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On the Integrability of the Abel and of the Extended Liénard Equations

Acta Mathematicae Applicatae Sinica 2019年第4期35卷 722-736页

作者： Man Kwong Mak tiberiu harko Departamento de Fsica Facultad de Ciencias Naturales Universidad de Atacama Copayapu 485 Copiapo Chile Department of Physics Babes-Bolyai University Kogalniceanu Street Cluj-Napoca 400084 Romania School of Physics Sun Yat-Sen University Guangzhou 510275 China

We present some exact integrability cases of the extended Liénard equation y′′+ f(y)(y′)n +k(y)(y′)m + g(y)y′+ h(y) = 0, with n > 0 and m > 0 arbitrary constants, while f(y), k(y), g(y), and h(y) are arbitrary functions. The solutions are obtained by transforming the equation Liénard equation to an equivalent first kind first order Abel type equation given bydv/dy= f(y)v3-n+ k(y)v3-m+ g(y)v2+ h(y)v3, with v = 1/y′.As a first step in our study we obtain three integrability cases of the extended quadratic-cubic Liénard equation,corresponding to n = 2 and m = 3, by assuming that particular solutions of the associated Abel equation are known. Under this assumption the general solutions of the Abel and Liénard equations with coefficients satisfying some differential conditions can be obtained in an exact closed form. With the use of the Chiellini integrability condition, we show that if a particular solution of the Abel equation is known, the general solution of the extended quadratic cubic Liénard equation can be obtained by quadratures. The Chiellini integrability condition is extended to generalized Abel equations with g(y) ≡ 0 and h(y) ≡ 0, and arbitrary n and m, thus allowing to obtain the general solution of the corresponding Liénard equation. The application of the generalized Chiellini condition to the case of the reduced Riccati equation is also considered.

关键词： Abel equation Liénard equation general solutions integrability conditions

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Fractal dimension and thermodynamic fluctuation properties of IDV light curves

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Research in Astronomy and Astrophysics 2011年第9期11卷 1031-1045页

作者： Chun-Sing Leung Jian-Yan Wei Zoltan Kovlics tiberiu harko National Astronomical Observatories Chinese Academy of Sciences Beijing 100012 China Graduate University of Chinese Academy of Sciences Beijing 100049 China Department of Physics and Center for Theoretical and Computational Physics The University ofHong Kong Pok Fu Lam Road Hong Kong Hong Kong SAR China

Fractals are a basic tool for phenomenologicalty describing natural ob- jects having a high degree of temporal or spatial variability. From a physical point of view the fractal properties of natural systems can also be interpreted by using the standard formalism of thermodynamical fluctuations. We introduce and analyze the fractal dimension of the Intra-Day-Variability （IDV） light-curves of BL Lac objects, in the optical, radio and X-ray bands. A general description of the fluctuation spec- trum of these systems based on general thermodynamical principles is also proposed. Based on the general fractal properties of a given physical system, we also introduce the predictability index for the IDV light curves. We have explicitly determined the fractal dimension for the R-band observations of five blazars, as well as for the ra- dio band observations of the compact extragalactic radio source J 1128＋5925 and of several X-ray sources. Our results show that the fractal dimension of the optical and X-ray observations indicates an almost pure ＂Brownian noise＂（random walk） spectrum, with a very low predictability index, while in the radio band the predictability index is much higher. We have also studied the spectral properties of the IDV light curves and we have shown that their spectral index is very closely correlated with the corresponding fractal dimension.

关键词： galaxies -- quasars: emission lines -- instabilities -- gravitation

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An Exact Anisotropic Quark Star Model

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Chinese Journal of Astronomy and Astrophysics 2002年第3期2卷 248-259页

作者： M.K.Mak T.harko DepartmentofPhysics TheHongKongUniversityofScienceandTechnologyClearWaterBayHongKong DepartmentofPhysics TheUniversityofHongKongPokfulamHongKong

We present an exact analytical solution of the gravitational field equations describing a static spherically symmetric anisotropic quark matter distribution. The radial pressure inside the star is assumed to obey a linear equation of state, while the tangential pressure is a complicated function of the radial coordinate. In order to obtain the general solution of the field equations a particular density profile inside the star is also assumed. The anisotropic pressure distribution leads to an increase in the maximum radius and mass of the quark star, which in the present model is around three solar masses.

关键词： Stars - quark: interior solution: anisotropy

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快速转动中子星的半解析方法

快速转动中子星的半解析方法

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中国天文学会高能分会2004年学术年会

作者：袁业飞 Prof.K.S.Cheng Dr.T.harko 中国科学技术大学天体物理中心 HKU HKU

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