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Öğe Cumulative growth with fibonacci approach, golden section and physics(Pergamon-Elsevier Science Ltd, 2009) Buyukkilic, F.; Demirhan, D.In this study, a physical quantity belonging to a physical system in its stages of orientation towards growth has been formulated using Fibonacci recurrence approximation. Fibonacci p-numbers emerging in this process have been expressed as a power law for the first time as far as we are aware. The golden sections sp are related to the growth percent rates lambda(p). With this mechanism, the physical origins of the mathematical forms of e(q)(x) and ln(q)(x) encountered in Tsallis thermostatistics have been clarified. It has been established that Fibonacci p-numbers could be taken as elements of generalized random Cantor set. The golden section random cantor set is used by M. S. El Naschie in his fundamental works in high energy physics and is also considered in the present work. Moreover, we conclude that the cumulative growth mechanism conveys the consequences of the discrete structure of space and memory effect. (C) 2008 Elsevier Ltd. All rights reserved.Öğe Cumulative growth with fibonacci approach, golden section and physics(Pergamon-Elsevier Science Ltd, 2009) Buyukkilic, F.; Demirhan, D.In this study, a physical quantity belonging to a physical system in its stages of orientation towards growth has been formulated using Fibonacci recurrence approximation. Fibonacci p-numbers emerging in this process have been expressed as a power law for the first time as far as we are aware. The golden sections sp are related to the growth percent rates lambda(p). With this mechanism, the physical origins of the mathematical forms of e(q)(x) and ln(q)(x) encountered in Tsallis thermostatistics have been clarified. It has been established that Fibonacci p-numbers could be taken as elements of generalized random Cantor set. The golden section random cantor set is used by M. S. El Naschie in his fundamental works in high energy physics and is also considered in the present work. Moreover, we conclude that the cumulative growth mechanism conveys the consequences of the discrete structure of space and memory effect. (C) 2008 Elsevier Ltd. All rights reserved.Öğe A fractional mathematical approach to the distribution functions of quantum gases: Cosmic Microwave Background Radiation problem is revisited(Elsevier Science Bv, 2009) Ertik, H.; Demirhan, D.; Sirin, H.; Buyukkilic, F.Efforts on the fundamentals of the nonextensive thermostatistical formulations of the realistic description of the physical systems have always been underway. In this context, the quantum systems of bosons and fermions are taken into consideration as g-ons. A new formalism of the unified distribution functions has been introduced using a fractional mathematical approach. With the purpose of verification of the theory, blackbody radiation problem has been investigated by making use of the generalized fractional Planck's distribution. In this context, the observed Cosmic Microwave Background Radiation (CMBR) energy density could be obtained exactly within nonextensive thermostatistical approach for the value alpha = 0.999983 of the order of the fractional derivative and for the blackbody temperature T = 2.72818 K. (C) 2009 Elsevier B.V. All rights reserved.Öğe A fractional mathematical approach to the distribution functions of quantum gases: Cosmic Microwave Background Radiation problem is revisited(Elsevier Science Bv, 2009) Ertik, H.; Demirhan, D.; Sirin, H.; Buyukkilic, F.Efforts on the fundamentals of the nonextensive thermostatistical formulations of the realistic description of the physical systems have always been underway. In this context, the quantum systems of bosons and fermions are taken into consideration as g-ons. A new formalism of the unified distribution functions has been introduced using a fractional mathematical approach. With the purpose of verification of the theory, blackbody radiation problem has been investigated by making use of the generalized fractional Planck's distribution. In this context, the observed Cosmic Microwave Background Radiation (CMBR) energy density could be obtained exactly within nonextensive thermostatistical approach for the value alpha = 0.999983 of the order of the fractional derivative and for the blackbody temperature T = 2.72818 K. (C) 2009 Elsevier B.V. All rights reserved.