General Analytical Solution Research Articles

Kepler’s problem of a two-body system perturbed by a third body

One of the most important problems in basic physics and astronomy is studying the motion of planets, satellites, and other celestial bodies. The solution to the two-body problem enables astronomers to predict the orbits of the Moon, satellites, and spaceships around the Earth. The general analytic solution for the three-body problem stands unsolved except in some special cases. This reduces the problem to a two-body problem. In this work, the authors present a closed-form approach to the three-body problem theoretically and numerically based on particle–particle vector analysis. The theoretical approach, which is based on the real Moon–Sun–Earth problem information, illustrates the perturbation of the Moon in the Sun–Earth problem and shows an expected orbital motion with a perturbation in the Sun–Earth orbit due to the revolution of the Moon. The numerical investigation uses the same information to study the same problem and calculate the angular momentums of each pair of objects. The two solutions show good agreement with the well-known Earth-Moon and Sun–Earth momentums. The Moon–Sun orbit is close to an elliptic shape with angular momentum of about 3.27 × 1038 J.s. This approach is the key to future studies for n-body problem solutions.

Impact of induced electric field on the motion of a charged particle in a uniform time-dependent magnetic field that is linearly increasing with time

The classical and quantum motion of a charged particle in a uniform constant magnetic field is widely studied in the scientific literature since it is a problem that allows an exact mathematical analytical solution. On the other hand the cases of a non-uniform, time-dependent or both non-uniform and time-dependent magnetic field lead to mathematical problems that do not have general exact analytical solutions except for a few very specific cases. We present in this work, the results for the two-dimensional motion of a charged particle in a uniform time-dependent magnetic field that is linearly increasing with time for a certain finite interval and then stabilizes to a constant value afterwards. The work draws attention to certain interesting features observed for the two-dimensional motion of the charged particle as a result of the interplay between the effects of an induced electric field and the time varying magnetic field.

Analytical Solution for the Problem of Point Location in Arbitrary Planar Domains

This paper presents a general analytical solution for the problem of locating points in planar regions with an arbitrary geometry at the boundary. The proposed methodology overcomes the traditional solutions used for polygonal regions. The method originated from the explicit evaluation of the contour integral using the Residue and Cauchy theorems, which then evolved toward a technique very similar to the winding number and, finally, simplified into a variant of ray-crossing approach slightly more informed and more universal than the classic approach, which had been used for decades. The very close relation of both techniques also emerges during the derivation of the solution. The resulting algorithm becomes simpler and potentially faster than the current state of the art for point locations in arbitrary polygons because it uses fewer operations. For polygonal regions, it is also applicable without further processing for special cases of degeneracy, and it is possible to use in fully integer arithmetic; it can also be vectorized for parallel computation. The major novelty, however, is the extension of the technique to virtually any shape or segment delimiting a planar domain, be it linear, a circular arc, or a higher order curve.

Accretion tori around rotating neutron stars I. Structure, shape, and size

We present a full general relativistic analytic solution for a radiation-pressure-supported equilibrium fluid torus orbiting a rotating neutron star (NS). We applied previously developed analytical methods that include the effects of both the NS's angular momentum and quadrupole moment in the Hartle-Thorne geometry. The structure, size, and shape of the torus are explored, with a particular focus on the critically thick solution -- the cusp tori. For the astrophysically relevant range of NS parameters, we examined how our findings differ from those obtained for the Schwarzschild space-time. The solutions for rotating stars display signatures of an interplay between relativistic and Newtonian effects where the impact of the NS angular momentum and quadrupole moment are almost counterbalanced at a given radius. Nevertheless, the space-time parameters still strongly influence the size of tori, which can be shown in a coordinate-independent way. Finally, we discuss the importance of the size of the central NS which determines whether or not a surrounding torus exists. We provide a set of tools in a Wolfram Mathematica code, which establishes a basis for further investigation of the impact of the NSs’ super-dense matter equation of state on the spectral and temporal behaviour of accretion tori.

