Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Conference Object The Hirota Method for Reaction-Diffusion Equations With Three Distinct Roots(American Institute of Physics, 2004) Tanoğlu, Gamze; Pashaev, OktayThe Hirota Method, with modified background is applied to construct exact analytical solutions of nonlinear reaction-diffusion equations of two types. The first equation has only nonlinear reaction part, while the second one has in addition the nonlinear transport term. For both cases, the reaction part has the form of the third order polynomial with three distinct roots. We found analytic one-soliton solutions and the relationships between three simple roots and the wave speed of the soliton. For the first case, if one of the roots is the mean value of other two roots, the soliton is static.We show that the restriction on three distinct roots to obtain moving soliton is removed in the second case by, adding nonlinear transport term to the first equation.Article Citation - WoS: 9Citation - Scopus: 9Exactly Solvable Quantum Sturm-Liouville Problems(American Institute of Physics, 2009) Atılgan Büyükaşık, Şirin; Pashaev, Oktay; Tığrak Ulaş, EsraThe harmonic oscillator with time-dependent parameters covers a broad spectrum of physical problems from quantum transport, quantum optics, and quantum information to cosmology. Several methods have been developed to quantize this fundamental system, such as the path integral method, the Lewis-Riesenfeld time invariant method, the evolution operator or dynamical symmetry method, etc. In all these methods, solution of the quantum problem is given in terms of the classical one. However, only few exactly solvable problems of the last one, such as the damped oscillator or the Caldirola-Kanai model, have been treated. The goal of the present paper is to introduce a wide class of exactly solvable quantum models in terms of the Sturm-Liouville problem for classical orthogonal polynomials. This allows us to solve exactly the corresponding quantum parametric oscillators with specific damping and frequency dependence, which can be considered as quantum Sturm-Liouville problems.Conference Object Citation - WoS: 1Citation - Scopus: 1The Extended Discrete (g'/g)-expansion Method and Its Application To the Relativistic Toda Lattice System(American Institute of Physics, 2009) Aslan, İsmailWe propose the extended discrete (G′/G)-expansion method for directly solving nonlinear differentialdifference equations. For illustration, we choose the relativistic Toda lattice system. We derive further discrete hyperbolic and trigonometric function traveling wave solutions, as well as discrete rational function solutions.