Browsing by Author "Shiri, Babak"

Now showing 1 - 13 of 13

A General Fractional Pollution Model for Lakes
(Springernature, 2022) Shiri, Babak; Baleanu, Dumitru; Baleanu, Dumitru; 56389
A model for the amount of pollution in lakes connected with some rivers is introduced. In this model, it is supposed the density of pollution in a lake has memory. The model leads to a system of fractional differential equations. This system is transformed into a system of Volterra integral equations with memory kernels. The existence and regularity of the solutions are investigated. A high-order numerical method is introduced and analyzed and compared with an explicit method based on the regularity of the solution. Validation examples are supported, and some models are simulated and discussed.
All linear fractional derivatives with power functions’ convolution kernel and interpolation properties
(Pergamon-elsevier Science Ltd, 2023) Shiri, Babak; Baleanu, Dumitru; Baleanu, Dumitru; 56389
Our attempt is an axiomatic approach to find all classes of possible definitions for fractional derivatives with three axioms. In this paper, we consider a special case of linear integro-differential operators with power functions' convolution kernel a(a)(t-s)b(a) of order a a (0,1). We determine analytic functions a(a) and b(a) such that when a-* 0+, the corresponding operator becomes identity operator, and when a-* 1- the corresponding operator becomes derivative operator. Then, a sequential operator is used to extend the fractional operator to a higher order. Some properties of the sequential operator in this regard also are studied. The singularity properties, Laplace transform and inverse of the new class of fractional derivatives are investigated. Several examples are provided to confirm theoretical achievements. Finally, the solution of the relaxation equation with diverse fractional derivatives is obtained and compared.
Collocation methods for terminal value problems of tempered fractional differential equations
(Elsevier, 2020) Shiri, Babak; Baleanu, Dumitru; Wu, Guo-Cheng; Baleanu, Dumitru; 56389
A class of tempered fractional differential equations with terminal value problems are investigated in this paper. Discretized collocation methods on piecewise polynomials spaces are proposed for solving these equations. Regularity results are constructed on weighted spaces and convergence order is studied. Several examples are supported the theoretical parts and compared with other methods. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.
Discrete fractional watermark technique
(2020) Baleanu, Dumitru; Shiri, Babak; Baleanu, Dumitru; 56389
The fractional logistic map holds rich dynamics and is adopted to generate chaotic series. A watermark image is then encrypted and inserted into the original images. Since the encryption image takes the fractional order within (0, 1], it increases the key space and becomes difficult to attack. This study provides a robust watermark method in the protection of the copyright of hardware, images, and other electronic files.
Discrete fractional watermark technique
(Zhejiang Univ, 2020) Wang, Zai-rong; Baleanu, Dumitru; Shiri, Babak; Baleanu, Dumitru; 56389
The fractional logistic map holds rich dynamics and is adopted to generate chaotic series. A watermark image is then encrypted and inserted into the original images. Since the encryption image takes the fractional order within (0, 1], it increases the key space and becomes difficult to attack. This study provides a robust watermark method in the protection of the copyright of hardware, images, and other electronic files.
Generalized fractional differential equations for past dynamic
(Amer inst Mathematical Sciences-aims, 2022) Baleanu, Dumitru; Baleanu, Dumitru; Shiri, Babak; 56389
Well-posedness of the terminal value problem for nonlinear systems of generalized fractional differential equations is studied. The generalized fractional operator is formulated with a classical operator and a related weighted space. The terminal value problem is transformed into weakly singular Fredholm and Volterra integral equations with delay. A lower bound for the well-posedness of the corresponding problem is introduced. A collocation method covering all problems with generalized derivatives is introduced and analyzed. Illustrative examples for validation and application of the proposed methods are supported. The effects of various fractional derivatives on the solution, wellposedness, and fitting error are studied. An application for estimating the population of diabetes cases in the past is introduced.
New fractional signal smoothing equations with short memory and variable order
(Elsevier Gmbh, 2020) Ma, Chang-You; Baleanu, Dumitru; Shiri, Babak; Wu, Guo-Cheng; Baleanu, Dumitru; 56389
In this paper, systems of fuzzy fractional differential equations with a lateral type of the Hukuhara derivative and the generalized Hukuhara derivative are numerically studied. Collocation method on discontinuous piecewise polynomial spaces is proposed. Convergence of the proposed method is analyzed. The superconvergent results on the graded mesh are studied. Examples are provided to support theoretical results. Finally, the effect of uncertainty in a diabetes model and its resulting complications is investigated as a practical application.