Ferroelectric frontiers: Navigating phase portraits, chaos, multistability and sensitivity in thin-film dynamics

This research includes the study of the non-linear dynamics of thin-film ferroelectric materials governed by an equation of wave dynamics within the material. This equation plays a key role in both physics and the study of aqueous flow. The work is planned as examining the symmetries group analysis drops, studying the dynamical system features through bifurcation phase portraits, and carrying dynamic phenomena in chaos theory. Diverse techniques are taken, such as Lyapunov exponent, 2D, and 3D phase portraits, Poincaré maps, time series analysis, and sensitivity to multistability under the different conditions of the initial state. In addition, the study involves using the extended hyperbolic function method to obtain the general analytical solutions via which various kinds of solitary wave solutions are produced including trigonometric and hyperbolic functions and periodic, bright, and singular soliton solutions. These solutions are followed by a list of constraint conditions in the form of equations. Visual data of 2D, 3D, and contour plots are presented, with parameters carefully set to reflect various scenarios. Sensitivity analysis is performed using alternative initial conditions, and stability analysis is demonstrated graphically. To fully grasp the dynamic features of these systems and accurately predict outcomes, it is essential to advance new technologies and methodologies that can further enhance our understanding and predictive capabilities in complex systems.

Isles of regularity in a sea of chaos amid the gravitational three-body problem

Context. The three-body problem (3BP) poses a longstanding challenge in physics and celestial mechanics. Despite the impossibility of obtaining general analytical solutions, statistical theories have been developed based on the ergodic principle. This assumption is justified by chaos, which is expected to fully mix the accessible phase space of the 3BP. Aims. This study probes the presence of regular (i.e. non-chaotic) trajectories within the 3BP and assesses their impact on statistical escape theories. Methods. Using three-body simulations performed with the accurate, regularized code TSUNAMI, we established criteria for identifying regular trajectories and analysed their impact on statistical outcomes. Results. Our analysis reveals that regular trajectories occupy a significant fraction of the phase space, ranging from 28% to 84% depending on the initial setup, and their outcomes defy the predictions of statistical escape theories. The coexistence of regular and chaotic regions at all scales is characterized by a multi-fractal behaviour. Integration errors manifest as numerical chaos, artificially enhancing the mixing of the phase space and affecting the reliability of individual simulations, yet preserving the statistical correctness of an ensemble of realizations. Conclusions. Our findings underscore the challenges in applying statistical escape theories to astrophysical problems, as they may bias results by excluding the outcome of regular trajectories. This is particularly important in the context of formation scenarios of gravitational wave mergers, where biased estimates of binary eccentricity can significantly impact estimates of coalescence efficiency and detectable eccentricity.

Deep learning for derivatives pricing: a comparative study of asymptotic and quasi-process corrections

AbstractIn this study, we compare two methods for using neural networks to efficiently learn the price of derivatives. The first method, proposed by Funahashi (Quant Financ 21(4):575–592, 2021a), involves training the neural networks to learn the difference between the derivative price and its asymptotic expansion, rather than learning the derivative price directly. The target derivative price is then obtained by adding the approximate solution with the predicted value of neural networks. This method reduces the required amount of learning data, often by a factor of one hundred to one thousand, compared to the case where the derivative price is directly learned through neural networks. In the second method, established in this paper, the neural networks learn the difference between the derivative price written on the underlying asset price that follows the target complex stochastic process and the derivative price written on the underlying asset price that has a relatively simple stochastic process that has a closed-form solution for the target derivative prices. This method provides an alternative valuation method when no efficient approximate solution for the derivative value is observed and if one can arbitrarily determine the model parameters of the quasi-process that approximates the original process. We also propose a unified method to determine the model parameters of quasi-processes from underlying asset processes. These methods prove valuable in cases where general analytic solutions are absent, as seen in widely used financial models such as the stochastic volatility models. These cases involve time-consuming numerical calculations to generate learning data, highlighting the value of the two methods, which significantly compress calculation times.

Anisotropic functionally graded nano-beam models and closed-form solutions in plane gradient elasticity

This study delves into the investigation of exact analytical solutions for the plane stress and displacement fields within linear homogeneous anisotropic nano-beam models of gradient elasticity. It focuses on solving the Helmholtz equation, which encompasses a second-order non-homogeneous linear partial differential equation in plane gradient elasticity theory, utilizing polynomial series-type solutions. The analysis centers on the utilization of gradient Airy stress functions to derive stress fields in the gradient theory. The research yields closed-form analytical solutions for Airy stress functions, stress, and strain fields in both classical and gradient theories. The study considers five distinct types of two-dimensional functionally graded cantilever beams with various boundary conditions: a cantilever anisotropic nano-beam subjected to a concentrated force at the free end, a cantilever anisotropic nano-beam under a uniform load, a simple anisotropic nano-beam under a uniform load, a propped cantilever anisotropic nano-beam under a uniform load, and a fixed end cantilever anisotropic nano-beam under a uniform load. General analytical solutions for the gradient stress and displacement fields of two-dimensional and one-dimensional anisotropic nano-beams under different boundary conditions are provided. The study showcases significant strain gradient size effects at the nano-scale through the derived analytical solutions for anisotropic beams. Additionally, it demonstrates that the strain gradient theory results for the limit case of the gradient coefficient c precisely align with results for isotropic and anisotropic materials in elasticity theory and classical theory. Furthermore, real-world applications are discussed, considering stress and displacement fields in real anisotropic materials such as TiSi2 single crystals and orthotropic materials, which are special cases of anisotropic materials like wood and epoxy as documented in the literature.

