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A generalized delta derivative on time scale with applications
Author(s) -
Singh Negi Shekhar,
Abbas Syed,
Malik Muslim
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6601
Subject(s) - derivative (finance) , mathematics , exponential function , scale (ratio) , generalizations of the derivative , function (biology) , time derivative , notation , calculus (dental) , second derivative , mathematical analysis , physics , arithmetic , economics , medicine , dentistry , quantum mechanics , evolutionary biology , financial economics , biology
Inspired by a well‐known “Hilger‐derivative” on time scale introduced by Stephan Hilger, we introduce a new derivative on the time scale, called “black delta (or in notation ▴ )‐derivative.” It is an adaptive unified approach of derivative of continuous and discrete calculus. Several important nontrivial results concerning this derivative are obtained. Also, a necessary and sufficient condition for the existence of this new derivative and its connection with Hilger‐derivative are discussed. In addition, a new general a ‐exponential function (for fixed a ≥ 1) is introduced to obtain the unique solution of the initial value problem on time scales with this derivative. Furthermore, a relation between the a ‐exponential function with the existing exponential function on the time scale is obtained. We elucidate the results by considering some interesting examples. We expect that this derivative will be applicable for future investigations on various studies in the field of mathematical modeling, engineering applications.