arXiv subjects

M. K. Mak

Publications and source records attributed to M. K. Mak.

At least 19 recordsLinked to original sources

arXiv2017

Approximate Analytical Solution for the Dynamic Model of Large Amplitude Non-Linear Oscillations Arising in Structural Engineering

In this work we obtain an approximate solution of the strongly nonlinear second order differential equation $\frac{d^{2}u}{dt^{2}}+ω^{2}u+αu^{2}\frac{d^{2}u}{dt^{2}}+αu\left( \frac{du}{dt}\right)^{2}+βω^{2}u^{3}=0$, describing the large amplitude free vibrations of a uniform cantilever beam, by using a method based on the Laplace transform, and the convolution theorem. By reformulating the initial differential equation as an integral equation, with the use of an iterative procedure, an approximate solution of the nonlinear vibration equation can be obtained in any order of approximation. The iterative approximate solutions are compared with the exact numerical solution of the vibration equation.

math.CA↗

arXiv2017

Hybrid metric-Palatini stars

We consider the internal structure and the physical properties of specific classes of neutron, quark and Bose-Einstein Condensate stars in the hybrid metric-Palatini gravity theory, which is a combination of the metric and Palatini $f(R)$ formalisms. The theory is very successful in accounting for the observed phenomenology, since it unifies local constraints at the Solar System level and the late-time cosmic acceleration, even if the scalar field is very light. We derive the equilibrium equations for a spherically symmetric configuration (mass continuity and Tolman-Oppenheimer-Volkoff) in the framework of hybrid metric-Palatini theory, and we investigate their solutions numerically for different equations of state of neutron and quark matter, by adopting for the scalar field potential a Higgs-type form. Stellar models, described by the stiff fluid, radiation-like, the bag model and the Bose-Einstein Condensate equations of state are explicitly constructed in both General Relativity and hybrid metric-Palatini gravity, thus allowing an in depth comparison between the predictions of these two gravitational theories. As a general result it turns out that for all the considered equations of state, hybrid metric-Palatini gravity stars are more massive than their general relativistic counterparts. Furthermore, two classes of stellar models corresponding to fixed forms of the scalar field are also investigated. Interestingly enough, in the case of a constant scalar field the equation of state of the matter takes the form of the bag model equation of state describing quark matter. As a possible astrophysical application of the obtained results we suggest that stellar mass black holes, with masses in the range of $3.8M_{\odot}$ and $6M_{\odot}$, respectively, could be in fact hybrid metric-Palatini gravity neutron or quark stars.

gr-qc↗

arXiv2016

Exact Scalar-Tensor Cosmological Models

Scalar-tensor gravitational theories are important extensions of standard general relativity, which can explain both the initial inflationary evolution, as well as the late accelerating expansion of the Universe. In the present paper we investigate the cosmological solution of a scalar-tensor gravitational theory, in which the scalar field $ϕ$ couples to the geometry via an arbitrary function $F(ϕ$). The kinetic energy of the scalar field as well as its self-interaction potential $V(ϕ)$ are also included in the gravitational action. By using a standard mathematical procedure, the Lie group approach, and Noether symmetry techniques, we obtain several exact solutions of the gravitational field equations describing the time evolutions of a flat Friedman-Robertson-Walker Universe in the framework of the scalar-tensor gravity. The obtained solutions can describe both accelerating and decelerating phases during the cosmological expansion of the Universe.

gr-qc↗

arXiv2016

Exact power series solutions of the structure equations of the general relativistic isotropic fluid stars with linear barotropic and polytropic equations of state

Obtaining exact solutions of the spherically symmetric general relativistic gravitational field equations describing the interior structure of an isotropic fluid sphere is a long standing problem in theoretical and mathematical physics. The usual approach to this problem consists mainly in the numerical investigation of the Tolman-Oppenheimer-Volkoff and of the mass continuity equations, which describes the hydrostatic stability of the dense stars. In the present paper we introduce an alternative approach for the study of the relativistic fluid sphere, based on the relativistic mass equation, obtained by eliminating the energy density in the Tolman-Oppenheimer-Volkoff equation. Despite its apparent complexity, the relativistic mass equation can be solved exactly by using a power series representation for the mass, and the Cauchy convolution for infinite power series. We obtain exact series solutions for general relativistic dense astrophysical objects described by the linear barotropic and the polytropic equations of state, respectively. For the polytropic case we obtain the exact power series solution corresponding to arbitrary values of the polytropic index $n$. The explicit form of the solution is presented for the polytropic index $n=1$, and for the indexes $n=1/2$ and $n=1/5$, respectively. The case of $n=3$ is also considered. In each case the exact power series solution is compared with the exact numerical solutions, which are reproduced by the power series solutions truncated to seven terms only. The power series representations of the geometric and physical properties of the linear barotropic and polytropic stars are also obtained.