Öğe The influence of fractality on the time evolution of the diffusion process(Elsevier Science Bv, 2010) Sirin, H.; Buyukkilic, F.; Ertik, H.; Demirhan, D.In the literature, the deviations from standard behaviors of the solutions of the kinetic equation and the analogous diffusion equation are put forward by investigations which are carried out in the frame of fractional mathematics and nonextensive physics On the other hand, the physical origins of the order of derivative namely a in fractional mathematics and the entropy index q in nonextensive physics are a topic of interest in scientific media. In this study, the solutions of the diffusion equation which have been obtained in the framework of fractional mathematics and nonextensive physics are revised. The diffusion equation is solved by the cumulative diminuation/growth method which has been developed by two of the present authors and physical nature of the parameters a and q are enlightened in connection with fractality of space and the memory effect It has been emphasized that the mathematical basis of deviations from standard behavior in the distribution functions could be established by fractional mathematics where as the physical mechanism could be revealed using the cumulative diminuation/growth method (C) 2010 Elsevier B.V. All rights reservedÖğe Investigation of the cumulative diminution process using the Fibonacci method and fractional calculus(Elsevier Science Bv, 2016) Buyukkilic, F.; Bayrakdar, Z. Ok; Demirhan, D.In this study, we investigate the cumulative diminution phenomenon for a physical quantity and a diminution process with a constant acquisition quantity in each step in a viscous medium. We analyze the existence of a dynamical mechanism that underlies the success of fractional calculus compared with standard mathematics for describing stochastic processes by proposing a Fibonacci approach, where we assume that the complex processes evolves cumulatively in fractal space and discrete time. Thus, when the differential-integral order a is attained, this indicates the involvement of the viscosity of the medium in the evolving process. The future value of the diminishing physical quantity is obtained in terms of the Mittag-Leffler function (MLF) and two rheological laws are inferred from the asymptotic limits. Thus, we conclude that the differential-integral calculus of fractional mathematics implicitly embodies the cumulative diminution mechanism that occurs in a viscous medium. (C) 2015 Elsevier B.V. All rights reserved.Öğe Solution of Schrodinger equation for two different potentials using extended Nikiforov-Uvarov method and polynomial solutions of biconfluent Heun equation(Amer Inst Physics, 2018) Karayer, H.; Demirhan, D.; Buyukkilic, F.Exact solutions of the Schrodinger equation for two different potentials are presented by using the extended Nikiforov-Uvarov method. The first one is the inverse square root potential which is a long-range potential and the second one is a combination of Coulomb, linear, and harmonic potentials which is often used to describe quarkonium. Eigenstate solutions are obtained in a systematicway without using any ansatz or transformation. Eigenfunctions for considered potentials are given in terms of biconfluent Heun polynomials. Published by AIP Publishing.Öğe Solutions of local fractional sine-Gordon equations(Taylor & Francis Ltd, 2019) Karayer, H.; Demirhan, D.; Buyukkilic, F.In this paper, we study sine-Gordon equation in order to obtain exact solitary wave solutions in the domain of fractional calculus. By using the definition of conformable fractional derivative, we obtain analytical solutions of time, space and time-space fractional sine-Gordon equations. We analyze graphically the effect of fractional order on evolution of the kink and antikink type solitons.Öğe SOME SPECIAL SOLUTIONS OF BICONFLUENT AND TRICONFLUENT HEUN EQUATIONS IN ELEMENTARY FUNCTIONS BY EXTENDED NIKIFOROV-UVAROV METHOD(Pergamon-Elsevier Science Ltd, 2015) Karayer, H.; Demirhan, D.; Buyukkilic, F.We report some special solutions of biconfluent Heun equation and triconfluent Heun equation via extended Nikiforov-Uvarov (NU) method which is developed by changing the boundary conditions of NU method [8]. We demonstrate that eigenvalue solutions of these confluent forms of Heun equation can be achieved exactly. Furthermore we give eigenvalue solution of a two-electron quantum dot model problem as a physical application of extended NU method.
