Form Of Series Expansion Research Articles

Heat equation for Sturm–Liouville operator with singular propagation and potential

Abstract This article considers the initial boundary value problem for the heat equation with the time-dependent Sturm–Liouville operator with singular potentials. To obtain a solution by the method of separation of variables, the problem is reduced to the problem of eigenvalues of the Sturm–Liouville operator. Further on, the solution to the initial boundary value problem is constructed in the form of a Fourier series expansion. A heterogeneous case is also considered. Finally, we establish the well-posedness of the equation in the case when the potential and initial data are distributions, also for singular time-dependent coefficients.

Analysis of the thermo-hydro-mechanical coupled dynamic response of sediments under eccentric driving of deep-sea mining vehicle

This study establishes a coupled thermo-hydro-mechanical dynamic model(THMD) for saturated porous sediments under the eccentric motion of a deep-sea mining vehicle, considering the characteristics of deep-sea sediments. We applied the method of complex Fourier series expansion to simplify the coupled equations into ordinary differential equations and obtained the complex Fourier series expansion forms of various physical quantities of sediments during the eccentric movement of the mining vehicle. The undetermined constants are determined based on boundary conditions, yielding a series representation of the solution for THMD of deep-sea sediments. Subsequently, the solution was utilized to investigate the influence of mining vehicle parameters on the dynamic response of deep-sea sediments. This study demonstrates that the vertical displacement, vertical stress, and excess pore water pressure of sediments increase as the speed of the mining vehicle increases. Furthermore, the vertical displacement, vertical stress, and excess pore water pressure increase with the increment of load amplitude, and the differences in the curves at different eccentricities gradually become larger as the load amplitude increases. The proposed model can be applied to geotechnical engineering problems in saturated porous foundations, providing a necessary theoretical basis and analytical approach.

Gravitational waves with dark matter minispikes: Fourier-domain waveforms of eccentric intermediate-mass-ratio-inspirals

Abstract An intermediate mass black hole (IMBH) may have a dark matter (DM) minihalo around it and develop a spiky structure called DM minispike. Gravitational waves (GWs) can be produced if a stellar compact object, such as a black hole (BH) or neutron star, inspirals into the IMBH. This kind of systems are known as itermediate-mass-ratio-inspirals (IMRIs) and may be observed by space-based gravitational wave detectors including LISA, Taiji and Tianqin. In this paper, we lay the foundations for the construction of analytic expressions for Fourier-domain gravitational waves produced by eccentric IMRIs with DM minispikes in a post-circular or small-eccentricity approximation (e < 0.4). We take the effect of dynamical friction from the DM as a perturbation, and decompose the dynamical equations into perturbed part and unperturbed part. The equations are solved in a series expansion form about zero initial eccentricity to eighth order. The time-dependent, “plus” and “cross” polarizations are expanded in Bessel functions, which are then self-consistently reexpanded in a power series about zero initial eccentricity. The stationary-phase approximation is then employed to obtain the explicit DM-modified analytic expressions for the Fourier transform of the post-circular expanded, time-domain signal. We exemplify this framework by considering a typical IMRI with a DM minispike and find the GW detectability strongly depends on the radial profile of the DM distribution. When the density of DM is large enough, the signal to noise ratio (SNR) will be degraded significantly and a detection loss may occur if we use a template without the effect of DM to treat a signal including the DM effect. With the Fourier-domain gravitational waveforms we also estimate the accuracy of the measurement of the DM minispike parameters in our reference model. Our framework hold the promise to construct a “ready-to-use” Fourier-domain waveforms for data analysis of eccentric IMRIs with DM minispikes.

On the regularity of the flow of analytic vector fields

The flow of vector field X is the solution to the system of linear partial differential equations ∂_t u(t,z)=X(u(t,z)) with initial condition u(0,z)=z. It is known that for an analytic vector field, its flow is given by a convergent Lie series. Recently, Carillo gave a quick and elementary proof of this fact by introducing a special family of norms. This paper gives an even more concise proof by directly estimating the formal series expansion and using a family of majorant functions studied by Lax.