Nonlinear higher order fractional terminal value problems
(Amer inst Mathematical Sciences-aims, 2022) Baleanu, Dumitru; Baleanu, Dumitru; Shiri, Babak; 56389
Terminal value problems for systems of fractional differential equations are studied with an especial focus on higher-order systems. Discretized piecewise polynomial collocation methods are used for approximating the exact solution. This leads to solving a system of nonlinear equations. For solving such a system an iterative method with a required tolerance is introduced and analyzed. The existence of a unique solution is guaranteed with the aid of the fixed point theorem. Order of convergence for the given numerical method is obtained. Numerical experiments are given to support theoretical results.
Numerical solution of a new mathematical model for intravenous drug administration
(Springer Heidelberg, 2024) Alijani, Zahra; Baleanu, Dumitru; Shiri, Babak; Perfilieva, Irina; Baleanu, Dumitru; 56389
We develop and analyze a new mathematical model for intravenous drug administration and the associated diffusion process. We use interval analysis to obtain a system of weakly singular Volterra integral equations over ordinary functions. We then use the operational method based on Chebyshev polynomials for obtaining an approximate solution of the numerical form. We show that for a certain class of fuzzy number valued functions, their generalized Hukuhara derivatives can be reduced to the derivatives of ordinary real-valued functions. By using our approach, we are able to estimate numerical solutions very accurately.
Pathological study on uncertain numbers and proposed solutions for discrete fuzzy fractional order calculus
(de Gruyter Poland Sp Z O O, 2023) Shiri, Babak; Baleanu, Dumitru; Baleanu, Dumitru; Ma, Chang-You; 56389
A pathological study in the definition of uncertain numbers is carried out, and some solutions are proposed. Fundamental theorems for uncertain discrete fractional and integer order calculus are established. The concept of maximal solution for obtaining a unique uncertain solution is introduced. The solutions of uncertain discrete relaxation equations for the integer and the fractional order are obtained. Various numerical examples are accompanied to clarify the theoretical results and study of uncertain system behavior.
Spline collocation methods for systems of fuzzy fractional differential equations
(Pergamon-elsevier Science Ltd, 2020) Alijani, Zahra; Baleanu, Dumitru; Baleanu, Dumitru; Shiri, Babak; Wu, Guo-Cheng; 56389
In this paper, systems of fuzzy fractional differential equations with a lateral type of the Hukuhara derivative and the generalized Hukuhara derivative are numerically studied. Collocation method on discontinuous piecewise polynomial spaces is proposed. Convergence of the proposed method is analyzed. The superconvergent results on the graded mesh are studied. Examples are provided to support theoretical results. Finally, the effect of uncertainty in a diabetes model and its resulting complications is investigated as a practical application. (C) 2019 Elsevier Ltd. All rights reserved.
Terminal value problems for the nonlinear systems of fractional differential equations
(Elsevier, 2021) Shiri, Babak; Baleanu, Dumitru; Wu, Guo-Cheng; Baleanu, Dumitru; 56389
Terminal value problems of fractional nonlinear systems are studied in this paper. The existence and uniqueness are given. The regularity of the solution is obtained in the weighted spaces. Discretized piecewise polynomial collocation methods are proposed on the graded mesh. A convergence analysis and the order are presented. Numerical examples for supporting theoretical results and applications for population models are illustrated. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.
Visco-elastic dampers in structural buildings and numerical solution with spline collocation methods
(Springer Heidelberg, 2020) Dadkhah, Ehsan; Baleanu, Dumitru; Shiri, Babak; Ghaffarzadeh, Hosein; Baleanu, Dumitru; 56389
The dynamic system of a structure utilized with visco-elastic dampers can be modeled by fractional differential equations. All the resulted systems of fractional differential equations can be represented in a state space and can be transformed into a system of multi-term fractional differential equations of order 1. Considering the presence of indeterministic exogenous force like earthquake we need powerful, convergent and reliable numerical methods to simulate the response of this dynamical systems. Therefore, spline collocations method has been proposed and studied for solving system of multi-term fractional differential equations of order 1. A rigorous mathematical analysis is provided to show the efficiency and effectiveness of the method. To this end, we apply a functional analysis framework to obtain convergence and superconvergence properties of the proposed methods on the graded mesh. Some numerical experiments are provided to confirm the theoretical results. Finally, this method is used for simulating the response of a 4-story building under El Centro earthquake excitation.