gr-qc↗

arXiv2015

Exact Scalar-Tensor Cosmological Solutions via Noether Symmetry

In this paper, we investigate the Noether symmetries of a generalized scalar-tensor, Brans-Dicke type cosmological model, in which we consider explicit scalar field dependent couplings to the Ricci scalar, and to the scalar field kinetic energy, respectively. We also include the scalar field self-interaction potential into the gravitational action. From the condition of the vanishing of the Lie derivative of the gravitational cosmological Lagrangian with respect to a given vector field we obtain three cosmological solutions describing the time evolution of a spatially flat Friedman-Robertson-Walker Universe filled with a scalar field. The cosmological properties of the solutions are investigated in detail, and it is shown that they can describe a large variety of cosmological evolutions, including models that experience a smooth transition from a decelerating to an accelerating phase.

gr-qc↗

arXiv2015

Exact travelling wave solutions of non-linear reaction-convection-diffusion equations -- an Abel equation based approach

We consider quasi-stationary (travelling wave type) solutions to a general nonlinear reaction-convection-diffusion equation with arbitrary, autonomous coefficients. The second order nonlinear equation describing one dimensional travelling waves can be reduced to a first kind first order Abel type differential equation By using two integrability conditions for the Abel equation (the Chiellini lemma and the Lemke transformation), several classes of exact travelling wave solutions of the general reaction--convection-diffusion equation are obtained, corresponding to different functional relations imposed between the diffusion, convection and reaction functions. In particular, we obtain travelling wave solutions for two non-linear second order partial differential equations, representing generalizations of the standard diffusion equation and of the classical Fisher--Kolmogorov equation, to which they reduce for some limiting values of the model parameters. The models correspond to some specific, power law type choices of the reaction and convection functions, respectively. The travelling wave solutions of these two classes of differential equations are investigated in detail by using both numerical and semi-analytical methods.

math-ph↗

arXiv2014

Wormhole geometries supported by quark matter at ultra-high densities

A fundamental ingredient in wormhole physics is the presence of exotic matter, which involves the violation of the null energy condition. In this context, we investigate the possibility that wormholes could be supported by quark matter at extreme densities. Theoretical and experimental investigations of the structure of baryons show that strange quark matter, consisting of the u, d and s quarks, is the most energetically favorable state of baryonic matter. Moreover, at ultra-high densities, quark matter may exist in a variety of superconducting states, namely, the Color-Flavor-Locked (CFL) phase. Motivated by these theoretical models, we explore the conditions under which wormhole geometries may be supported by the equations of state considered in the theoretical investigations of quark-gluon interactions. For the description of the normal quark matter we adopt the Massachusetts Institute of Technology (MIT) bag model equation of state, while the color superconducting quark phases are described by a first order approximation of the free energy. By assuming specific forms for the bag and gap functions, several wormhole models are obtained for both normal and superconducting quark matter. The effects of the presence of an electrical charge are also taken into account.

gr-qc↗

arXiv2014

Bianchi type I cosmological models in Eddington-inspired Born-Infeld gravity

We consider the dynamics of a barotropic cosmological fluid in an anisotropic, Bianchi type I space-time in Eddington-inspired Born-Infeld (EiBI) gravity. By assuming an isotropic pressure distribution, we obtain the general solution of the field equations in an exact parametric form. The behavior of the geometric and thermodynamic parameters of the Bianchi type I Universe is studied, by using both analytical and numerical methods, for some classes of high density matter, described by the stiff causal, radiation, and pressureless fluid equations of state. In all cases the study of the models with different equations of state can be reduced to the integration of a highly nonlinear second order ordinary differential equation for the energy density. The time evolution of the anisotropic Bianchi type I Universe strongly depends on the initial values of the energy density and of the Hubble function. An important observational parameter, the mean anisotropy parameter is also studied in detail, and we show that for the dust filled Universe the cosmological evolution always ends into an isotropic phase, while for high density matter filled universes the isotropization of Bianchi type I universes is essentially determined by the initial conditions of the energy density.

