132 results on '"Baleanu, Dumitru"'
Search Results
2. EDITORIAL SPECIAL ISSUE: PART IV-III-II-I SERIES.
- Author
-
KARACA, YELİZ, BALEANU, DUMITRU, MOONIS, MAJAZ, ZHANG, YU-DONG, and GERVASI, OSVALDO
- Subjects
- *
SYSTEMS theory , *MULTIDISCIPLINARY design optimization , *CHAOS theory , *ARTIFICIAL intelligence , *MATHEMATICAL analysis , *COMPUTER science , *SCIENTIFIC computing , *FRACTIONAL programming - Abstract
Complex systems, as interwoven miscellaneous interacting entities that emerge and evolve through self-organization in a myriad of spiraling contexts, exhibit subtleties on global scale besides steering the way to understand complexity which has been under evolutionary processes with unfolding cumulative nature wherein order is viewed as the unifying framework. Indicating the striking feature of non-separability in components, a complex system cannot be understood in terms of the individual isolated constituents' properties per se, it can rather be comprehended as a way to multilevel approach systems behavior with systems whose emergent behavior and pattern transcend the characteristics of ubiquitous units composing the system itself. This observation specifies a change of scientific paradigm, presenting that a reductionist perspective does not by any means imply a constructionist view; and in that vein, complex systems science, associated with multiscale problems, is regarded as ascendancy of emergence over reductionism and level of mechanistic insight evolving into complex system. While evolvability being related to the species and humans owing their existence to their ancestors' capability with regards to adapting, emerging and evolving besides the relation between complexity of models, designs, visualization and optimality, a horizon that can take into account the subtleties making their own means of solutions applicable is to be entailed by complexity. Such views attach their germane importance to the future science of complexity which may probably be best regarded as a minimal history congruent with observable variations, namely the most parallelizable or symmetric process which can turn random inputs into regular outputs. Interestingly enough, chaos and nonlinear systems come into this picture as cousins of complexity which with tons of its components are involved in a hectic interaction with one another in a nonlinear fashion amongst the other related systems and fields. Relation, in mathematics, is a way of connecting two or more things, which is to say numbers, sets or other mathematical objects, and it is a relation that describes the way the things are interrelated to facilitate making sense of complex mathematical systems. Accordingly, mathematical modeling and scientific computing are proven principal tools toward the solution of problems arising in complex systems' exploration with sound, stimulating and innovative aspects attributed to data science as a tailored-made discipline to enable making sense out of voluminous (-big) data. Regarding the computation of the complexity of any mathematical model, conducting the analyses over the run time is related to the sort of data determined and employed along with the methods. This enables the possibility of examining the data applied in the study, which is dependent on the capacity of the computer at work. Besides these, varying capacities of the computers have impact on the results; nevertheless, the application of the method on the code step by step must be taken into consideration. In this sense, the definition of complexity evaluated over different data lends a broader applicability range with more realism and convenience since the process is dependent on concrete mathematical foundations. All of these indicate that the methods need to be investigated based on their mathematical foundation together with the methods. In that way, it can become foreseeable what level of complexity will emerge for any data desired to be employed. With relation to fractals, fractal theory and analysis are geared toward assessing the fractal characteristics of data, several methods being at stake to assign fractal dimensions to the datasets, and within that perspective, fractal analysis provides expansion of knowledge regarding the functions and structures of complex systems while acting as a potential means to evaluate the novel areas of research and to capture the roughness of objects, their nonlinearity, randomness, and so on. The idea of fractional-order integration and differentiation as well as the inverse relationship between them lends fractional calculus applications in various fields spanning across science, medicine and engineering, amongst the others. The approach of fractional calculus, within mathematics-informed frameworks employed to enable reliable comprehension into complex processes which encompass an array of temporal and spatial scales notably provides the novel applicable models through fractional-order calculus to optimization methods. Computational science and modeling, notwithstanding, are oriented toward the simulation and investigation of complex systems through the use of computers by making use of domains ranging from mathematics to physics as well as computer science. A computational model consisting of numerous variables that characterize the system under consideration allows the performing of many simulated experiments via computerized means. Furthermore, Artificial Intelligence (AI) techniques whether combined or not with fractal, fractional analysis as well as mathematical models have enabled various applications including the prediction of mechanisms ranging extensively from living organisms to other interactions across incredible spectra besides providing solutions to real-world complex problems both on local and global scale. While enabling model accuracy maximization, AI can also ensure the minimization of functions such as computational burden. Relatedly, level of complexity, often employed in computer science for decision-making and problem-solving processes, aims to evaluate the difficulty of algorithms, and by so doing, it helps to determine the number of required resources and time for task completion. Computational (-algorithmic) complexity, referring to the measure of the amount of computing resources (memory and storage) which a specific algorithm consumes when it is run, essentially signifies the complexity of an algorithm, yielding an approximate sense of the volume of computing resources and seeking to prove the input data with different values and sizes. Computational complexity, with search algorithms and solution landscapes, eventually points toward reductions vis à vis universality to explore varying degrees of problems with different ranges of predictability. Taken together, this line of sophisticated and computer-assisted proof approach can fulfill the requirements of accuracy, interpretability, predictability and reliance on mathematical sciences with the assistance of AI and machine learning being at the plinth of and at the intersection with different domains among many other related points in line with the concurrent technical analyses, computing processes, computational foundations and mathematical modeling. Consequently, as distinctive from the other ones, our special issue series provides a novel direction for stimulating, refreshing and innovative interdisciplinary, multidisciplinary and transdisciplinary understanding and research in model-based, data-driven modes to be able to obtain feasible accurate solutions, designed simulations, optimization processes, among many more. Hence, we address the theoretical reflections on how all these processes are modeled, merging all together the advanced methods, mathematical analyses, computational technologies, quantum means elaborating and exhibiting the implications of applicable approaches in real-world systems and other related domains. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