Efficient solutions for fractional Tsunami shallow-water mathematical model: A comparative study via semi analytical techniques

In this study, we delve into the effective solution of the fractional tsunami shallow-water mathematical model. Tsunami propagation and inundation modeling is of paramount importance for hazard assessment and disaster preparedness. By employing two distinct semi-analytical techniques: the residual power series method and the modified Taylor series method, we obtain approximate solutions for the tsunami mathematical model. Solutions for varying coastal inclinations and ocean depths have been acquired for the given model taking into consideration the changes in coastal gradient and ocean depth and how to impact tsunami wave velocity and wave amplification at different values of the fractional order derivative and different steps of time. The utilized semi analytical techniques generate the solution in the form of power series expansion, so the convergence study is implemented. In order to prove the efficiency of the proposed methods, several tables for comparisons between our results and other numerical results from literature are presented. Also graphs of the obtained solutions introduced in two and three dimensions.

Thermodynamics of the electron–positron plasma at very high temperatures and sources of annihilation γ-quanta in our Galaxy

Thermodynamic properties and equations of state of the electron–positron plasma (or gas) at high and very high temperatures (T≥170 keV) are derived and investigated. We have derived a number of simple analytical formulas for the Fermi–Dirac distribution functions, which can be applied to various Fermi gases and plasmas in different cases. Almost all these formulas are represented in the form of series expansions. The coefficients in these expansions are relatively simple functions of the μT ratio, where T is the temperature and μ is the chemical potential of this Fermi system. Our approach works well for high-temperature electron–positron plasmas, which are in thermal equilibrium with the photon gas of annihilation γ-quanta, and for the model gas of fermions, where there is no radiation at all. The first case corresponds to the ultra-relativistic limit for high-temperature electron–positron plasma, while the second case represents a model Fermi gas of particles, which has some non-zero chemical potential. By investigating sources of annihilation γ-quanta in our Galaxy, we have arrived to a remarkable conclusion about the high-temperature limit in our regular (photon) optics.

Poisson distributions in stochastic dynamics of gene expression: What events do they count?

The Poisson distribution is the probability distribution of the number of independent events in a given period. Although the Poisson distribution appears ubiquitously in various stochastic dynamics of gene expression, both as time-dependent distributions and stationary distributions, underlying independent events that give rise to such distributions have not been clear, especially in the presence of the degradation of gene products, which is not a Poisson process. I show that the variable following the Poisson distribution is the number of independent events where biomolecules are created, destined to survive until the end of a given time duration. This new viewpoint enables me to rederive the Poisson distribution as a time-dependent probability distribution for molecule numbers in various monomolecular reaction models of stochastic gene dynamics. Additionally, it allows me to derive an analytic form of the time-dependent probability distribution for multispecies monomolecular reaction models with species whose lifetimes follow nonexponential distributions, which is the convolution of the Poisson distribution with the multinomial distribution. This distribution is then utilized for deriving a novel series expansion form of a time-dependent distribution for a model with a stochastic production rate.

Pupil aberrations correction of the afocal telescope for the TianQin project

TianQin is a planned Chinese space-based gravitational wave (GW) observatory with a frequency band of 10−4–1 Hz. Optical telescopes are essential for the delivery of the measurement beam to support a precise distance measurement between pairs of proof masses. As the design is driven by the interferometric displacement sensitivity requirements, the stability control of optical path length (OPL) is extremely important beyond the traditional requirement of diffraction-limited imaging quality. The recurring tilt-to-length (TTL) coupling noise arises from the OPL variation due to the wavefront deformation and angular misalignment. Reducing the residual chief ray aberration in the optical design helps suppress TTL coupling noise. To correct the pupil aberrations, we derive primary pupil aberrations in a series expansion form, and then refine the formulation of merit function by combining the pupil aberration theory and traditional image aberration theory. The automatic correction of pupil aberrations is carried out by using the macro programming in the commercial optical software Zemax, leading to a high performance telescope design. The design results show that on one side the pupil aberrations have been corrected, and on the other side, its optical performance meets the requirements for TianQin project. The RMS wavefront error over the science field of view (FOV) is less than λ/200 and the maximum TTL coupling noise over the entire ±300 μrad FOV is 0.0152 nm µrad−1 . We believe that our design approach can be a good guide for the space telescope design in any other space-based GW detection project, as well as other similar optical systems.