An investigation on the torsional vibration of a FG strain gradient nanotube

AbstractIn this work, an attention is paid to the prediction of torsional vibration frequencies of functionally graded porous nanotubes based on the Lam strain gradient elasticity theory. The nanotubes are formed of functionally graded porous nanomaterials that vary in the radial direction. This study also aims to obtain the analytical solution of the strain gradient model presented by Lam for torsional vibration response, in a simple manner, for different rigid or restrained boundary conditions. The torsion angle of a functionally graded nanotube is defined by an infinite Fourier series. Then, the Stokes’ transformation is applied to force the boundary conditions to the desired state. An eigenvalue problem is established with the help of the two systems of equations obtained. This eigenvalue problem, which includes deformable springs at both ends of the nanotube, appears as a general analytical solution that can find torsional vibration frequencies. It is shown that the vibrational responses can be significantly influenced by the through‐radius gradings of material, material length scale parameters and deformable springs of the functionally graded nanotubes and consequently can be predicted by giving proper values to torsional spring parameters.

Numerical modeling of underground flow through porous media using volumes scheme

The Richards equation attracts the attention of several researchers due to its importance in the hydrogeology field especially the flow through porous soil. It is a non linear partial differential equation that has no general analytic solution. Thus, the need to use numerical methods to solve it. In this work, a finite volumes scheme is used to simulate the pressure form and the mixed form of Richards equation in one dimension. Euler explicit and implicit schemes are used for the time discretization and a condition of scheme’s stability is given. Different test cases are done to validate the accuracy and the efficiency of the proposed numerical model

A general analytical solution for axisymmetric electro‐osmotic consolidation of unsaturated soil with semi‐permeable boundary

AbstractThis study proposes a closed‐form solution for axisymmetric electro‐osmotic consolidation of unsaturated soil under semi‐permeable boundary conditions. The governing equations are formulated to allow for vertical and radial flows of liquid and air phases. The techniques of eigenfunction expansion and Laplace transformation are employed to develop the exact solution for excess pore‐air (EPAP) and pore‐water pressures (EPWP). The proposed solution is first validated by comparing it to an existing solution, followed by verification through finite element simulations. Both methods of validation confirm the accuracy of the analytical solution. Then, based on the obtained solution, the effects of vertical flow, semi‐permeable boundary conditions, electrical voltage, electro‐osmosis conductivity and spacing ratio re/rw on the consolidation profile have been further investigated. Parametric studies show that the EPWP at the steady state depends on the electro‐osmosis conductivity and applied electricity gradient. In addition, the dissipation rates of EPWP and EPAP in the axisymmetric electro‐osmotic consolidation would be underestimated if the vertical flows are neglected. The semi‐permeable boundary conditions have great influences on the dissipation rate and the steady‐state solution. The proposed solution could serve as a theoretical basis for axisymmetric electro‐osmotic consolidation of unsaturated soil.

Computational approach in obtaining analytic solutions of a generalized nonlinear breaking soliton equation with applications in engineering and physics

Higher-order nonlinear wave models have been a source of attraction to a huge number of researchers in recent times as a result of their significance in mathematical physics, other nonlinear sciences as well as engineering. In consequence, we outline in this paper the analytical studies entrenched on a generalized structure of a nonlinear breaking soliton equation with higher-order nonlinearity in four variables which have applications in science as well as engineering. Lie group theory is utilized to generate an 11-dimensional Lie algebra associated with the equation under consideration and in addition one parametric group of transformations related to the algebra is calculated. Besides, the technique is further invoked in performing reductions of the various subalgebras of the understudy model. Moreover, in conjunction with the theory, the direct integration technique is engaged to secure an analytic solution of the equation and as a result, a general analytic solution with regard to the second-kind elliptic-integral function is furnished. Moreover, we engage the novel simplest equation technique to gain more general solutions to the equation. In consequence, solitonic solutions comprising periodic, dark-bright, topological kink as well as singular solutions are achieved. We complemented that by exhibiting the dynamics of the secured solutions with the aid of graphical representations. In conclusion, we calculate conserved quantities associated with the aforementioned equation by invoking the well-celebrated classical Noether theorem technique.