3. On the solutions for generalised multiorder fractional partial differential equations arising in physics.
- Author
-
Purohit, Sunil Dutt, Baleanu, Dumitru, and Jangid, Kamlesh
- Subjects
- *
FRACTIONAL differential equations , *SCHRODINGER equation , *HEAT equation , *BROWNIAN motion , *FRACTIONAL calculus , *REACTION-diffusion equations , *PARTIAL differential equations , *WAVE equation - Abstract
In this article, we have studied solutions of a generalised multiorder fractional partial differential equations involving the Caputo time‐fractional derivative and the Riemann–Liouville space fractional derivatives using Laplace–Fourier transform technique. Proposed generalised multiorder fractional partial differential equation is reducible to Schrödinger equation, wave equation and diffusion equation in a more general sense, and hence, solutions of these equations are specifically noted. Not only this, solutions of equation proposed in the stochastic resetting theory in the context of Brownian motion can also be found in a general regime. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
4. On a Fractional Parabolic Equation with Regularized Hyper-Bessel Operator and Exponential Nonlinearities.
- Author
-
Baleanu, Dumitru, Binh, Ho Duy, and Nguyen, Anh Tuan
- Subjects
- *
FRACTIONAL differential equations , *PARTIAL differential equations , *SOBOLEV spaces , *ELLIPTIC operators , *EQUATIONS , *HILBERT space , *FOURIER series - Abstract
Recent decades have witnessed the emergence of interesting models of fractional partial differential equations. In the current work, a class of parabolic equations with regularized Hyper-Bessel derivative and the exponential source is investigated. More specifically, we examine the existence and uniqueness of mild solutions in Hilbert scale-spaces which are constructed by a uniformly elliptic symmetry operator on a smooth bounded domain. Our main argument is based on the Banach principle argument. In order to achieve the necessary and sufficient requirements of this argument, we have smoothly combined the application of the Fourier series supportively represented by Mittag-Leffler functions, with Hilbert spaces and Sobolev embeddings. Because of the presence of the fractional operator, we face many challenges in handling proper integrals which appear in the representation of mild solutions. Besides, the source term of an exponential type also causes trouble for us when deriving the desired results. Therefore, powerful embeddings are used to limit the growth of nonlinearity. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
5. On a New Modification of the Erdélyi-Kober Fractional Derivative.
- Author
-
Odibat, Zaid and Baleanu, Dumitru
- Subjects
- *
FRACTIONAL calculus , *FRACTIONAL differential equations , *MATHEMATICAL models , *DERIVATIVES (Mathematics) , *DIFFERENTIAL calculus - Abstract
In this paper, we introduce a new Caputo-type modification of the Erdélyi-Kober fractional derivative. We pay attention to how to formulate representations of Erdélyi-Kober fractional integral and derivatives operators. Then, some properties of the new modification and relationships with other Erdélyi-Kober fractional derivatives are derived. In addition, a numerical method is presented to deal with fractional differential equations involving the proposed Caputotype Erdélyi-Kober fractional derivative. We hope the presented method will be widely applied to simulate such fractional models. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
6. EDITORIAL.
- Author
-
Karaca, Yeliz, Baleanu, Dumitru, Moonis, Majaz, Muhammad, Khan, Zhang, Yu-Dong, and Gervasi, Osvaldo
- Subjects
- *
SPACE sciences , *QUINTIC equations , *APPLIED sciences , *ALGORITHMS , *FRACTIONAL differential equations , *ARTIFICIAL intelligence - Published
- 2021
- Full Text
- View/download PDF
7. Optimal solutions for singular linear systems of Caputo fractional differential equations.
- Author
-
Dassios, Ioannis and Baleanu, Dumitru
- Subjects
- *
LINEAR systems , *MATHEMATICAL optimization , *FRACTIONAL differential equations , *LEAST squares - Abstract
In this article, we focus on a class of singular linear systems of fractional differential equations with given nonconsistent initial conditions (IC). Because the nonconsistency of the IC can not lead to a unique solution for the singular system, we use two optimization techniques to provide an optimal solution for the system. We use two optimization techniques to provide the optimal solution for the system because a unique solution for the singular system cannot be obtained due to the non‐consistency of the IC. These two optimization techniques involve perturbations to the non‐consistent IC, specifically, an l2 perturbation (which seeks an optimal solution for the system in terms of least squares), and a second‐order optimization technique at an l1 minimum perturbation, (which includes an appropriate smoothing). Numerical examples are given to justify our theory. We use the Caputo (C) fractional derivative and two recently defined alternative versions of this derivative, the Caputo‐Fabrizio (CF) and the Atangana‐Baleanu (AB) fractional derivative. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
8. Fractional calculus in the sky.
- Author
-
Baleanu, Dumitru and Agarwal, Ravi P.
- Subjects
- *
FRACTIONAL differential equations , *FRACTIONAL calculus , *MATHEMATICIANS - Abstract
Fractional calculus was born in 1695 on September 30 due to a very deep question raised in a letter of L'Hospital to Leibniz. The prophetical answer of Leibniz to that deep question encapsulated a huge inspiration for all generations of scientists and is continuing to stimulate the minds of contemporary researchers. During 325 years of existence, fractional calculus has kept the attention of top level mathematicians, and during the last period of time it has become a very useful tool for tackling the dynamics of complex systems from various branches of science and engineering. In this short manuscript, we briefly review the tremendous effect that the main ideas of fractional calculus had in science and engineering and briefly present just a point of view for some of the crucial problems of this interdisciplinary field. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
9. Numerical solution of two‐dimensional time fractional cable equation with Mittag‐Leffler kernel.
- Author
-
Kumar, Sachin and Baleanu, Dumitru
- Subjects
- *
FRACTIONAL differential equations , *PARTIAL differential equations , *NUMERICAL integration , *EQUATIONS , *DIFFERENTIAL operators , *CABLES , *TWO-dimensional models - Abstract
The main motive of this article is to study the recently developed Atangana‐Baleanu Caputo (ABC) fractional operator that is obtained by replacing the classical singular kernel by Mittag‐Leffler kernel in the definition of the fractional differential operator. We investigate a novel numerical method for the nonlinear two‐dimensional cable equation in which time‐fractional derivative is of Mittag‐Leffler kernel type. First, we derive an approximation formula of the fractional‐order ABC derivative of a function tk using a numerical integration scheme. Using this approximation formula and some properties of shifted Legendre polynomials, we derived the operational matrix of ABC derivative. In the author of knowledge, this operational matrix of ABC derivative is derived the first time. We have shown the efficiency of this newly derived operational matrix by taking one example. Then we solved a new class of fractional partial differential equations (FPDEs) by the implementation of this ABC operational matrix. The two‐dimensional model of the time‐fractional model of the cable equation is solved and investigated by this method. We have shown the effectiveness and validity of our proposed method by giving the solution of some numerical examples of the two‐dimensional fractional cable equation. We compare our obtained numerical results with the analytical results, and we conclude that our proposed numerical method is feasible and the accuracy can be seen by error tables. We see that the accuracy is so good. This method will be very useful to investigate a different type of model that have Mittag‐Leffler fractional derivative. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