Weierstrass elliptic function solutions and degenerate solutions of a variable coefficient higher-order Schrödinger equation

In this paper, the auxiliary equation method is used to study the Weierstrass elliptic function solutions and degenerate solutions of the variable coefficient higher order Schrödinger equation, including Jacobian elliptic function solutions, trigonometric function solutions and hyperbolic function solutions. The types of solutions of the variable coefficient higher-order Schrödinger equation are enriched, and the method of seeking precise and accurate solutions is extended. It is concluded that the types of degenerate solutions are related to the coefficients of the equation itself when the degenerate solutions are obtained from the solutions of the Weierstrass elliptic functions. In addition, the solutions form of the equation is extended from the power series expansion form to the Laurent series expansion form, and the corresponding solutions are obtained. After the conversion formula between the Weierstrass elliptic function solutions and the Jacobian elliptic function solutions is constructed, the Jacobian elliptic function solutions of the higher order Schrödinger equation with variable coefficients are also obtained. These have not been previously studied.

Efficient Solution of Fractional System Partial Differential Equations Using Laplace Residual Power Series Method

In this paper, we present an efficient solution method for solving fractional system partial differential equations (FSPDEs) using the Laplace residual power series (LRPS) method. The LRPS method is a powerful technique for solving FSPDEs, as it allows for the efficient computation of the solution in the form of a power series expansion. The method is based on the Laplace transform and the residual power series, and is applied to a system of coupled FSPDEs. The method is validated using several test problems, and the results show that the LRPS method is a reliable and efficient method for solving FSPDEs.

On the Analytical Solution of Fractional SIR Epidemic Model

This article presents the solution of the fractional SIR epidemic model using the Laplace residual power series method. We introduce the fractional SIR model in the sense of Caputo’s derivative; it is presented by three fractional differential equations, in which the third one depends on the first coupled equations. The Laplace residual power series method (LRPSM) is implemented in this research to solve the proposed model, in which we present the solution in a form of convergent series expansion that converges rapidly to the exact one. We analyze the results and compare the obtained approximate solutions to those obtained from other methods. Figures and tables are illustrated to show the efficiency of the LRPSM in handling the proposed SIR model.

A semi-analytical framework for comprehensive vibration analysis of segment-coupled plates with embedded acoustic black holes

This study introduces a semi-analytical framework for modeling and analyzing the vibration behavior of the acoustic black hole (ABH) structures, which have been drawing an increasing attention because of their outstanding vibration control and energy harvesting abilities. The proposed model combining with the segment-coupling strategy is developed to address the main issue in modeling ABH structures: the non-uniform wavelength distribution on the whole structure domain. The flexural displacement field, on basis of a set of modified Fourier series, is respectively assumed on uniform and ABH segments to avoid the non-uniform wavelength distribution. Such modified form of Fourier series expansion makes the solution applicable to any boundary conditions and available to the segment-coupling strategy. This framework allows one to study ABH vibration-concentrating capability with varying system parameters in a fast and efficient way. By means of the proposed segment-coupling strategy, the modeling of the periodic ABH structure can be easily achieved. Under the general Rayleigh–Ritz framework, the extremalization of the Hamiltonian over Lagrange’s equation yields the system governing equation, which can be solved for structural responses. The consistent with the FEM results and diverse numerical examples demonstrate that the proposed model is capable of providing an accurate prediction and flexible analysis of the plates with embedded acoustic black holes. Time–space transient response in forms of 2D visualization images is presented to show the vibration-concentrating effect. Free, forced, and transient vibration analysis are conducted to illustrate the typical ABH phenomena with varying ABH parameters, which may provide theoretical support for the acoustic design of damping structures based on acoustic black holes.