Revisiting the observability and identifiability properties of a popular HIV model

This paper revisits the observability and identifiability properties of a popular ODE model commonly adopted to characterize the HIV dynamics in HIV-infected patients with antiretroviral treatment. These properties are determined by using the general analytical solution of the unknown input observability problem, introduced very recently in Martinelli (2022). This solution provides the systematic procedures able to determine the state observability and the parameter identifiability of any ODE model, in particular, even in the presence of time varying parameters. Four variants of the HIV model are investigated. They differ because some of their parameters are considered constant or time varying. Fundamental new properties, which also highlight an error in the scientific literature, are automatically determined and discussed. Additionally, for each variant, the paper provides a quantitative answer to the following practical question: What is the minimal external information (external to the available measurements of the system outputs) required to make observable the state and identifiable all the model parameters? The answer to this fundamental question is obtained by exploiting the concept of continuous symmetry, recently introduced in Martinelli (2019). This concept allows us to determine a first preliminary general result which is then applied to the HIV model. Finally, for each variant, the paper concludes by providing a redefinition of the state and of the parameters in order to obtain a full description of the system only in terms of a state which is observable and a set of parameters which are identifiable (both constant and time varying).

UTILIZATION OF THOMAS MODEL TO PREDICT THEBREAKTHROUGH CURVES FOR ADSORPTION AND IONEXCHANGE

Fixed bed sorption processes such as adsorption and ion exchange do not operate at steady state. The design equations are differential that usually require numerical methods to solve. Thomas gives a general analytical solution for these equations. It is based on second-order reaction kinetics and the assumption of Langmuir isotherm. Computer program for the solution of Thomas model was designed using MATLAB 7.0. Four sets of experimental data are tested to show the capability of Thomas model to predict the breakthrough curves for adsorption and ion exchange processes. These data represent the sorption of o-cresol from water by activated carbon, acetic acid from air by activated carbon, phenol from water by natural zeolite, and trichloroethylene from air by zeolite- 5A.The results show that there are a good agreement between the experimental data and the model.

Allocating Cost of Uncertainties From Renewable Generation in Stochastic Electricity Market: General Mechanism and Analytical Solution

Allocating Cost of Uncertainties From Renewable Generation in Stochastic Electricity Market: General Mechanism and Analytical Solution

Non-linear dynamics of a test particle near the Lagrange points L4 and L5 (Earth-Moon and Sun-Earth case)

The two-bodies problem can be fully solved, and was solved by Kepler (1609) and Newton (1687). The general three-body problem is often given as an example of a mathematical problem that ‘can’t be solved’. So, there is no general analytical solution. This problem can be significant and a special case of this problem is the Circular Restricted Three-Body Problem (CRTBP), which can be applied to the Earth-Moon system with a spacecraft, the Sun-Earth system with an asteroid, etc. In this paper, let’s focus on the motion of a test particle near the triangular Lagrange points L4 and L5 in the Earth-Moon and the Sun-Earth systems. Studying the movement of an object around these points is especially important for space mission design. To generate a trajectory around these points, the non-linear equations of motion for the circular restricted three-body problem were numerically integrated into MATLAB® 2023 software and the results are presented in the plane (x, y) and the phase plane (x, vx) and (y, vy). By numerical orbit integration, it is possible to investigate what happens when the displacement is relatively large or short from the Lagrange points. Then the small astronomical body may vibrate around these points. The results in this paper are shown in the rotating and inertia axes. Various initial positions near the Lagrange points and velocities are used to produce various paths the test particle can take. The same examples of numerical studies of trajectories associated with Lagrange points are shown in the inertial and the rotating coordinates system and are discussed. From the results of the numerical tests performed in MATLAB® 2023, it is possible to saw that there are different types of periodic, quasi-periodic, and chaotic orbits

A general analytical solution for fluid flow and heat convection through arbitrary-shaped triangular ducts: A variational analysis

