Tag Archives: Classical Analysis and ODEs

On a General Method for Resolving Integrals of Multiple Spherical Bessel Functions Against Power Laws into Distributions [CL]

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http://arxiv.org/abs/2112.07809


We here present a method of performing integrals of products of spherical Bessel functions (SBFs) weighted by a power-law. Our method, which begins with double-SBF integrals, exploits a differential operator $\hat{D}$ defined via Bessel’s differential equation. Application of this operator raises the power-law in steps of two. We also here display a suitable base integral expression to which this operator can be applied for both even and odd cases. We test our method by showing that it reproduces previously-known solutions. Importantly, it also goes beyond them, offering solutions in terms of singular distributions, Heaviside functions, and Gauss’s hypergeometric,$\;_2{\rm F}_1$ for $all$ double-SBF integrals with positive semi-definite integer power-law weight. We then show how our method for double-SBF integrals enables evaluating $arbitrary$ triple-SBF overlap integrals, going beyond the cases currently in the literature. This in turn enables reduction of arbitrary quadruple, quintuple, and sextuple-SBF integrals and beyond into tractable forms.

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K. Meigs and Z. Slepian
Thu, 16 Dec 21
43/83

Comments: 20 pages, 1 figure, submitted

An Exact Integral-to-Sum Relation for Products of Bessel Functions [CL]

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http://arxiv.org/abs/2104.10169


A useful identity relating the infinite sum of two Bessel functions to their infinite integral was discovered in Dominici et al. (2012). Here, we extend this result to products of $N$ Bessel functions, and show it can be straightforwardly proven using the Abel-Plana theorem. For $N=2$, the proof is much simpler than that of the former work, and significantly enlarges the range of validity.

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O. Philcox and Z. Slepian
Thu, 22 Apr 2021
2/44

Comments: 13 pages, 1 figure. Comments welcome

A Brief Introduction to the Adomian Decomposition Method, with Applications in Astronomy and Astrophysics [IMA]

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http://arxiv.org/abs/2102.10511


The Adomian Decomposition Method (ADM) is a very effective approach for solving broad classes of nonlinear partial and ordinary differential equations, with important applications in different fields of applied mathematics, engineering, physics and biology. It is the goal of the present paper to provide a clear and pedagogical introduction to the Adomian Decomposition Method and to some of its applications. In particular, we focus our attention to a number of standard first-order ordinary differential equations (the linear, Bernoulli, Riccati, and Abel) with arbitrary coefficients, and present in detail the Adomian method for obtaining their solutions. In each case we compare the Adomian solution with the exact solution of some particular differential equations, and we show their complete equivalence. The second order and the fifth order ordinary differential equations are also considered. An important extension of the standard ADM, the Laplace-Adomian Decomposition Method is also introduced through the investigation of the solutions of a specific second order nonlinear differential equation. We also present the applications of the method to the Fisher-Kolmogorov second order partial nonlinear differential equation, which plays an important role in the description of many physical processes, as well as three important applications in astronomy and astrophysics, related to the determination of the solutions of the Kepler equation, of the Lane-Emden equation, and of the general relativistic equation describing the motion of massive particles in the spherically symmetric and static Schwarzschild geometry.

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M. Mak, C. Leung and T. Harko
Tue, 23 Feb 21
58/79

Comments: 41 pages, no figures, accepted for publication in the Romanian Astronomical Journal

The theory of figures of Clairaut with focus on the gravitational rigidity modulus: inequalities and an improvement in the Darwin-Radau equation [EPA]

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http://arxiv.org/abs/1811.07759


This paper contains a review of Clairaut’s theory with focus on the determination of a gravitational rigidity modulus $\gamma$ defined as $\left(\frac{C-I_o}{I_o}\right)\gamma=\frac{2}{3}\Omega^2$, where $C$ and $I_o$ are the polar and mean moment of inertia of the body and $\Omega$ is the body spin.The constant $\gamma$ is related to the static fluid Love number $k_2= \frac{3I_o G}{R^5} \frac{1}{\gamma}$, where $R$ is the body radius and $G$ is the gravitational constant. The new results are: a variational principle for $\gamma$, upper and lower bounds on the ellipticity that improve previous bounds by Chandrasekhar (1963) and a semi-empirical procedure for estimating $\gamma$ from the knowledge of $m$, $I_o$, and $R$, where $m$ is the mass of the body. The main conclusion is that for $0.2\le I_o/(mR^2)\le 0.4$ the approximation $\gamma\approx G \sqrt{ \frac{2^7}{5^5}\frac{m^5}{I_o^3}}= \gamma_I$ is a better estimate for $\gamma$ than that obtained from the Darwin-Radau equation, denoted as $\gamma_{DR}$. Moreover, within the range of applicability of the Darwin-Radau equation $0.32\le I_o/(mR^2)\le 0.4$ the relative difference between the two estimates, $|\gamma_{DR}/\gamma_I -1|$, is less than $0.05\%$.

