D. Demirhan

Generalization of the Maier–Saupe theory of the nematics within Tsallis thermostatistics

In this study, one of the mean-field theories in nematics, the Maier–Saupe theory (MST), is generalized within Tsallis thermostatistics (TT). The variation of the order parameter versus temperature has been investigated and compared with experimental data for p-azoxyanisole (PAA). So far, we believe that this is the first attempt at the application of TT in liquid crystals. It is well known that MST fails to explain the experimental data for some of the nematics, one of which is PAA. However, generalized MST is able to account for the experimental data for a long range of temperatures. Also, in this study, the effect of nonextensivity is shown for various values of the entropic index.

Extension of Nikiforov-Uvarov method for the solution of Heun equation

We report an alternative method to solve second order differential equations which have at most four singular points. This method is developed by changing the degrees of the polynomials in the basic equation of Nikiforov-Uvarov (NU) method. This is called extended NU method for this paper. The eigenvalue solutions of Heun equation and confluent Heun equation are obtained via extended NU method. Some quantum mechanical problems such as Coulomb problem on a 3-sphere, two Coulombically repelling electrons on a sphere, and hyperbolic double-well potential are investigated by this method.

Nonextensive thermostatistical investigation of the blackbody radiation

Abstract Thermodynamical quantities of the blackbody radiation such as free energy, entropy, total radiation energy, specific heat are calculated within the Tsallis thermostatistics where factorization method is incorporated. It is shown that basic thermodynamical relation of the blackbody radiation is form invariant with respect to the nonextensivity entropic index q. Furthermore, the nonextensive thermodynamical quantities related to the blackbody radiation are separately obtained in terms of q and the standard thermodynamical quantities of the blackbody radiation. It is indicated that the formulation may give a way to determine q which determines the degree of the nonextensivity that is one of the aims of the present study.

Solution of Schrödinger equation for two different potentials using extended Nikiforov-Uvarov method and polynomial solutions of biconfluent Heun equation

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 systematic way without using any ansatz or transformation. Eigenfunctions for considered potentials are given in terms of biconfluent Heun polynomials.

Some Special Solutions of Biconfluent and Triconfluent Heun Equations in Elementary Functions by Extended Nikiforov–Uvarov Method

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.

FRACTIONAL MATHEMATICAL INVESTIGATION OF BOSE–EINSTEIN CONDENSATION IN DILUTE 87Rb, 23Na AND 7Li ATOMIC GASES

Although atomic Bose gases are experimentally investigated in the dilute regime, interparticle interactions play an important role on the transition temperatures of Bose–Einstein condensation. In this study, Bose–Einstein condensation is handled using fractional calculus for a Bose gas consisting of interacting bosons which are trapped in a three-dimensional harmonic oscillator. In this frame, in order to introduce the nonextensive effect, fractionally generalized Bose–Einstein distribution function which features Mittag–Leffler function is adopted. The dependence of the transition temperature of Bose–Einstein condensation on α (a measure of fractality of space) has been established. The transition temperatures for the dilute 87Rb, 23Na and 7Li atomic gases have been obtained in consistent with experimental data and the nature of the interactions in the Bose–Einstein condensate has been enlightened. In the course of our investigations, we have arrived to the conclusion that for α 1 repulsive interactions are predominant.

Solutions of local fractional sine-Gordon equations

ABSTRACT 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.

Landau diamagnetism within nonextensive statistical thermodynamics

Abstract In this paper, Landau diamagnetism is revisited in the context of nonextensive statistical thermodynamics. Exact as well as approximate expressions for partition function, magnetization and susceptibility are obtained by using Hilhorst integral transformation which provides exact solutions in the framework of Tsallis statistics. The results have been reassessed with the standard results to illustrate the effect of nonextensivity.