Browsing by Author "Golmankhaneh, Alireza K."

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Sumudu Transform in Fractal Calculus
(Elsevier Science inc, 2019) Golmankhaneh, Alireza K.; Tunc, Cemil
The C-eta-Calculus includes functions on fractal sets, which are not differentiable or integrable using ordinary calculus. Sumudu transforms have an important role in control engineering problems because of preserving units, the scaling property of domains, easy visualization, and transforming linear differential equations to algebraic equations that can be easily solved. Analogues of the Laplace and Sumudu transforms in C-eta-Calculus are defined and the corresponding theorems are proved. The generalized Laplace and Sumudu transforms involve functions with totally disconnected fractal sets in the real line. Linear differential equations on Cantor-like sets are solved utilizing fractal Sumudu transforms. The results are summarized in tables and figures. Illustrative examples are solved to give more details. (C) 2019 Elsevier Inc. All rights reserved.
Stochastic Differential Equations on Fractal Sets
(Taylor & Francis Ltd, 2020) Golmankhaneh, Alireza K.; Tunc, Cemil
In this manuscript, we review fractal calculus and random processes. Random variables and processes on totally disconnected fractal sets are defined. Random walks on fractal middle-xi Cantor sets are suggested and corresponding variances are given which are power laws. The mean square stochastic calculus is generalized on fractal sets, which can lead to the standard case by setting dimension . Furthermore, we solve a fractal stochastic differential equation using the Frobenius method. Graphs are presented to give more details.
On the Lipschitz Condition in the Fractal Calculus
(Pergamon-elsevier Science Ltd, 2017) Golmankhaneh, Alireza K.; Tunc, Cemil
In this paper, the existence and uniqueness theorems are proved for the linear and non-linear fractal differential equations. The fractal Lipschitz condition is given on the F-alpha-calculus which applies for the non-differentiable function in the sense of the standard calculus. More, the metric spaces associated with fractal sets and about functions with fractal supports are defined to build fractal Cauchy sequence. Furthermore, Picard iterative process in the F-alpha-calculus which have important role in the numerical and approximate solution of fractal differential equations is explored. We clarify the results using the illustrative examples. (C) 2016 Elsevier Ltd. All rights reserved.