Generalized 2D Extension of the q-Bessel polynomials

The article aims at introducing a generalized 2D extension of the q-Bessel polynomials. The generating equation, series expansion and determinant form for the generalized 2D q-Bessel polynomials are established. An example providing the continuous 2D q-Hermite--Bessel polynomials is considered. The recurrence relation and other properties for the continuous 2D q-Hermite--Bessel polynomials are derived. The graphs and plots depicting the behaviour of the continuous 2D Hermite polynomials, continuous 2D Hermite--Bessel polynomials and continuous 2D q-Hermite--Bessel polynomials are drawn. The distribution of zeros for these polynomials is also presented.

Sunset similarity solution for a receding hydraulic fracture

This paper derives approximate ‘sunset’ similarity solutions for receding plane strain and radially symmetric hydraulic fractures in permeable elastic media close to the point of closure. Local analysis is used to show that a receding hydraulic fracture has a linear aperture asymptote$\hat {w}\sim \hat {s}$in the fracture tip, where$\hat {s}$is the distance from the fracture front. Due to the regularity of the linear asymptote, it is possible to determine similarity solutions in the form of power series expansions, which, for integers$N\ge 2$and values of the radius decay exponent$\gamma =1/N$, can be shown to terminate to yield polynomial solutions for the fracture aperture of degree$N$. Of this countable infinity of polynomial solutions, the final aperture profile as the fracture approaches closure is associated with the second-degree polynomial with$\gamma =1/2$called the sunset solution. For the reverse time$t^{\prime }$measured from closure, the sunset solution is characterized by$w\sim t^{\prime }$and$R\sim t^{\prime 1/2}$. Of all the admissible polynomial similarity solutions, the sunset solution is shown to form an attractor, as$t^{\prime }\rightarrow 0$, for receding hydraulic fractures associated with a wide variety of points in parametric space. Using the sunset solution, it is possible to estimate the duration of recession, assuming that the fracture aperture and radius at the start of recession are given, and determine how it scales with a dimensionless shut-in parameter. As the fracture approaches closure, the term responsible for coupling the elastic force balance and fluid conservation becomes subdominant to the other terms in the lubrication equation, which reduces to a local kinematic relation between the decaying fracture aperture and the leak-off velocity. This fundamental decoupling of dynamics from kinematics results in the sunset solution being dependent on only a single material parameter – namely the leak-off coefficient. This isolation of the leak-off coefficient by the sunset solution opens the possibility to determine this parameter from laboratory or field measurements.

Static Thin Disks with Power-law Density Profiles * * Dedicated to our teacher, Professor Jiří Bičák, on the occasion of his 80th birthday. For O.S., Jiří's interest in thin disks was, 30 yr ago (Bičák et al. ), the first motivation to study the topic.

The task of finding the potential of a thin circular disk with power-law radial density profile is revisited. The result, given in terms of infinite Legendre-type series in the above reference, has now been obtained in closed form thanks to the method of Conway employing Bessel functions. Starting from a closed-form expression for the potential generated by the elementary density term ρ 2l , we cover more generic—finite solid or infinite annular—thin disks using superposition and/or inversion with respect to the rim. We check several specific cases against the series-expansion form by numerical evaluation at particular locations. Finally, we add a method to obtain a closed-form solution for finite annular disks whose density is of “bump” radial shape, as modeled by a suitable combination of several powers of radius. Density and azimuthal pressure of the disks are illustrated on several plots, together with radial profiles of free circular velocity.

Analytical representation and efficient computation of the effective conductivity of two‐phase composite materials

AbstractMany engineered materials display ordered or disordered microstructures. Such materials exhibit transport properties which are unmatched by their single‐phase homogeneous counterparts. These properties are obtained by the mixture of two or more phases typically characterized by a large contrast in their properties. For the development of these materials, it is critical to develop a robust computational framework in order to provide a fundamental understanding of how microstructure affects performance. This hinges on predicting their macroscopic properties, given the constitutive laws and spatial distribution of their constituents. To this end, this work presents a computational framework based on formulating periodic conduction transport problems in terms of boundary integral equations whose kernel is expressed in terms of Weierstrass zeta‐function. The components of the effective conductivity tensor are then sought in the form of power series expansions of a conductivity contrast parameter. To accelerate their convergence, these expansions are transformed into Padé approximants. Presently restricted to the case of two‐dimensional, two‐phase microstructures, this framework is shown to yield accurate results over the entire range of the contrast parameter. Representation of the kernel as a lattice sum allows the use the fast multipole method, thereby making computations significantly more efficient.