This paper presents an analytical solution for fluid flow and heat transfer inside arbitrarily-shaped triangular ducts for the first time. The former analytical solutions are limited to the special case of isosceles triangular ducts. The literature has no report about the analytical solution for the general case of arbitrarily-shaped triangular ducts. Due to the significant role of fluid flow through non-circular channels in industry and the large number of triangular shapes, a method for solving the heat transfer problem for all triangular shapes is needed. The heat transfer of a fluid flow through a channel with an arbitrary triangular cross-section for the case of constant heat flux at the walls is solved in this work for the first time, considering viscous dissipation. Here, the functionals of flow and heat transfer equations are derived, and the resulting Euler–Lagrange equations are solved using the Ritz method. The effect of the duct geometry on the velocity profile and friction coefficient is studied in detail. The effect of the Brinkman number on the temperature distribution and Nusselt number is investigated for both cooling and heating cases. The results reveal that the critical Brinkman Number distinguishes between the cooling and heating cases and represents the critical point at which the Nusselt number approaches infinity. The value of the Nusselt number decreases with the increase of the Brinkman number in both the wall cooling and heating modes. It is also found that the equilateral triangle exhibits the minimum friction coefficient and the maximum value of the Poiseuille number.

Conformable Triple Sumudu Transform with Applications

One of the important topics in applied mathematics is the topic of integral transformations, due to their importance in electrical engineering applications, including communications in particular, and other sciences. In this work, one of the most important transformations in its three dimensions was presented, which is the triple Sumudu transform, including solving some real-life applications of physics, some of which have not been solved using such an integral transform before. In this work, we extend the Sumudu transform formula to the conformable fractional order, as well as other interesting and significant rules. The general analytical solution of a singular and nonlinear conformable fractional differential equation based on the conformable fractional Sumudu transform is also presented in this paper. The general solutions of several linear and nonhomogeneous conformable fractional differential equations can be obtained using the method we’ve proposed. As a result, our results reveal that our proposed method is an efficient one that can be used for solving all conformable fractional differential equations. The relationship between the Sumudu integral transform and other important and recently proposed integral transforms are also discussed. Finally, the triple Sumudu transform is used to solve boundary value problems, such as the heat equation with boundary values. The triple Sumudu integral transform is also used to solve linear partial integro-differential equations. The transform capability to handle such equations has been proven via its utilization in three applications.

Non-slipping adhesive contact between dissimilar elastic solids under normal and tangential loads

In this paper, a model of two-dimensional non-slipping adhesive contact between dissimilar elastic solids under normal and tangential loads is proposed, and the general analytical solution is derived. The interaction of adhesion and friction along the contacting interface is studied. Specifically, (i) the shear-off force of adhesive contact is proposed; (ii) the effect of tangential load on the puff-off force is presented; (iii) the influence of the normal load on the shear-off force is obtained; (iv) in the context of these results, the nominal static friction coefficient is defined and analyzed, and it is shown that the static friction coefficient is a function of external normal load, which can well explain some experimental results in the literature. The proposed model appears to be consistent with the available experimental data, which offers new insight into the interplay mechanism between friction and adhesion. These results provide the foundation for understanding and modelling some tribological phenomena at the macro-, micro- and nano- scales.

Bridging Scales in Black Hole Accretion and Feedback: Magnetized Bondi Accretion in 3D GRMHD

Fueling and feedback couple supermassive black holes (SMBHs) to their host galaxies across many orders of magnitude in spatial and temporal scales, making this problem notoriously challenging to simulate. We use a multi-zone computational method based on the general relativistic magnetohydrodynamic (GRMHD) code KHARMA that allows us to span 7 orders of magnitude in spatial scale, to simulate accretion onto a non-spinning SMBH from an external medium with a Bondi radius of R B ≈ 2 × 105 GM •/c 2, where M • is the SMBH mass. For the classic idealized Bondi problem, spherical gas accretion without magnetic fields, our simulation results agree very well with the general relativistic analytic solution. Meanwhile, when the accreting gas is magnetized, the SMBH magnetosphere becomes saturated with a strong magnetic field. The density profile varies as ∼r −1 rather than r −3/2 and the accretion rate is consequently suppressed by over 2 orders of magnitude below the Bondi rate . We find continuous energy feedback from the accretion flow to the external medium at a level of . Energy transport across these widely disparate scales occurs via turbulent convection triggered by magnetic field reconnection near the SMBH. Thus, strong magnetic fields that accumulate on horizon scales transform the flow dynamics far from the SMBH and naturally explain observed extremely low accretion rates compared to the Bondi rate, as well as at least part of the energy feedback.