Fen - Edebiyat Fakültesi

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Analytical Approximate Solutions of (n+1)-Dimensional Fractal Heat-Like and Wave-Like Equations
(MDPI, 2017) Baleanu, Dumitru; Baleanu, Dumitru; Al Qurashi, Maysaa Mohamed; Sakar, Mehmet Giyas; 56389; Matematik
In this paper, we propose a new type (n + 1)-dimensional reduced differential transform method (RDTM) based on a local fractional derivative (LFD) to solve (n + 1)-dimensional local fractional partial differential equations (PDEs) in Cantor sets. The presented method is named the (n + 1)-dimensional local fractional reduced differential transform method (LFRDTM). First the theories, their proofs and also some basic properties of this procedure are given. To understand the introduced method clearly, we apply it on the (n + 1)-dimensional fractal heat-like equations (HLEs) and wave-like equations (WLEs). The applications show that this new technique is efficient, simply applicable and has powerful effects in (n + 1)-dimensional local fractional problems.
Dark-bright optical solitary waves and modulation instability analysis with (2+1)-dimensional cubic-quintic nonlinear Schrodinger equation
(Taylor&Francis LTD, 2019) Baleanu, Dumitru; Aliyu, Aliyu Isa; Yusuf, Abdullahi; Baleanu, Dumitru; 56389; Matematik
This paper addresses the (2+1)-dimensional cubic-quintic nonlinear Schrodinger equation (CQNLS) that serves as the model to study the light propagation through nonlinear optical media and non-Kerr crystals. A dark-bright optical solitary wave solution of this equation is retrieved by adopting the complex envelope function ansatz. This type of solitary wave describes the properties of bright and dark optical solitary waves in the same expression. The integration naturally lead to a constraint condition placed on the solitary wave parameters which must hold for the solitary waves to exist. Additionally, the modulation instability (MI) analysis of the model is studied based on the standard linear stability analysis and the MI gain spectrum is got. Numerical simulation and physical interpretations of the obtained results are demonstrated. It is hoped that the results reported in this paper can enrich the nonlinear dynamical behaviors of the CQNLS.
On (2+1)-dimensional physical models endowed with decoupled spatial and temporal memory indices(star)
(Springer Heidelberg, 2019) Baleanu, Dumitru; Jaradat, Imad; Alquran, Marwan; Yousef, Feras; Momani, Shaher; 56389; Matematik
The current work concerns the development of an analytical scheme to handle (2 + 1) -dimensional partial differential equations endowed with decoupled spatial and temporal fractional derivatives (abbreviated by (alpha,beta) -models). For this purpose, a new bivariate fractional power series expansion has been integrated with the differential transform scheme. The mechanism of the submitted scheme depends mainly on converting the (alpha,beta) -model to a recurrence-differential equation that can be easily solved by virtue of an iterative procedure. This, in turn, reduces the computational cost of the Taylor power series method and consequently introduces a significant refinement for solving such hybrid models. To elucidate the novelty and efficiency of the proposed scheme, several (alpha,beta) -models are solved and the presence of remnant memory, due to the fractional derivatives, is graphically illustrated.
On fractional derivatives with generalized Mittag-Leffler kernels
(Pushpa Publishing House, 2018) Abdeljawad, Thabet; Baleanu, Dumitru; 56389; Matematik
Fractional derivatives with three parameter generalized Mittag-Leffler kernels and their properties are studied. The corresponding integral operators are obtained with the help of Laplace transforms. The action of the presented fractional integrals on the Caputo and Riemann type derivatives with three parameter Mittag-Leffler kernels is analyzed. Integration by parts formulas in the sense of Riemann and Caputo are proved and then used to formulate the fractional Euler-Lagrange equations with an illustrative example. Certain nonconstant functions whose fractional derivatives are zero are determined as well.
On the solutions of a fractional boundary value problem
(Scientific Technical Research Council Turkey-Tubitak, 2018) Uğurlu, Ekin; Baleanu, Dumitru; Taş, Kenan; 238990; 4971; 56389; Matematik
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.
Symmetry in Applied Continuous Mechanics
(MDPI, 2019) Baleanu, Dumitru; Baleanu, Dumitru; Vlase, Sorin; 234808; Matematik
Engineering practice requires the use of structures containing identical components or parts, which are useful from several points of view: less information is needed to describe the system, design is made quicker and easier, components are made faster than a complex assembly, and finally the time to achieve the structure and the cost of manufacturing decreases. Additionally, the subsequent maintenance of the system becomes easier and cheaper. This Special Issue is dedicated to this kind of mechanical structure, describing the properties and methods of analysis of these structures. Discrete or continuous structures in static and dynamic cases are considered. Theoretical models, mathematical methods, and numerical analysis of the systems, such as the finite element method and experimental methods, are expected to be used in the research. Such applications can be used in most engineering fields including machine building, automotive, aerospace, and civil engineering.
The (k, s)-fractional calculus of k-Mittag-Leffler function
(Springer Open, 2017) Baleanu, Dumitru; Rahman, G.; Baleanu, Dumitru; Mubeen, S.; Arshad, Muhammad; 56389; Matematik
In this paper, we introduce the (k, s)-fractional integral and differential operators involving k-Mittag-Leffler function E-k,rho,beta(delta) (z) as its kernel. Also, we establish various properties of these operators. Further, we consider a number of certain consequences of the main results.
The extended fractional Caputo-Fabrizio derivative of order 0 <= sigma < 1 on C-R[0,1] and the existence of solutions for two higher-order series-type differential equations
(Springer Open, 2018) Baleanu, Dumitru; Mousalou, Asef; Rezapour, Shahram; 56389; Matematik
We extend the fractional Caputo-Fabrizio derivative of order 0 <= sigma < 1 on C-R[0,1] and investigate two higher-order series-type fractional differential equations involving the extended derivation. Also, we provide an example to illustrate one of the main results.

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