizama, The Poisson distribution, abstract fractional difference equations, and stability, P. Am. Math. Soc., 145 (2017), 3809-3827. doi: 10.1090/proc/12895. Google Scholar
[34]	C. Lizama and M. Murillo-Arcila, Well posedness for semidiscrete fractional Cauchy problems with finite delay, J. Comput. Appl. Math., 339 (2018), 356-366. doi: 10.1016/j.cam.2017.07.027. Google Scholar
[35]	R. Wong and R. Beals, Special Functions: A Graduate Text, Cambridge University Press, New York, 2010. doi: 10.1017/CBO9780511762543. Google Scholar

Figure 1. Graphical representation of the set $ \mathscr{E}_{k, 0.1} $ for $ k \le 72 $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. Graphical representation of the set $ \mathscr{E}_{k, 0.01} $ for $ k \le 72 $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. Graphical representation of the set $ \mathscr{E}_{k, 0.001} $ for $ k \le 72 $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4. Graphical representation of the set $ \mathscr{E}_{k, 0.0001} $ for $ k \le 72 $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5. Graphical representation of the set $ \mathscr{E}_{k, 0.00001} $ for $ k \le 72 $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6. Heat maps for the cardinality of the set $ \{ k\ : \ (\mu, \nu ) \in \mathscr{E}_{k,\varepsilon}\} $ for $ \varepsilon = 0.01, 0.001, 0.0001, 0.00001, 0.000001, 0.0000001 $. The cardinality increases from small (dark blue) to large (dark red) and the actual cardinalities are shown along the sidebar of each subplot

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7. Heat maps for the cardinality of the set $ \{ k\ : \ (\mu, \nu ) \in \mathscr{E}_{k,\varepsilon}\} $ for $ \varepsilon = 1/100, 1/150, 1/400, 1/650, 1/900, 1/1000 $. These correspond to the interval of $ \varepsilon $ reflected in the top two subplots of Figure 6. Notice the change of cardinality values as $ \varepsilon $ decreases

Figure Options

Download full-size image

Download as PowerPoint slide

[1]	Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268
[2]	Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296
[3]	Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020319
[4]	Baoli Yin, Yang Liu, Hong Li, Zhimin Zhang. Approximation methods for the distributed order calculus using the convolution quadrature. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1447-1468. doi: 10.3934/dcdsb.2020168
[5]	Manuel Friedrich, Martin Kružík, Jan Valdman. Numerical approximation of von Kármán viscoelastic plates. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 299-319. doi: 10.3934/dcdss.2020322
[6]	Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020443
[7]	Madhurima Mukhopadhyay, Palash Sarkar, Shashank Singh, Emmanuel Thomé. New discrete logarithm computation for the medium prime case using the function field sieve. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020119
[8]	Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033
[9]	Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213
[10]	Xiaoli Lu, Pengzhan Huang, Yinnian He. Fully discrete finite element approximation of the 2D/3D unsteady incompressible magnetohydrodynamic-Voigt regularization flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 815-845. doi: 10.3934/dcdsb.2020143
[11]	Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020466
[12]	Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020355
[13]	Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366
[14]	Guo-Niu Han, Huan Xiong. Skew doubled shifted plane partitions: Calculus and asymptotics. Electronic Research Archive, 2021, 29 (1) : 1841-1857. doi: 10.3934/era.2020094
[15]	Yueh-Cheng Kuo, Huey-Er Lin, Shih-Feng Shieh. Asymptotic dynamics of hermitian Riccati difference equations. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020365
[16]	Bo Tan, Qinglong Zhou. Approximation properties of Lüroth expansions. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020389
[17]	George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003
[18]	Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020440
[19]	Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020463
[20]	Bilal Al Taki, Khawla Msheik, Jacques Sainte-Marie. On the rigid-lid approximation of shallow water Bingham. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 875-905. doi: 10.3934/dcdsb.2020146

2019 Impact Factor: 1.105

Tools

Metrics

Tools

Article outline

Figures and Tables

2019 Impact Factor: 1.105

Tools

Figures and Tables

2019 Impact Factor: 1.105

American Institute of Mathematical Sciences

Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors