Matematik Bölümü

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A fractional finite difference inclusion
(Eudoxus Press, 2016) Baleanu, Dumitru; Rezapour, Shahram; Salehi, Saeid
In this manuscript we investigated the fractional finite difference inclusion Delta(mu)(mu-2) x(t) is an element of F(t, x(t), Delta x(t)) via the boundary conditions Delta x(b + mu) = A and x(mu - 2) = B, where 1 <= 2, A,B is an element of R and F :N-mu-2(b+mu+2) x R -> 2(R) is a compact valued multifunction.
A note on fractional neutral integro-differential inclusions with state-dependent delay in Banach spaces
(Eudoxus Press, 2016) Baleanu, Dumitru; Baleanu, Dumitru; Arjunan, M. Mallika
We have applied different fixed point theorems to examine the existence results for fractional neutral integro-differential inclusions (FNIDI) with state-dependent delay (SDD) in Banach spaces. We tend to conjointly discuss the cases once the multivalued nonlinear term takes convex values further as nonconvex values. An example is offered to demonstrate the obtained results.
An algorithm for Hopf bifurcation analysis of a delayed reaction-diffusion model
(Springer, 2017) Kayan, Şeyma; Merdan, H.; 49206
We present an algorithm for determining the existence of a Hopf bifurcation of a system of delayed reaction-diffusion equations with the Neumann boundary conditions. The conditions on parameters of the system that a Hopf bifurcation occurs as the delay parameter passes through a critical value are determined. These conditions depend on the coefficients of the characteristic equation corresponding to linearization of the system. Furthermore, an algorithm to obtain the formulas for determining the direction of the Hopf bifurcation, the stability, and period of the periodic solution is given by using the Poincare normal form and the center manifold theorem. Finally, we give several examples and some numerical simulations to show the effectiveness of the algorithm proposed.
Digital processing of thermographic images for medical applications
(Chiminform Data S A, 2016) Baleanu, Dumitru; Guzman-Sepulveda, Jose Rafael; Gonzalez Parada, Adrian; Rosales Garcia, Juan; Torres Cisneros, Miguel; Baleanu, Dumitru
Breast cancer is the second leading cause of death in women worldwide with an average probability for a woman to develop breast cancer in her life of about 12%. Among the large variety of medical assessment techniques, thermography has attracted attention in applications related to detection and diagnosis due to its capability to provide valuable information on the physiological variations typical of early stages in cancer development thus making possible to diagnose patients in early stages so more thorough examinations can be done in proper time and manner. This paper presents a digital processing approach that allows identification and subsequent isolation of the region of interest in thermograms based texture analysis of the image. This algorithm was tested on case studies thermograms exhibiting different types of cancer and the results showed successful identification and extraction of the region of interest in all cases. Results are presented with different types of cancer in men and women and different image angles showing the robustness of the proposed method
Dirac systems with regular and singular transmission effects
(Scientific Technical Research Council Turkey-TUBİTAK, 2017) Uğurlu, Ekin; 238990
In this paper, we investigate the spectral properties of singular eigenparameter dependent dissipative problems in Weyl's limit-circle case with finite transmission conditions. In particular, these transmission conditions are assumed to be regular and singular. To analyze these problems we construct suitable Hilbert spaces with special inner products and linear operators associated with these problems. Using the equivalence of the Lax-Phillips scattering function and Sz-Nagy-Foias characteristic functions we prove that all root vectors of these dissipative operators are complete in Hilbert spaces.
Einstein field equations within local fractional calculus
(Editura Acad Romane, 2015) Baleanu, Dumitru; Yang, Xiao-Jun; Baleanu, Dumitru
In this paper, we introduce the local fractional Christoffel index symbols of the first and second kind. The divergence of a local fractional contravariant vector and the curl of local fractional covariant vector are defined. The fractional intrinsic derivative is given. The local fractional Riemann-Christoffel and Ricci tensors are obtained. Finally, the Einstein tensor and Einstein field are generalized by involving the fractional derivatives. Illustrative examples are presented
Numerical study for fractional euler-lagrange equations of a harmonic oscillator on a moving platform
(Polish Acad Sciences Inst Physics, 2016) Baleanu, Dumitru; Blaszczyk, Tomasz; Asad, Jihad H.; Alipour, Mohsen
We investigate the fractional harmonic oscillator on a moving platform. We obtained the fractional Euler-Lagrange equation from the derived fractional Lagrangian of the system which contains left Caputo fractional derivative. We transform the obtained differential equation of motion into a corresponding integral one and then we solve it numerically. Finally, we present many numerical simulations.
On modeling the groundwater flow within a confined aquifer
(Editura Acad Romane, 2015) Baleanu, Dumitru; Baleanu, Dumitru
The groundwater flow equation is used to simulate the movement of water under the confined aquifer. In this paper we study a modification of the groundwater flow equation within a newly proposed derivative. We numerically solve the generalized groundwater flow equation with the Crank-Nicholson scheme. We also analytically solve the generalized equation via the method of separation of variable.
Relaxation and diffusion models with non-singular kernels
(Elsevier Science Bv., 2017) Baleanu, Dumitru
Anomalous relaxation and diffusion processes have been widely quantified by fractional derivative models, where the definition of the fractional-order derivative remains a historical debate due to its limitation in describing different kinds of non-exponential decays (e.g. stretched exponential decay). Meanwhile, many efforts by mathematicians and engineers have been made to overcome the singularity of power function kernel in its definition. This study first explores physical properties, of relaxation and diffusion models where the temporal derivative was defined recently using an exponential kernel. Analytical analysis shows that the Caputo type derivative model with an exponential kernel cannot characterize non-exponential dynamics well-documented in anomalous relaxation and diffusion. A legitimate extension of the previous derivative is then proposed by replacing the exponential kernel with a stretched" exponential kernel. Numerical tests show that the Caputo type derivative model with the stretched exponential kernel can describe a much wider range of anomalous diffusion than the exponential kernel, implying the potential applicability of the new derivative in quantifying real-world, anomalous relaxation and diffusion processes.
Stability analysis of Caputo-like discrete fractional systems
(Elsevier Science Bv., 2017) Baleanu, Dumitru
This study investigates stability of Caputo delta fractional difference equations. Solutions' monotonicity and asymptotic stability of a linear fractional difference equation are discussed. A stability theorem for a discrete fractional Lyapunov direct method is proved. Furthermore, an inequality is extended from the continuous case and a sufficient condition is given. Some linear, nonlinear and time varying examples are illustrated and the results show wide prospects of the stability theorems in fractional control systems of discrete time.
The first integral method for the (3+1)-dimensional modified korteweg-de vries-zakharov-kuznetsov and hirota equations
(Editura Academiei Romane, 2015) Baleanu, Dumitru; Kılıç, B.; Uğurlu, Y.; İnç, Mustafa
The first integral method is applied to get the different types of solutions of the (3+1)-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov and Hirota equations. We obtain envelope, bell shaped, trigonometric, and kink soliton solutions of these nonlinear evolution equations. The applied method is an effective one to obtain different types of solutions of nonlinear partial differential equations

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