10. On a strong-singular fractional differential equation.
- Author
-
Baleanu, Dumitru, Ghafarnezhad, Khadijeh, Rezapour, Shahram, and Shabibi, Mehdi
- Subjects
- *
FRACTIONAL differential equations , *BOUNDARY value problems , *DIFFERENTIAL equations , *CAPUTO fractional derivatives , *INTEGRO-differential equations , *LAPLACIAN operator - Abstract
It is important we try to solve complicate differential equations specially strong singular ones. We investigate the existence of solutions for a strong-singular fractional boundary value problem under some conditions. In this way, we provide a new technique for our study. We provide an example to illustrate our main result. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
11. A fractional differential equation model for the COVID-19 transmission by using the Caputo–Fabrizio derivative.
- Author
-
Baleanu, Dumitru, Mohammadi, Hakimeh, and Rezapour, Shahram
- Subjects
- *
FRACTIONAL differential equations , *COVID-19 , *CAPUTO fractional derivatives , *FIXED point theory , *LAPLACE transformation - Abstract
We present a fractional-order model for the COVID-19 transmission with Caputo–Fabrizio derivative. Using the homotopy analysis transform method (HATM), which combines the method of homotopy analysis and Laplace transform, we solve the problem and give approximate solution in convergent series. We prove the existence of a unique solution and the stability of the iteration approach by using fixed point theory. We also present numerical results to simulate virus transmission and compare the results with those of the Caputo derivative. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
12. A Novel 2-Stage Fractional Runge–Kutta Method for a Time-Fractional Logistic Growth Model.
- Author
-
Arshad, Muhammad Sarmad, Baleanu, Dumitru, Riaz, Muhammad Bilal, and Abbas, Muhammad
- Subjects
- *
RUNGE-Kutta formulas , *FRACTIONAL differential equations , *EULER method - Abstract
In this paper, the fractional Euler method has been studied, and the derivation of the novel 2-stage fractional Runge–Kutta (FRK) method has been presented. The proposed fractional numerical method has been implemented to find the solution of fractional differential equations. The proposed novel method will be helpful to derive the higher-order family of fractional Runge–Kutta methods. The nonlinear fractional Logistic Growth Model is solved and analyzed. The numerical results and graphs of the examples demonstrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
13. A fractional derivative with two singular kernels and application to a heat conduction problem.
- Author
-
Baleanu, Dumitru, Jleli, Mohamed, Kumar, Sunil, and Samet, Bessem
- Subjects
- *
HEAT conduction , *FRACTIONAL differential equations - Abstract
In this article, we suggest a new notion of fractional derivative involving two singular kernels. Some properties related to this new operator are established and some examples are provided. We also present some applications to fractional differential equations and propose a numerical algorithm based on a Picard iteration for approximating the solutions. Finally, an application to a heat conduction problem is given. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
14. Applications of some fixed point theorems for fractional differential equations with Mittag-Leffler kernel.
- Author
-
Afshari, Hojjat and Baleanu, Dumitru
- Subjects
- *
FRACTIONAL differential equations , *MATHEMATICAL mappings , *FIXED point theory - Abstract
Using some fixed point theorems for contractive mappings, including α-γ-Geraghty type contraction, α-type F-contraction, and some other contractions in F -metric space, this research intends to investigate the existence of solutions for some Atangana–Baleanu fractional differential equations in the Caputo sense. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
15. A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions.
- Author
-
Baleanu, Dumitru, Etemad, Sina, and Rezapour, Shahram
- Subjects
- *
FIXED point theory , *THERMOSTAT , *BOUNDARY value problems , *SET-valued maps , *DIFFERENTIAL equations , *CAPUTO fractional derivatives , *FRACTIONAL differential equations - Abstract
We provide an extension for the second-order differential equation of a thermostat model to the fractional hybrid equation and inclusion versions. We consider boundary value conditions of this problem in the form of the hybrid conditions. To prove the existence of solutions for our hybrid fractional thermostat equation and inclusion versions, we apply the well-known Dhage fixed point theorems for single-valued and set-valued maps. Finally, we give two examples to illustrate our main results. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
16. New generalized integral transform via Dzherbashian-Nersesian fractional operator.
- Author
-
Belgacem, Rachid, Bokhari, Ahmed, Baleanu, Dumitru, and Djilali, Salih
- Subjects
- *
GENERALIZED integrals , *INTEGRAL transforms , *FRACTIONAL differential equations - Abstract
In this paper, we derive a new generalized integral transform on Dzherbashian-Nersesian fractional operator and give some special cases. We make a generalization of the application of integral transformations to different fractional operators, where several previous results can be invoked from a single relation. We also use the new results obtained to solve some fractional differential equations involving the recent revival of Dzherbashian-Nersesian fractional operators. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
17. Solving PDEs of fractional order using the unified transform method.
- Author
-
Fernandez, Arran, Baleanu, Dumitru, and Fokas, Athanassios S.
- Subjects
- *
PARTIAL differential equations , *FRACTIONAL calculus , *FRACTIONAL differential equations , *MATHEMATICS , *NUMERICAL analysis - Abstract
Abstract We consider the unified transform method, also known as the Fokas method, for solving partial differential equations. We adapt and modify the methodology, incorporating new ideas where necessary, in order to apply it to solve a large class of partial differential equations of fractional order. We demonstrate the applicability of the method by implementing it to solve a model fractional problem. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
18. Caputo and related fractional derivatives in singular systems.
- Author
-
Dassios, Ioannis K. and Baleanu, Dumitru I.
