Monotonicity Results For Discrete Caputo-Fabrizio Fractional Operators
Abstract
Nearly every theory in mathematics has a discrete equivalent that simplifies it theoretically and practically so that it may be used in modeling real-world issues. With discrete calculus, for instance, it is possible to find the "difference" of any function from the first order up to the n-th order. On the other hand, it is also feasible to expand this theory using discrete fractional calculus and make n any real number such that the 1⁄2-order difference is properly defined. This article is divided into five chapters, each of which develops the most straightforward discrete fractional variational theory while illustrating some fundamental concepts and features of discrete fractional calculus. It is also investigated how the idea may be applied to the development of tumors. The first section provides a succinct introduction to the discrete fractional calculus and several key mathematical concepts that are utilized often in the subject. We demonstrate in section 2 that if the Caputo-Fabrizio nabla fractional difference operator of order and commencing at is positive for then is -increasing. On the other hand, if is rising and , then . Additionally, a result of monotonicity for the Caputo-type fractional difference operator is established. We show a fractional difference version of the mean-value theorem as an application and contrast it to the traditional discrete fractional instance.
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References
[2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, elsevier 204 (2006).
[3] T. Abdeljawad, On Riemann and Caputo fractional differences, Computers and Mathematics with Applications 62 (3) (2011) 1602-1611.
[4] F. Atici, P. Eloe, Initial value problems in discrete fractional calculus, Proceedings of the American Mathematical Society 137 (3) (2009) 981-989.
[5] G. C. Wu, D. Baleanu, S. D. Zeng, G. D. Zhen, Discrete fractional diffusion equation, Nonlinear Dynamics 80 (1) (2015) 281-286.
[6] G. C. Wu, D. Baleanu, S. D. Zeng, G. D. Zhen, Several fractional differences and their applications to discrete maps, Journal of Applied Nonlinear Dynamics 4 (2015) 339-348.
[7] M. Caputo, F. Mauro, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications 1 (2) (2015) 1-13.
[8] T. Abdeljawad, D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Reports on Mathematical Physics 80 (1) (2017) 11-27.
[9] R. Dahal, C. S. Goodrich, A monotonicity result for discrete fractional difference operators, Archiv der Mathematik. 102 (3) (2014) 293-299.
[10] B. Jia, L. Erbe, A. Peterson, Two monotonicity results for nabla and delta fractional differences, Archiv der Mathematik 104 (6) (2015) 589-597.
[11] F. M. Atici, M. Uyanik, Analysis of discrete fractional operators, Applicable Analysis and Discrete Mathematics (2015) 139-149.
[12] T. Abdeljawad, M. A. Ferhan, On the definitions of nabla fractional operators, In Abstract and Applied Analysis 2012.
[13] T. Abdeljawad, On delta and nabla Caputo fractional differences and dual identities, Discrete Dynamics in Nature and Society 2013.
[14] R. Dahal, S. G. Christopher, A monotonicity result for discrete fractional difference operators, Archiv der Mathematik . 102 (3) (2014) 293-299.
[15] S. G. Christopher, A convexity result for fractional differences, Applied Mathematics Letters 35 (2014) 58-62.
[16] F. M. Atici, P. Eloe, Discrete fractional calculus with the nabla operator, Electronic Journal of Qualitative Theory of Differential Equations 2009.
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