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C. Ragazzo
Tue, 20 Nov 18
1/73

Comments: N/A

Existence and Classification of Asymptotic Solutions for Tolman-Oppenheimer-Volkoff Systems [CL]

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http://arxiv.org/abs/1809.02281


The Tolman-Oppenheimer-Volkoff (TOV) equations are a partially uncoupled system of nonlinear and non-autonomous ordinary differential equations which describe the structure of isotropic spherically symmetric static fluids. Nonlinearity makes explicit solutions of TOV systems very rare. We show, however, that there are at least a two-parameter table of TOV systems which are asymptotically integrable. The solutions are presented explicitly and classified according to the nature of the matter (ordinary or exotic) and to the existence of cavities and singularities.

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Y. Martins, L. Campos, D. Teixeira, et. al.
Mon, 10 Sep 18
41/58

Comments: N/A

Tensor calculus in spherical coordinates using Jacobi polynomials. Part-I: Mathematical analysis and derivations [CL]

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http://arxiv.org/abs/1804.10320


This paper presents a method for the accurate and efficient computations on scalar, vector and tensor fields in three-dimensional spherical polar coordinates. The methods uses spin-weighted spherical harmonics in the angular directions and rescaled Jacobi polynomials in the radial direction. For the 2-sphere, spin-weighted harmonics allow for automating calculations in a fashion as similar to Fourier series as possible. Derivative operators act as wavenumber multiplication on a set of spectral coefficients. After transforming the angular directions, a set of orthogonal tensor rotations put the radially dependent spectral coefficients into individual spaces each obeying a particular regularity condition at the origin. These regularity spaces have remarkably simple properties under standard vector-calculus operations, such as \textit{grad} and \textit{div}. We use a hierarchy of rescaled Jacobi polynomials for a basis on these regularity spaces. It is possible to select the Jacobi-polynomial parameters such that all relevant operators act in a minimally banded way. Altogether, the geometric structure allows for the accurate and efficient solution of general partial differential equations in the unit ball.

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G. Vasil, D. Lecoanet, K. Burns, et. al.
Mon, 30 Apr 18
-122/63

Comments: Submitted to JCP simultaneously with Part-II

Approximate solution to the fractional Lane-Emden type equations [CL]

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http://arxiv.org/abs/1705.03749


In this paper, approximate solutions for a class of fractional Lane – Emden type equations based on the series expansion method are presented. Various examples are introduced and discussed. The recurrence relation for the components of the approximate solution is constructed. In the standard integer order derivative, it is shown that the results are the same as those obtained by Adomian decomposition method, homotopy perturbation method, modified Laplace decomposition method and variational iteration method.

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M. Nouh and E. Abdel-Salam
Thu, 11 May 17
20/54

Comments: N/A

Periodic and quasi-periodic attractors for the spin-orbit evolution of Mercury with a realistic tidal torque [CL]

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http://arxiv.org/abs/1703.01189


Mercury is entrapped in a 3:2 resonance: it rotates on its axis three times for every two revolutions it makes around the Sun. It is generally accepted that this is due to the large value of the eccentricity of its orbit. However, the mathematical model originally introduced to study its spin-orbit evolution proved not to be entirely convincing, because of the expression commonly used for the tidal torque. Only recently, in a series of papers mainly by Efroimsky and Makarov, a different model for the tidal torque has been proposed, which has the advantages of being more realistic, and of providing a higher probability of capture in the 3:2 resonance with respect to the previous models. On the other hand, a drawback of the model is that the function describing the tidal torque is not smooth and consists of a superposition of kinks, so that both analytical and numerical computations turn out to be rather delicate: indeed, standard perturbation theory based on power series expansion cannot be applied and the implementation of a fast algorithm to integrate the equations of motion numerically requires a high degree of care. In this paper, we make a detailed study of the spin-orbit dynamics of Mercury, as predicted by the realistic model: In particular, we present numerical and analytical results about the nature of the librations of Mercury’s spin in the 3:2 resonance. The results provide evidence that the librations are quasi-periodic in time.