- Subjects
- *
CAPUTO fractional derivatives , *FRACTIONAL differential equations , *COEFFICIENTS (Statistics) , *LINEAR systems , *NUMERICAL analysis - Abstract
By using the Caputo (C) fractional derivative and two recently defined alternative versions of this derivative, the Caputo–Fabrizio (CF) and the Atangana–Baleanu (AB) fractional derivative, firstly we focus on singular linear systems of fractional differential equations with constant coefficients that can be non-square matrices, or square & singular. We study existence of solutions and provide formulas for the case that there do exist solutions. Then, we study the existence of unique solution for given initial conditions. Several numerical examples are given to justify our theory. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
19. Fractional discrete-time diffusion equation with uncertainty: Applications of fuzzy discrete fractional calculus.
- Author
-
Huang, Lan-Lan, Baleanu, Dumitru, Mo, Zhi-Wen, and Wu, Guo-Cheng
- Subjects
- *
FRACTIONAL calculus , *DISCRETE choice models , *DECISION making , *DISCRETE geometry , *FRACTIONAL differential equations - Abstract
This study provides some basics of fuzzy discrete fractional calculus as well as applications to fuzzy fractional discrete-time equations. With theories of r -cut set, fuzzy Caputo and Riemann–Liouville fractional differences are defined on a isolated time scale. Discrete Leibniz integral law is given by use of w -monotonicity conditions. Furthermore, equivalent fractional sum equations are established. Fuzzy discrete Mittag-Leffler functions are obtained by the Picard approximation. Finally, fractional discrete-time diffusion equations with uncertainty is investigated and exact solutions are expressed in form of two kinds of fuzzy discrete Mittag-Leffler functions. This paper suggests a discrete time tool for modeling discrete fractional systems with uncertainty. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
20. A survey on fuzzy fractional differential and optimal control nonlocal evolution equations.
- Author
-
Agarwal, Ravi P., Baleanu, Dumitru, Nieto, Juan J., Torres, Delfim F.M., and Zhou, Yong
- Subjects
- *
FRACTIONAL differential equations , *OPTIMAL control theory , *EVOLUTION equations , *FEEDBACK control systems , *BANACH spaces - Abstract
We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
21. Novel Mittag-Leffler stability of linear fractional delay difference equations with impulse.
- Author
-
Wu, Guo-Cheng, Baleanu, Dumitru, and Huang, Lan-Lan
- Subjects
- *
FRACTIONAL differential equations , *DISCRETE-time systems , *DIFFERENTIAL equations , *FRACTIONAL calculus , *SYSTEM analysis - Abstract
In this letter we propose a class of linear fractional difference equations with discrete-time delay and impulse effects. The exact solutions are obtained by use of a discrete Mittag-Leffler function with delay and impulse. Besides, we provide comparison principle, stability results and numerical illustration. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
22. Numerical solutions of fractional parabolic equations with generalized Mittag–Leffler kernels.
- Author
-
Alomari, Abedel‐Karrem, Abdeljawad, Thabet, Baleanu, Dumitru, Saad, Khaled M., and Al‐Mdallal, Qasem M.
- Subjects
- *
FRACTIONAL differential equations , *NONLINEAR equations , *FRACTIONAL integrals , *EQUATIONS , *LAPLACIAN operator , *PARABOLIC operators - Abstract
In this article, we investigate the generalized fractional operator Caputo type (ABC) with kernels of Mittag–Lefller in three parameters Eα,μγ(λt) and its fractional integrals with arbitrary order for solving the time fractional parabolic nonlinear equation. The generalized definition generates infinitely many problems for a fixed fractional derivative α. We utilize this operator with homotopy analysis method for constructing the new scheme for generating successive approximations. This procedure is used successfully on two examples for finding the solutions. The effectiveness and accuracy are verified by clarifying the convergence region in the ℏ‐curves as well as by calculating the residual error and the results were accurate. Based on the experiment, we verify the existence of the solution for the new parameters. Depending on these results, this treatment can be used to find approximate solutions to many fractional differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
23. Comprehending themodel of omicron variant using fractional derivatives.
- Author
-
Sharma, Shivani, Goswami, Pranay, Baleanu, Dumitru, and Dubey, Ravi Shankar
- Subjects
- *
SARS-CoV-2 Omicron variant , *BASIC reproduction number , *FRACTIONAL differential equations , *CORONAVIRUSES , *MATHEMATICAL models - Abstract
The world is grappled with an unprecedented challenges due to Corona virus. We are all battling this epidemic together, but we have not been able to defeat this epidemic yet. A new variant of this virus, named 'Omicron' is spreading these days. The fractional differential equations are providing us with better tools to study the mathematical model with memory effects. In this paper, we will consider an extended SER mathematical model with quarantined and vaccinated compartment to speculate the Omicron variant. This extended Susceptible Exposed Infected Recovered SER model involves equations that associate with the group of individuals those are susceptible (S), exposed (E): this class includes the individuals who are infected but not yet infectious, infectious (W): this class includes the individuals who are infected but not yet Quarantined, quarantined (Q): this class includes those group of people who are infectious, confirmed and quarantined, recovered (R) this class includes the group of individuals who have recovered, and vaccinated (V): this class includes the group of individuals who have been vaccinated. The non-negativity and of the extended SER model is analysed, the equilibrium points and the basic reproduction number are also calculated. The proposed model is then extended to the mathematical model using AB derivative operator. Proof for the existence and the uniqueness for the solution of fractional mathematical model in sense of AB fractional derivative is detailed and a numerical method is detailed to obtain the numerical solutions. Further we have discussed the efficiency of the vaccine against the Omicron variant via graphical representation. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
24. On the solutions of a fractional boundary value problem.
- Author
-
UĞURLU, Ekin, BALEANU, Dumitru, and TAŞ, Kenan
- Subjects
- *
NUMERICAL solutions to boundary value problems , *NONLINEAR equations , *FRACTIONAL differential equations , *EXISTENCE theorems , *UNIQUENESS (Mathematics) , *FIXED point theory - Abstract
This paper is devoted to showing the existence and uniqueness of solution of a regular second-order nonlinear fractional differential equation subject to the ordinary boundary conditions. The Banach fixed point theorem is used to prove the results. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