Solution for optimal control of general nonlinear systems using generating functions

AbstractIn this paper, we present an analytical local solution for optimal control problems applicable to a wide class of general nonlinear systems. We first introduce the optimal control problem for a general nonlinear system and formulate the associated Hamilton–Jacobi–Bellman equation. Starting from the necessary conditions for optimality represented by the Hamiltonian system of the optimal control problem, we solve the Hamilton–Jacobi equation for the associated generating functions. The generating functions are obtained in the form of series expansions by using canonical transformation theory and the semi‐tensor product of matrices. The coefficients of the generating functions satisfy simply the algebraic Riccati equation and linear algebraic equations for the second‐degree homogeneous component and higher degree components, respectively, from which an algorithm and data structure can be derived. This enables us to obtain analytical optimal feedback controllers and the corresponding Lyapunov function. Through theoretical analysis, it is established that the proposed methodology guarantees stability of the closed‐loop system. This approach is not affected by the complexity of the performance index and state equations, and most importantly, it does not require any initial admissible control law. A simulation example is presented to illustrate the effectiveness and applicability of the proposed approach.

APPLICATION OF THE THEORY OF THE MULTICOMPONENT DRY FRICTION IN SOME OF CONTROL ROBOT SYSTEMS

The implementations of the theory of multicomponent dry friction [1-19] for analyze the dynamics of some robotic systems, such as a butterfly robot [16-18, 20] or a humanoid robot is proposed. Since the main controlled element of these systems is a spherical, elastic composite shell, it is required to calculate the distribution of normal contact stresses inside the contact spot. The contact pressure distribution for such elements is constructed using the S. A. Ambartsumyan’s equation for a transversally isotropic spherical shell. This equation is modified by introducing the averaged contact pressure and normal displacements for the shell. The construction of the resolving integral equation for the contact pressure is based on the principle of superposition and the method of Green's functions. For this, the corresponding Green's function is constructed, which is the normal displacement of the shell as a solution to the problem of the effect of concentrated pressure. Green's function as well as the contact pressure, it is sought in the form of series expansions in Legendre polynomials, taking into account additional relations for the reduced contact pressure and normal displacements. Using the Green's function, an integral equation solving the problem is constructed. As a result, the problem is reduced to determining the expansion coefficients in a series of the reduced contact pressure. Restricting ourselves to a finite number of terms in the series of expansions, using the discretization of the contact area and the properties of Legendre polynomials, the problem is reduced to solving a system of algebraic equations for the expansion coefficients for the reduced pressure. After that, from the additional relation, the coefficients of the required expansion of the contact pressure in a series in Legendre polynomials are determined. To describe the conditions of shell contact with the surface, the theory of multicomponent anisotropic dry friction is used, taking into account the combined kinematics of shell motion (simultaneous sliding, rotation and rolling). The coefficients of the dry friction model can be calculated using simple explicit formulas [1-19] based on numerical experiments.

Order theory for discrete gradient methods

The discrete gradient methods are integrators designed to preserve invariants of ordinary differential equations. From a formal series expansion of a subclass of these methods, we derive conditions for arbitrarily high order. We derive specific results for the average vector field discrete gradient, from which we get P-series methods in the general case, and B-series methods for canonical Hamiltonian systems. Higher order schemes are presented, and their applications are demonstrated on the Hénon–Heiles system and a Lotka–Volterra system, and on both the training and integration of a pendulum system learned from data by a neural network.

Inverse source problem for an equation of mixed parabolic-hyperbolic type with the time fractional derivative in a cylindrical domain

This article is devoted to the study of an inverse source problem for a mixed type equation with a fractional diffusion equation in the parabolic part and a wave equation in the hyperbolic part of a cylindrical domain. The solution is obtained in the form of FourierBessel series expansion using an orthogonal set of Bessel functions. The theorems of uniqueness and existence of a solution are proved.