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M. Bartuccelli, J. Deane and G. Gentile
Mon, 6 Mar 17
38/47

Comments: 32 pages, 8 figures, 5 tables

Second-order variational equations for N-body simulations [EPA]

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http://arxiv.org/abs/1603.03424


First-order variational equations are widely used in N-body simulations to study how nearby trajectories diverge from one another. These allow for efficient and reliable determinations of chaos indicators such as the Maximal Lyapunov characteristic Exponent (MLE) and the Mean Exponential Growth factor of Nearby Orbits (MEGNO).
In this paper we lay out the theoretical framework to extend the idea of variational equations to higher order. We explicitly derive the differential equations that govern the evolution of second-order variations in the N-body problem. Going to second order opens the door to new applications, including optimization algorithms that require the first and second derivatives of the solution, like the classical Newton’s method. Typically, these methods have faster convergence rates than derivative-free methods. Derivatives are also required for Riemann manifold Langevin and Hamiltonian Monte Carlo methods which provide significantly shorter correlation times than standard methods. Such improved optimization methods can be applied to anything from radial-velocity/transit-timing-variation fitting to spacecraft trajectory optimization to asteroid deflection.
We provide an implementation of first and second-order variational equations for the publicly available REBOUND integrator package. Our implementation allows the simultaneous integration of any number of first and second-order variational equations with the high-accuracy IAS15 integrator. We also provide routines to generate consistent and accurate initial conditions without the need for finite differencing.

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H. Rein and D. Tamayo
Mon, 14 Mar 16
32/47

Comments: 11 pages, accepted for publication in MNRAS, code available at this https URL, figures can be reproduced interactively with binder at this http URL

Stochastic Eternal Inflation in a Bianchi Type I Universe [CL]

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http://arxiv.org/abs/1508.02670


In this paper, we analyze a Bianchi Type I model with a scalar field in a chaotic inflation potential, $V(\phi) = \frac{1}{2}\phi^2$ in the context of stochastic eternal inflation. We use the typical slow-roll approximation in combination with expansion-normalized variables in an orthonormal frame approach to obtain a dynamical system which describes the dynamics of the shear anisotropy and the inflaton field. We first show that the dynamics of the inflaton field can be decoupled from the dynamics of the shear anisotropy. We then use a fixed-points analysis in combination with global techniques from topological dynamical systems theory to prove that the cosmological model under consideration isotropizes irrespective of an inflationary epoch, which has also described by other authors who have investigated a Bianchi Type I model under similar configurations. We then show that for inflation to occur, the amount of anisotropy must be very small.
We also give a description of the stochastic dynamics of the inflaton field by using techniques from stochastic calculus. We show that the Klein-Gordon equation becomes a stochastic differential equation with a highly nonlinear drift term. In this case, the deceleration parameter itself becomes a random variable, and we give details regarding when such a model can undergo inflation. We finally derive the form of the long-term, stationary probability distribution of the inflaton field, and show that it has the form of a double-well potential. We then calculate the probability of inflation occurring based on this approach. We conclude the paper by performing some numerical simulations of the stochastic differential equation describing the dynamics of the inflaton field. We conjecture that even in the case of stochastic eternal inflation, one requires precise initial conditions for inflation to occur.

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I. Kohli and M. Haslam
Wed, 12 Aug 15
50/50

Comments: N/A

Finite-Time Singularities in $k=0$ FLRW Cosmologies [CL]

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http://arxiv.org/abs/1507.02241


In this paper, we consider a spatially flat FLRW cosmological model with matter obeying a barotropic equation of state $p = w \mu$, $-1<w\leq1$, and a cosmological constant, $\Lambda$. We use Osgood’s criterion to establish three cases when such models admit finite-time singularities. The first case is for an arbitrary initial condition, with a negative cosmological constant, and phantom energy $w < -1$. We show that except for a very fine-tuned choice of the initial condition $\theta_{0}$, the universe will develop a finite-time singularity. The second case we consider is for a nonnegative cosmological constant, phantom energy, and the expansion scalar being larger than that of the flat-space de Sitter solution, and show that such models only expand forever for $\Lambda = 0$. In all other cases, the universe model develops a finite-time singularity. The final case we consider is for a nonnegative cosmological constant, a matter source with $-1 < w \leq 1$, and an expansion scalar that is asymptotically that of the de Sitter universe. We show that such models will only expand forever when $\Lambda = 0$, otherwise, they will develop a finite-time singularity. This is significant, since the inflationary epoch is a subset of this domain. However, as we show, the inclusion of a bulk viscosity term in the Einstein field equations eliminates this singularity, and the universe expands forever. This could have interesting implications for the role of bulk viscosity in dynamical models of the universe.