25. Finite Difference Method for Time-Space Fractional Advection-Diffusion Equations with Riesz Derivative.
- Author
-
Arshad, Sadia, Baleanu, Dumitru, Jianfei Huang, Mohamed Al Qurashi, Maysaa, Yifa Tang, and Yue Zhao
- Subjects
- *
ADVECTION-diffusion equations , *DERIVATIVES (Mathematics) , *RIESZ spaces , *CAPUTO fractional derivatives , *FRACTIONAL differential equations , *INTEGRAL equations - Abstract
In this article, a numerical scheme is formulated and analysed to solve the time-space fractional advection-diffusion equation, where the Riesz derivative and the Caputo derivative are considered in spatial and temporal directions, respectively. The Riesz space derivative is approximated by the second-order fractional weighted and shifted Grünwald-Letnikov formula. Based on the equivalence between the fractional differential equation and the integral equation, we have transformed the fractional differential equation into an equivalent integral equation. Then, the integral is approximated by the trapezoidal formula. Further, the stability and convergence analysis are discussed rigorously. The resulting scheme is formally proved with the second order accuracy both in space and time. Numerical experiments are also presented to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
26. Stability analysis of impulsive fractional difference equations.
- Author
-
Wu, Guo–Cheng and Baleanu, Dumitru
- Subjects
- *
FRACTIONAL differential equations , *DIFFERENCE equations , *MATHEMATICAL induction , *NUMERICAL solutions to differential equations , *VOLTERRA equations - Abstract
We revisit motivation of the fractional difference equations and some recent applications to image encryption. Then stability of impulsive fractional difference equations is investigated in this paper. The fractional sum equation is considered and impulsive effects are introduced into discrete fractional calculus. A class of impulsive fractional difference equations are proposed. A discrete comparison principle is given and asymptotic stability of nonlinear fractional difference equation are discussed. Finally, an impulsive Mittag–Leffler stability is defined. The numerical result is provided to support the analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
27. A RELIABLE MIXED METHOD FOR SINGULAR INTEGRO-DIFFERENTIAL EQUATIONS OF NON-INTEGER ORDER.
- Author
-
BALEANU, DUMITRU, DARZI, RAHMAT, and AGHELI, BAHRAM
- Subjects
- *
FRACTIONAL calculus , *FRACTIONAL differential equations , *BOUNDARY value problems , *COLLOCATION methods , *INTEGRO-differential equations - Abstract
It is our goal in this article to apply a method which is based on the assumption that combines two methods of conjugating collocation and multiple shooting method. The proposed method can be used to find the numerical solution of singular fractional integro-differential boundary value problems (SFIBVPs) [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
28. Existence results of fractional differential equations with Riesz-Caputo derivative.
- Author
-
Chen, Fulai, Baleanu, Dumitru, and Wu, Guo-Cheng
- Subjects
- *
BOUNDARY value problems , *GRONWALL inequalities , *FRACTIONAL differential equations , *FRACTIONAL calculus , *DIFFERENTIAL equations - Abstract
This paper is concerned with a class of boundary value problems for fractional differential equations with the Riesz-Caputo derivative, which holds two-sided nonlocal effects. By means of a new fractional Gronwall inequalities and some fixed point theorems, we obtained some existence results of solutions. Three examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
29. Fractional differential equations of Caputo–Katugampola type and numerical solutions.
- Author
-
Zeng, Shengda, Baleanu, Dumitru, Bai, Yunru, and Wu, Guocheng
- Subjects
- *
FRACTIONAL differential equations , *NUMERICAL solutions to differential equations , *DISCRETIZATION methods , *STOCHASTIC convergence , *CAPUTO fractional derivatives - Abstract
This paper is concerned with a numerical method for solving generalized fractional differential equation of Caputo–Katugampola derivative. A corresponding discretization technique is proposed. Numerical solutions are obtained and convergence of numerical formulae is discussed. The convergence speed arrives at O ( Δ T 1 − α ) . Numerical examples are given to test the accuracy. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
30. Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions.
- Author
-
Fernandez, Arran, Baleanu, Dumitru, and Srivastava, H.M.
- Subjects
- *
RIEMANN integral , *HENSTOCK-Kurzweil integral , *FRACTIONAL calculus , *FRACTIONAL differential equations , *SEMIGROUP algebras - Abstract
Highlights • Prabhakar and related operators can be expressed as series of Riemann–Liouville operators. • Fundamental properties of Prabhakar operators are recovered from the series formulae. • The product and chain rules hold for Prabhakar fractional-calculus operators. • Fractional iteration for these operators is discussed. Abstract We consider an integral transform introduced by Prabhakar, involving generalised multi-parameter Mittag-Leffler functions, which can be used to introduce and investigate several different models of fractional calculus. We derive a new series expression for this transform, in terms of classical Riemann–Liouville fractional integrals, and use it to obtain or verify series formulae in various specific cases corresponding to different fractional-calculus models. We demonstrate the power of our result by applying the series formula to derive analogues of the product and chain rules in more general fractional contexts. We also discuss how the Prabhakar model can be used to explore the idea of fractional iteration in connection with semigroup properties. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
31. ANALYTICAL TREATMENTS TO SYSTEMS OF FRACTIONAL DIFFERENTIAL EQUATIONS WITH MODIFIED ATANGANA–BALEANU DERIVATIVE.
- Author
-
AL-REFAI, MOHAMMED, SYAM, MUHAMMED I., and BALEANU, DUMITRU
- Subjects
- *
CAPUTO fractional derivatives , *LINEAR systems , *COLLOCATION methods , *NONLINEAR systems , *FRACTIONAL differential equations - Abstract
The solutions of systems of fractional differential equations depend on the type of the fractional derivative used in the system. In this paper, we present in closed forms the solutions of linear systems involving the modified Atangana–Baleanu derivative that has been introduced recently. For the nonlinear systems, we implement a numerical scheme based on the collocation method to obtain approximate solutions. The applicability of the results is tested through several examples. We emphasize here that certain systems with the Atangana–Baleanu derivative admit no solutions which is not the case with the modified derivative. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