gr-qc↗

arXiv2014

Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach

We consider quasi-stationary (travelling wave type) solutions to a nonlinear reaction-diffusion equation with arbitrary, autonomous coefficients, describing the evolution of glioblastomas, aggressive primary brain tumors that are characterized by extensive infiltration into the brain and are highly resistant to treatment. The second order nonlinear equation describing the glioblastoma growth through travelling waves can be reduced to a first order Abel type equation. By using the integrability conditions for the Abel equation several classes of exact travelling wave solutions of the general reaction-diffusion equation that describes glioblastoma growth are obtained, corresponding to different forms of the product of the diffusion and reaction functions. The solutions are obtained by using the Chiellini lemma and the Lemke transformation, respectively, and the corresponding equations represent generalizations of the classical Fisher--Kolmogorov equation. The biological implications of two classes of solutions are also investigated by using both numerical and semi-analytical methods for realistic values of the biological parameters.

q-bio.TO↗

arXiv2014

A Riccati equation based approach to isotropic scalar field cosmologies

Gravitationally coupled scalar fields $ϕ$, distinguished by the choice of an effective self-interaction potential $V(ϕ)$, simulating a temporarily non-vanishing cosmological term, can generate both inflation and late time acceleration. In scalar field cosmological models the evolution of the Hubble function is determined, in terms of the interaction potential, by a Riccati type equation. In the present work we investigate scalar field cosmological models that can be obtained as solutions of the Riccati evolution equation for the Hubble function. Four exact integrability cases of the field equations are presented, representing classes of general solutions of the Riccati evolution equation. The solutions correspond to cosmological models in which the Hubble function is proportional to the scalar field potential plus a linearly decreasing function of time, models with the time variation of the scalar field potential proportional to the potential minus its square, models in which the potential is the sum of an arbitrary function and the square of the function integral, and models in which the potential is the sum of an arbitrary function and the derivative of its square root, respectively. The cosmological properties of all models are investigated in detail, and it is shown that they can describe the inflationary or the late accelerating phase in the evolution of the Universe

gr-qc↗

arXiv2014

New further integrability cases for the Riccati equation

New further integrability conditions of the Riccati equation $dy/dx=a(x)+b(x)y+c(x)y^{2}$ are presented. The first case corresponds to fixed functional forms of the coefficients $a(x)$ and $c(x)$ of the Riccati equation, and of the function $F(x)=a(x)+[f(x)-b^{2}(x)]/4c(x)$, where $f(x)$ is an arbitrary function. The second integrability case is obtained for the "reduced" Riccati equation with $b(x)\equiv 0$. If the coefficients $a(x)$ and $c(x)$ satisfy the condition $\pm d\sqrt{f(x)/c(x)}/dx=a(x)+f(x)$, where $f(x)$ is an arbitrary function, then the general solution of the "reduced" Riccati equation can be obtained by quadratures. The applications of the integrability condition of the "reduced" Riccati equation for the integration of the Schrödinger and Navier-Stokes equations are briefly discussed.

math-ph↗

arXiv2014

Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates

In this paper, the exact analytical solution of the Susceptible-Infected-Recovered (SIR) epidemic model is obtained in a parametric form. By using the exact solution we investigate some explicit models corresponding to fixed values of the parameters, and show that the numerical solution reproduces exactly the analytical solution. We also show that the generalization of the SIR model, including births and deaths, described by a nonlinear system of differential equations, can be reduced to an Abel type equation. The reduction of the complex SIR model with vital dynamics to an Abel type equation can greatly simplify the analysis of its properties. The general solution of the Abel equation is obtained by using a perturbative approach, in a power series form, and it is shown that the general solution of the SIR model with vital dynamics can be represented in an exact parametric form.

q-bio.PE↗

arXiv2014

Dark matter density profile and galactic metric in Eddington-inspired Born-Infeld gravity

We consider the density profile of pressureless dark matter in Eddington-inspired Born-Infeld (EiBI) gravity. The gravitational field equations are investigated for a spherically symmetric dark matter galactic halo, by adopting a phenomenological tangential velocity profile for test particles moving in stable circular orbits around the galactic center. The density profile and the mass distribution, as well as the general form of the metric tensor is obtained by numerically integrating the gravitational field equations, and in an approximate analytical form by using the Newtonian limit of the theory. In the weak field limit the dark matter density distribution is described by the Lane-Emden equation with polytropic index $n=1$, and is non-singular at the galactic center. The parameter $κ$ of the theory is determined so that that the theory could provide a realistic description of the dark matter halos. The gravitational properties of the dark matter halos are also briefly discussed in the Newtonian approximation.