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I. Kohli
Thu, 9 Jul 15
7/50

Comments: arXiv admin note: text overlap with arXiv:1505.07770

On Singularities in Cosmic Inflation [CL]

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http://arxiv.org/abs/1505.07770


In this paper, we examine a flat FLRW spacetime with a scalar field potential and show by applying Osgood’s criterion to the Einstein field equations that all such models, irrespective of the particular choice of potential develop finite-time singularities. That is, we show that solutions to the field equations rapidly diverge in finite time. This can have important implications for the role of inflation in cosmological models, since one of the implications of this is that within the inflationary epoch, a singularity develops in finite time, which would call into question the role of inflation in the dynamic evolution of our universe. We further point out that a possible reason for this behaviour is that the solutions to the field equations in such inflationary scenarios do not obey global existence and uniqueness properties, which is a typical characteristic of solutions that diverge in finite time.

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I. Kohli
Fri, 29 May 15
25/68

Comments: For submission to: Classical and Quantum Gravity

Novel representation of the general Heun's functions [CL]

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http://arxiv.org/abs/1405.6837


In the present article we introduce and study a novel representation of the general Heun’s functions. It is based on the symmetric form of the Heun’s differential equation and hopefully will simplify the solution of the existing basic open problems in the theory of these functions. The novel representation will stimulate also the development of new effective computational methods for calculations with the general Heun’s functions which at present is a quite problematic issue.

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P. Fiziev
Mon, 2 Jun 14
42/56

Comments: 11 pages LaTex file

Slepian Spatial-Spectral Concentration on the Ball [CL]

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http://arxiv.org/abs/1403.5553


We formulate and solve the Slepian spatial-spectral concentration problem on the three-dimensional ball. Both the standard Fourier-Bessel and also the Fourier-Laguerre spectral domains are considered since the latter exhibits a number of practical advantages (spectral decoupling and exact computation). The Slepian spatial and spectral concentration problems are formulated as eigenvalue problems, the eigenfunctions of which form an orthogonal family of concentrated functions. Equivalence between the spatial and spectral problems is shown. The spherical Shannon number on the ball is derived, which acts as the analog of the space-bandwidth product in the Euclidean setting, giving an estimate of the number of concentrated eigenfunctions and thus the dimension of the space of functions that can be concentrated in both the spatial and spectral domains simultaneously. Various symmetries of the spatial region are considered that reduce considerably the computational burden of recovering eigenfunctions, either by decoupling the problem into smaller subproblems or by affording analytic calculations. The family of concentrated eigenfunctions forms a Slepian basis that can be used be represent concentrated signals efficiently. We illustrate our results with numerical examples and show that the Slepian basis indeeds permits a sparse representation of concentrated signals.

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Z. Khalid, R. Kennedy and J. McEwen
Mon, 24 Mar 14
49/50

Exploring Vacuum Energy in a Two-Fluid Bianchi Type I Universe [CL]

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http://arxiv.org/abs/1402.1967


We use a dynamical systems approach based on the method of orthonormal frames to study the dynamics of a two-fluid, non-tilted Bianchi Type I cosmological model. In our model, one of the fluids is a fluid with bulk viscosity, while the other fluid assumes the role of a cosmological constant and represents nonnegative vacuum energy. We begin by completing a detailed fixed-point analysis of the system which gives information about the local sinks, sources and saddles. We then proceed to analyze the global features of the dynamical system by using topological methods such as finding Lyapunov and Chetaev functions, and finding the $\alpha$- and $\omega$-limit sets using the LaSalle invariance principle. The fixed points found were a flat Friedmann-LeMa\^{\i}tre-Robertson-Walker (FLRW) universe with no vacuum energy, a de Sitter universe, a flat FLRW universe with both vacuum and non-vacuum energy, and a Kasner quarter-circle universe. We also show in this paper that the vacuum energy we observe in our present-day universe could actually be a result of the bulk viscosity of the ordinary matter in the universe, and proceed to calculate feasible values of the bulk viscous coefficient based on observations reported in the Planck data. We conclude the paper with some numerical experiments that shed further light on the global dynamics of the system.

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I. Kohli and M. Haslam
Tue, 11 Feb 14
43/55

Attractive interaction between ions inside a quantum plasma structure [CL]

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http://arxiv.org/abs/1311.6875


We construct the model of a quantum spherically symmetric plasma structure based on radial oscillations of ions. We suppose that ions are involved in ion-acoustic waves. We find the exact solution of the Schrodinger equation for an ion moving in the self-consistent oscillatory potential of an ion-acoustic wave. The system of ions is secondly quantized and its ground state is constructed. Then we consider the interaction between ions by the exchange of an acoustic wave. It is shown that this interaction can be attractive. We describe the formation of pairs of ions inside a plasma structure and demonstrate that such a plasmoid can exist in dense astrophysical medium. The application of our results for terrestrial plasmas is also discussed.

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Thu, 28 Nov 13
20/47