32. REGULAR FRACTIONAL DIFFERENTIAL EQUATIONS IN THE SOBOLEV SPACE.
- Author
-
Ugurlu, Ekin, Baleanu, Dumitru, and Tas, Kenan
- Subjects
- *
FRACTIONAL differential equations , *SOBOLEV spaces , *STURM-Liouville equation , *SYMMETRIC operators , *FRACTIONAL integrals - Abstract
In this paper regular fractional Sturm-Liouville boundary value problems are considered. In particular, new inner products are described in the Sobolev space and a symmetric operator is established in this space. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
33. Monotonicity results for fractional difference operators with discrete exponential kernels.
- Author
-
Abdeljawad, Thabet and Baleanu, Dumitru
- Subjects
- *
MONOTONIC functions , *FRACTIONAL differential equations , *MATHEMATICS theorems , *DISCRETE systems , *KERNEL (Mathematics) - Abstract
We prove that if the Caputo-Fabrizio nabla fractional difference operator $({}^{\mathrm{CFR}}_{a-1}\nabla^{\alpha}y)(t)$ of order $0<\alpha\leq1$ and starting at $a-1$ is positive for $t=a,a+1,\ldots$ , then $y(t)$ is α-increasing. Conversely, if $y(t)$ is increasing and $y(a)\geq0$ , then $({}^{\mathrm{CFR}}_{a-1}\nabla^{\alpha}y)(t)\geq0$ . A monotonicity result for the Caputo-type fractional difference operator is proved as well. As an application, we prove a fractional difference version of the mean-value theorem and make a comparison to the classical discrete fractional case. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
34. A generalisation of the Malgrange-Ehrenpreis theorem to find fundamental solutions to fractional PDEs.
- Author
-
Baleanu, Dumitru and Fernandez, Arran
- Subjects
- *
NONLINEAR differential equations , *FRACTIONAL differential equations , *ARBITRARY constants , *FRACTIONAL calculus , *DERIVATIVES (Mathematics) - Abstract
We present and prove a new generalisation of the Malgrange-Ehrenpreis theorem to fractional partial differential equations, which can be used to construct fundamental solutions to all partial differential operators of rational order and many of arbitrary real order. We demonstrate with some examples and mention a few possible applications. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
35. A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputo-Fabrizio derivative.
- Author
-
Baleanu, Dumitru, Mousalou, Asef, and Rezapour, Shahram
- Subjects
- *
APPROXIMATION theory , *FRACTIONAL differential equations , *CARTOGRAPHY , *INTEGRALS , *MATHEMATICAL analysis - Abstract
We present a new method to investigate some fractional integro-differential equations involving the Caputo-Fabrizio derivation and we prove the existence of approximate solutions for these problems. We provide three examples to illustrate our main results. By checking those, one gets the possibility of using some discontinuous mappings as coefficients in the fractional integro-differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
36. ANALYSIS OF THE NEW TECHNIQUE TO SOLUTION OF FRACTIONAL WAVE- AND HEAT-LIKE EQUATION.
- Author
-
BALEANU, DUMITRU, AGHELI, BAHRAM, and DARZI, RAHMAT
- Subjects
- *
NUMERICAL solutions to wave equations , *FRACTIONAL differential equations , *NUMERICAL solutions to heat equation , *HOMOTOPY theory , *PERTURBATION theory , *POWER series - Abstract
We have applied the new approach of homotopic perturbation method (NHPM) for wave- and heat-like equation featuring time-fractional derivative. A combination of NHPM and multiple fractional power series form has been used the first time to present analytical solution. In order to illustrate the simplicity and ability of the suggested approach, some specific and clear examples have been given. All computations were done using Mathematica. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
37. Analysis and some applications of a regularized [formula omitted]–Hilfer fractional derivative.
- Author
-
Jajarmi, Amin, Baleanu, Dumitru, Sajjadi, Samaneh Sadat, and Nieto, Juan J.
- Subjects
- *
FRACTIONAL differential equations , *DIFFERENTIAL equations , *CAPUTO fractional derivatives - Abstract
The main purpose of this research is to present a generalization of Ψ –Hilfer fractional derivative, called as regularized Ψ –Hilfer, and study some of its basic characteristics. In this direction, we show that the ψ –Riemann–Liouville integral is the inverse operation of the presented regularized differentiation by means of the same function ψ. In addition, we consider an initial-value problem comprising this generalization and analyze the existence as well as the uniqueness of its solution. To do so, we first present an approximation sequence via a successive substitution approach; then we prove that this sequence converges uniformly to the unique solution of the regularized Ψ –Hilfer fractional differential equation (FDE). To solve this FDE, we suggest an efficient numerical scheme and show its accuracy and efficacy via some real-world applications. Simulation results verify the theoretical consequences and show that the regularized Ψ –Hilfer fractional mathematical system provides a more accurate model than the other kinds of integer- and fractional-order differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
38. On some new properties of fractional derivatives with Mittag-Leffler kernel.
- Author
-
Baleanu, Dumitru and Fernandez, Arran
- Subjects
- *
FRACTIONAL calculus , *DERIVATIVES (Mathematics) , *KERNEL (Mathematics) , *SEMIGROUPS (Algebra) , *FRACTIONAL differential equations - Abstract
We establish a new formula for the fractional derivative with Mittag-Leffler kernel, in the form of a series of Riemann–Liouville fractional integrals, which brings out more clearly the non-locality of fractional derivatives and is easier to handle for certain computational purposes. We also prove existence and uniqueness results for certain families of linear and nonlinear fractional ODEs defined using this fractional derivative. We consider the possibility of a semigroup property for these derivatives, and establish extensions of the product rule and chain rule, with an application to fractional mechanics. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