gr-qc↗

arXiv2014

Arbitrary scalar field and quintessence cosmological models

The mechanism of the initial inflationary scenario of the universe and of its late-time acceleration can be described by assuming the existence of some gravitationally coupled scalar fields $ϕ$, with the inflaton field generating inflation and the quintessence field being responsible for the late accelerated expansion. Various inflationary and late-time accelerated scenarios are distinguished by the choice of an effective self-interaction potential $V(ϕ)$, which simulates a temporarily non-vanishing cosmological term. In this work, we present a new formalism for the analysis of scalar fields in flat isotropic and homogeneous cosmological models. The basic evolution equation of the models can be reduced to a first order non-linear differential equation. Approximate solutions of this equation can be constructed in the limiting cases of the scalar field kinetic energy and potential energy dominance, respectively, as well as in the intermediate regime. Moreover, we present several new accelerating and decelerating exact cosmological solutions, based on the exact integration of the basic evolution equation for scalar field cosmologies. More specifically, exact solutions are obtained for exponential, generalized cosine hyperbolic, and power law potentials, respectively. Cosmological models with power law scalar field potentials are also analyzed in detail.

gr-qc↗

arXiv2014

A class of exact solutions of the Liénard type ordinary non-linear differential equation

A class of exact solutions is obtained for the Liénard type ordinary non-linear differential equation. As a first step in our study the second order Liénard type equation is transformed into a second kind Abel type first order differential equation. With the use of an exact integrability condition for the Abel equation (the Chiellini lemma), the exact general solution of the Abel equation can be obtained, thus leading to a class of exact solutions of the Liénard equation, expressed in a parametric form. We also extend the Chiellini integrability condition to the case of the general Abel equation. As an application of the integrability condition the exact solutions of some particular Liénard type equations, including a generalized van der Pol type equation, are explicitly obtained.

math-ph↗

arXiv2013

Analytical solutions of the Riccati equation with coefficients satisfying integral or differential conditions with arbitrary functions

Ten new exact solutions of the Riccati equation $dy/dx=a(x)+b(x)y+c(x)y^{2}$ are presented. The solutions are obtained by assuming certain relations among the coefficients $a(x)$, $b(x)$ and $c(x)$ of the Riccati equation, in the form of some integral or differential expressions, also involving some arbitrary functions. By appropriately choosing the form of the coefficients of the Riccati equation, with the help of the conditions imposed on the coefficients, we obtain ten new integrability cases for the Riccati equation. For each case the general solution of the Riccati equation is also presented. The possibility of the application of the obtained mathematical results for the study of anisotropic general relativistic stellar models is also briefly considered.

math.CA↗

arXiv2013

Isotropic stars in general relativity

We present a general solution of the Einstein gravitational field equations for the static spherically symmetric gravitational interior spacetime of an isotropic fluid sphere. The solution is obtained by transforming the pressure isotropy condition, a second order ordinary differential equation, into a Riccati type first order differential equation, and using a general integrability condition for the Riccati equation. This allows us to obtain an exact non-singular solution of the interior field equations for a fluid sphere, expressed in the form of infinite power series. The physical features of the solution are studied in detail numerically by cutting the infinite series expansions, and restricting our numerical analysis by taking into account only $n=21$ terms in the power series representations of the relevant astrophysical parameters. In the present model all physical quantities (density, pressure, speed of sound etc.) are finite at the center of the sphere. The physical behavior of the solution essentially depends on the equation of state of the dense matter at the center of the star. The stability properties of the model are also analyzed in detail for a number of central equations of state, and it is shown that it is stable with respect to the radial adiabatic perturbations. The astrophysical analysis indicates that this solution can be used as a realistic model for static general relativistic high density objects, like neutron stars.

gr-qc↗

arXiv2013

A Chiellini type integrability condition for the generalized first kind Abel differential equation

The Chiellini integrability condition of the first order first kind Abel equation $dy/dx=f(x)y^2+g(x)y^3$ is extended to the case of the general Abel equation of the form $dy/dx=a(x)+b(x)y+f(x)y^{α-1}+g(x)y^α$, where $α\in \Re$, and $α> 1$. In the case $α=2$ the generalized Abel equations reduces to a Riccati type equation, for which a Chiellini type integrability condition is obtained.

nlin.SI↗