39. Existence and discrete approximation for optimization problems governed by fractional differential equations.
- Author
-
Bai, Yunru, Baleanu, Dumitru, and Wu, Guo–Cheng
- Subjects
- *
FRACTIONAL differential equations , *APPROXIMATION theory , *ALGORITHMS , *FIXED point theory , *NUMERICAL analysis - Abstract
We investigate a class of generalized differential optimization problems driven by the Caputo derivative. Existence of weak Carath e ´ odory solution is proved by using Weierstrass existence theorem, fixed point theorem and Filippov implicit function lemma etc. Then a numerical approximation algorithm is introduced, and a convergence theorem is established. Finally, a nonlinear programming problem constrained by the fractional differential equation is illustrated and the results verify the validity of the algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
40. Finite-time stability of discrete fractional delay systems: Gronwall inequality and stability criterion.
- Author
-
Wu, Guo–Cheng, Baleanu, Dumitru, and Zeng, Sheng–Da
- Subjects
- *
DISCRETE-time systems , *STABILITY theory , *GRONWALL inequalities , *FRACTIONAL differential equations , *MATHEMATICAL constants - Abstract
This study investigates finite-time stability of Caputo delta fractional difference equations. A generalized Gronwall inequality is given on a finite time domain. A finite-time stability criterion is proposed for fractional differential equations. Then the idea is extended to the discrete fractional case. A linear fractional difference equation with constant delays is considered and finite-time stable conditions are provided. One example is numerically illustrated to support the theoretical result. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
41. Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations.
- Author
-
Baleanu, Dumitru, Wu, Guo–Cheng, and Zeng, Sheng–Da
- Subjects
- *
FRACTIONAL differential equations , *CHAOS theory , *TAYLOR'S series , *FRACTIONAL calculus , *DECOMPOSITION method , *LYAPUNOV stability - Abstract
This paper investigates chaotic behavior and stability of fractional differential equations within a new generalized Caputo derivative. A semi–analytical method is proposed based on Adomian polynomials and a fractional Taylor series. Furthermore, chaotic behavior of a fractional Lorenz equation are numerically discussed. Since the fractional derivative includes two fractional parameters, chaos becomes more complicated than the one in Caputo fractional differential equations. Finally, Lyapunov stability is defined for the generalized fractional system. A sufficient condition of asymptotic stability is given and numerical results support the theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
42. A frational derivative in lusion problem via an integral boundary Condition.
- Author
-
Baleanu, Dumitru, Moghaddam, Mehdi, Mohammadi, Hakimeh, and Rezapour, Shahram
- Subjects
- *
FRACTIONAL differential equations , *BOUNDARY value problems , *CAPUTO fractional derivatives , *EXISTENCE theorems , *INTEGRALS - Abstract
We investigate the existence of solutions for the fractional differential inclusion cDα x(t) ∈ F(t, x(t)) equipped with the boundary value problems x(0) = 0 Mid x(0) = 0 and x(1) ..., where 0 < n < 1, 1 < α ≤ 2, cDα is the standard Caputo differentiation and F : [0,1] x R → 2R is a compact valued multifunction. An illustrative example is also discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2016
43. Fractional calculus and application of generalized Struve function.
- Author
-
Nisar, Kottakkaran, Baleanu, Dumitru, and Qurashi, Maysaa'
- Subjects
- *
FRACTIONAL calculus , *THEORY of distributions (Functional analysis) , *INTEGRAL transforms , *FRACTIONAL differential equations , *LAPLACE distribution - Abstract
A new generalization of Struve function called generalized Galué type Struve function (GTSF) is defined and the integral operators involving Appell's functions, or Horn's function in the kernel is applied on it. The obtained results are expressed in terms of the Fox-Wright function. As an application of newly defined generalized GTSF, we aim at presenting solutions of certain general families of fractional kinetic equations associated with the Galué type generalization of Struve function. The generality of the GTSF will help to find several familiar and novel fractional kinetic equations. The obtained results are general in nature and it is useful to investigate many problems in applied mathematical science. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
44. On two fractional differential inclusions.
- Author
-
Baleanu, Dumitru, Hedayati, Vahid, Rezapour, Shahram, and Al Qurashi, Maysaa'
- Subjects
- *
DIFFERENTIAL inclusions , *FIXED point theory , *FRACTIONAL differential equations , *MATHEMATICAL models , *DIFFERENTIABLE dynamical systems - Abstract
We investigate in this manuscript the existence of solution for two fractional differential inclusions. At first we discuss the existence of solution of a class of fractional hybrid differential inclusions. To illustrate our results we present an illustrative example. We study the existence and dimension of the solution set for some fractional differential inclusions. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
45. On existence results for impulsive fractional neutral stochastic integro-differential equations with nonlocal and state-dependent delay conditions.
- Author
-
Kalamani, Palaniyappan, Baleanu, Dumitru, Selvarasu, Siva, and Mallika Arjunan, Mani
- Subjects
- *
FRACTIONAL calculus , *DIFFERENTIAL equations , *AIRY differential equation , *STOCHASTIC partial differential equations - Abstract
This manuscript deals with a new set of sufficient conditions for the existence of solutions for a class of impulsive fractional neutral stochastic integro-differential systems (IFNSIDS) with nonlocal conditions (NLCs) and state-dependent delay (SDD) in Hilbert spaces. An example is provided to illustrate the obtained theory. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
46. Image encryption technique based on fractional chaotic time series.
- Author
-
Wu, Guo-Cheng, Baleanu, Dumitru, and Lin, Zhen-Xiang
- Subjects
- *
IMAGE encryption , *DIGITAL image processing , *FRACTIONAL calculus , *CHAOS theory , *FRACTIONAL differential equations - Abstract
Chaos in discrete fractional maps has been reported very recently. In this study, the chaotic time series of fractional order is used in the scrambling technique and a novel image encryption scheme is designed. The fractional difference order and the chaotic coefficient play crucial roles in controlling chaotic behaviors. The encrypted results are analyzed, which shows that the encryption scheme is highly secure. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
47. On the existence of solution for fractional differential equations of order $3<\delta_{1}\leq4$.
- Author
-
Baleanu, Dumitru, Agarwal, Ravi, Khan, Hasib, Khan, Rahmat, and Jafari, Hossein
- Subjects
- *
FRACTIONAL differential equations , *BOUNDARY value problems , *GREEN'S functions , *INTEGRALS , *LIOUVILLE'S theorem , *FRACTIONAL calculus - Abstract
In this paper, we deal with a fractional differential equation of order $\delta_{1}\in(3,4]$ with initial and boundary conditions, $\mathcal{D}^{\delta_{1}}\psi(x)=-\mathcal{H}(x,\psi(x))$, $\mathcal{D}^{\alpha_{1}} \psi(1)=0=\mathcal{I}^{3-\delta_{1}}\psi(0)= \mathcal{I}^{4-\delta_{1}}\psi(0)$, $\psi(1) = \frac{\Gamma(\delta_{1}-\alpha_{1})}{\Gamma(\nu_{1})}\mathcal{I}^{\delta _{1}-\alpha_{1}} \mathcal{H}(x,\psi(x))(1)$, where $x\in[0,1]$, $\alpha_{1} \in(1,2]$, addressing the existence of a positive solution (EPS), where the fractional derivatives $\mathcal{D}^{\delta_{1}}$, $\mathcal{D}^{\alpha_{1}}$ are in the Riemann-Liouville sense of the order $\delta_{1}$, $\alpha_{1}$, respectively. The function $\mathcal{H}\in C([0,1]\times{R} , {R})$ and $\mathcal{I}^{\delta_{1}-\alpha_{1}}\mathcal{H}(x,\psi(x))(1)=\frac {1}{\Gamma(\delta_{1}-\alpha_{1})} \int_{0}^{1}(1-z)^{\delta_{1}-\alpha_{1}-1}\mathcal{H}(z,\psi(z))\,dz$. To this aim, we establish an equivalent integral form of the problem with the help of a Green's function. We also investigate the properties of the Green's function in the paper which we utilize in our main result for the EPS of the problem. Results for the existence of solutions are obtained with the help of some classical results. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
48. Existence criterion for the solutions of fractional order p-Laplacian boundary value problems.
- Author
-
Jafari, Hossein, Baleanu, Dumitru, Khan, Hasib, Khan, Rahmat, and Khan, Aziz
- Subjects
- *
EXISTENCE theorems , *LAPLACIAN operator , *FRACTIONAL differential equations , *NUMERICAL solutions to boundary value problems , *DIFFERENTIAL equations - Abstract
The existence criterion has been extensively studied for different classes in fractional differential equations (FDEs) through different mathematical methods. The class of fractional order boundary value problems (FOBVPs) with p-Laplacian operator is one of the most popular class of the FDEs which have been recently considered by many scientists as regards the existence and uniqueness. In this scientific work our focus is on the existence and uniqueness of the FOBVP with p-Laplacian operator of the form: $D^{\gamma}(\phi_{p}(D^{\theta}z(t)))+a(t)f(z(t)) =0$, $3<{\theta}$, $\gamma\leq{4}$, $t\in[0,1]$, $z(0)=z'''(0)$, $\eta D^{\alpha}z(t)|_{t=1}= z'(0)$, $\xi z''(1)-z''(0)=0$, $0<\alpha<1$, $\phi_{p}(D^{\theta}z(t))|_{t=0}=0 =(\phi_{p}(D^{\theta}z(t)))'|_{t=0}$, $(\phi_{p}(D^{\theta} z(t)))''|_{t=1} = \frac{1}{2}(\phi_{p}(D^{\theta} z(t)))''|_{t=0}$, $(\phi_{p}(D^{\theta}z(t)))'''|_{t=0}=0$, where $0<\xi, \eta<{1}$ and $D^{\theta}$, $D^{\gamma}$, $D^{\alpha}$ are Caputo's fractional derivatives of orders θ, γ, α, respectively. For this purpose, we apply Schauder's fixed point theorem and the results are checked by illustrative examples. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
49. Duality of singular linear systems of fractional nabla difference equations.
- Author
-
Dassios, Ioannis K. and Baleanu, Dumitru I.
- Subjects
- *
DUALITY theory (Mathematics) , *MATHEMATICAL singularities , *LINEAR systems , *FRACTIONAL differential equations , *DIFFERENCE equations - Abstract
The main objective of this article is to provide a link between the solutions of an initial value problem of a linear singular system of fractional nabla difference equations, its proper dual system and its transposed dual system. By taking into consideration the case that the coefficients are square constant matrices with the leading coefficient singular, we study the prime system and by using the invariants of its pencil we give necessary and sufficient conditions for existence and uniqueness of solutions. After we prove that by using the pencil of the prime system we can study the existence and uniqueness of solutions of the proper dual system and the transposed dual system. Moreover their solutions, when they exist, can be explicitly represented without resorting to further processes of computations for each one separately. Finally, numerical examples are given based on a singular fractional nabla real dynamical system to justify our theory. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
50. On some self-adjoint fractional finite difference equations.
- Author
-
Baleanu, Dumitru, Rezapour, Shahram, and Salehi, Saeid
- Subjects
- *
FINITE difference method , *SELFADJOINT operators , *EXISTENCE theorems , *BOUNDARY value problems , *FRACTIONAL differential equations , *NUMERICAL analysis - Abstract
Recently, the existence of solution for the fractional self-adjoint equation Δν-1ν(pΔy)(t) = h(t) for order 0 < ν ≤ 1 was reported in [9]. In thispaper, we investigated the self-adjoint fractional finite difference equation Δν-2ν(pΔy)(t) = h(t,p(t + ν -- 2)Δy(t + ν -- 2)) via the boundary conditions y(ν -- 2) = 0, such that Δy (ν -- 2) = 0 and Δy(ν + b) = 0. Also, we analyzed the self-adjoint fractional finite difference equation Δν-2ν(pΔ²y)(t) = h(t,p(t + ν -- 3)Δ²y(t + ν -- 3)) via the boundary conditions y(ν -- 2) = 0, Δy(ν -- 2) = 0, Δ²y(ν -- 2) = 0 and Δ²y(ν + b) = 0. Finally, we conclude a result about the existence of solution for the general equation Δν-2937(pΔmy)(t) = h(t,p(t + ν -- m -- 1)Δmy(t + ν -- m -- 1)) via the boundary conditions y(ν -- 2) = Δy(ν -- 2) = Δ²y(ν -- 2) = ... = Δmy(ν -- 2) = 0 and Δmy(ν + b) = 0 for order 1 < ν ≤ 2. [ABSTRACT FROM AUTHOR]
- Published
- 2015
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.