Exact Numerical Solutions Research Articles

Two particles in one-dimensional confinement: the role of mass imbalance and Pauli's exclusion principle within short and long range interactions

Abstract We consider the problem of two interacting particles in one-dimensional harmonic confinement. By considering both particles are subject to the same harmonic curvature, we obtain exact numerical solutions in a straightforward way for any choice of interaction potential. The formulation is then applied in the situations of ({\it i}) non-identical particles with different masses $m_1$ and $m_2$; ({\it ii}) bosons and fermions; ({\it iii}) short and long range interactions; and ({\it iv}) combining all these ingredients within a time-dependent applied electric field. We analyze the role of the mass imbalance and Pauli's exclusion principle by investigating the formation of bound pairs, the expected value of the separation between the particles and the effects of interaction and exclusion principle on the density distributions.

Role of nonthermal solar wind protons in the excitation of electromagnetic proton-cyclotron instability: a kinetic theory based exact numerical investigation

Free transverse kinetic energy i.e. perpendicular temperature anisotropy of protons excite the electromagnetic ion/proton cyclotron instability which is pertained to waves associated with prevalent electromagnetic ion/proton cyclotron emissions in various natural regions of plasmas. The transverse dielectric response function of left hand circularly polarized electromagnetic proton cyclotron (EPC) instability is calculated for two models of nonthermal Cairns distributed plasmas. These models are distinguished according to the effective thermal velocities of protons. For the energetic nonthermal protons populations, nonthermality dependent effective temperature model is proposed which significantly contributes in the excitation of aforementioned plasma mode and cause an appreciable enhancement in the instability growth rate. Exact numerical solution of dispersion relation yields oscillatory real frequency and growth rate of instability. A comparative analysis is also carried out to examine the instability behavior in distinct nonthermal and thermal plasma models. Contemporary numerical investigations are highly beneficial to understand the intricate dynamics of space plasmas.

Impact of laterally mobile surface charge on diffusiophoresis of hydrophobic rigid colloids

The diffusiophoresis of charged hydrophobic nanoparticles (NPs) governed by an imposed ionic concentration gradient is analysed. The main objective is to elucidate the impact of the laterally mobile adsorbed surface ions at the interface on the propulsion of the hydrophobic NPs in diffusiophoresis. In addition, the dielectric polarization due to the difference in dielectric constant between the NPs and the suspension medium is also considered. The mobile surface ions create a friction as well as an electric force at the hydrophobic surface, which leads to a modification of the slip velocity condition and the slip length. We obtain an exact numerical solution of the governing electrokinetic equations in their full form by adopting a control volume formulation. The numerical model is supplemented by analytical solutions derived based on the Debye–Hückel linearization. We find that the lateral mobility of the surface ions obstruct the coions to diffuse from the higher concentration side to the lower concentration side, which results in a repulsive force to the particle leading to the occurrence of a negative mobility. Based on the numerical results and analytical solutions, we have shown that for a fully mobile surface charge, the diffusiophoresis of a hydrophobic NP is identical to the diffusiophoresis of a liquid droplet whose viscosity is related to the slip length of the hydrophobic particle. We establish that the dielectric polarization enhances the velocity of a hydrophobic particle, which has potential applications in the practical context.

Application of the frozen-modes approximation to classical harmonic oscillator systems

The problems of open classical systems usually correspond to a motion of a test particle that interacts with a large number of bath oscillators. Often, the test particle itself can be considered a harmonic oscillator. For such composite systems, exact numerical solutions are available, but they can become increasingly costly for a large number of bath oscillators. Here we take inspiration from the recent work on open quantum systems and investigate the applicability of the frozen-modes approximation to such classical systems. This approach assumes that some part of the low-frequency bath modes are frozen, thus only their initial values need to be considered. We show that by applying the frozen-modes approximation one can significantly increase the accuracy of the perturbative multiple-scales solution, especially for slow baths. This approach provides a good accuracy even for strong system–bath couplings, a regime that is not accessible to straightforward applications of the perturbation theory. We also suggest a rule for the splitting of spectral density to the fast and slow bath modes. We find that our approach gives excellent results for the ohmic spectral density, but it could be applied for other similar spectral densities as well.

The universal multiplicity function: counting haloes and voids

We present a novel combination of the excursion-set approach with the peak theory formalism in Lagrangian space and provide accurate predictions for halo and void statistics over a wide range of scales. The set-up is based on an effective moving barrier. Besides deriving the corresponding numerical multiplicity function, we introduce a new analytical formula reaching the percent level agreement with the exact numerical solution obtained via Monte Carlo realisations down to small scales, ∼ 1012 h -1M⊙. In the void case, we derive the dependence of the effective moving barrier on the void formation threshold, δ v, by comparison against the Lagrangian void size function measured in the DEMNUni simulations. We discuss the mapping from Lagrangian to Eulerian space for both haloes and voids; adopting the spherical symmetry approximation, we obtain a strong agreement at intermediate and large scales. Finally, using the effective moving barrier, we derive Lagrangian void density profiles accurately matching measurements from cosmological simulations, a major achievement towards using void profiles for precision cosmology with the next generation of galaxy surveys.

Euler Method for a Class of Linear Impulsive Neutral Differential Equations

This paper presents a new numerical scheme for a class of linear impulsive neutral differential equations with constant coefficients based on the Euler method. We rigorously establish the first-order convergence of the proposed numerical approach. Additionally, the asymptotical stability of the exact solutions and numerical solutions of impulsive neutral differential equations are studied. To substantiate our findings, two illustrative examples are provided, demonstrating the theoretical conclusions of this paper.

Two Schemes of Impulsive Runge–Kutta Methods for Linear Differential Equations with Delayed Impulses

In this paper, two different schemes of impulsive Runge–Kutta methods are constructed for a class of linear differential equations with delayed impulses. One scheme is convergent of order p if the corresponding Runge–Kutta method is p order. Another one in the general case is only convergent of order 1, but it is more concise and may suit for more complex differential equations with delayed impulses. Moreover, asymptotical stability conditions for the exact solution and numerical solutions are obtained, respectively. Finally, some numerical examples are provided to confirm the theoretical results.

Exact Numerical Solution of the Fully Connected Classical and Quantum Heisenberg Spin Glass.

We present the mean field solution of the quantum and classical Heisenberg spin glasses, using the combination of a high precision numerical solution of the Parisi full replica symmetry breaking equations and a continuous time quantum MonteCarlo algorithm. We characterize the spin glass order and its low-energy excitations down to zero temperature. The Heisenberg spin glass has a rougher energy landscape than its Ising analog, and exhibits a very slow temperature evolution of its dynamical properties. We extend our analysis to the doped, metallic Heisenberg spin glass, which displays unexpectedly slow spin dynamics, reflecting the proximity to the melting quantum critical point and its associated Sachdev-Ye-Kitaev Planckian dynamics.

Dynamic analysis of the three-phase magneto-electro-elastic (MEE) structures with the finite element method enriched by proper enrichment functions

In this study, a carefully designed enriched finite element method (EFEM) is presented to improve the solution accuracy of the conventional FEM by analyzing the dynamic behavior of the magnetic-electric-elastic (MEE) composite structures, which are frequently used in designing various smart and intelligent devices. By formulating the proper EFEM with ideal numerical performance, different enrichment functions are considered and the corresponding solution quality of different versions of the EFEM is compared and examined in great detail. When the Lagrange polynomial basis functions together with the harmonic trigonometric functions are used as enrichment functions, the obtained EFEM shows extremely powerful and ideal numerical performance, which is obviously better than the other versions of EFEM and the conventional FEM, in studying the free vibration and harmonic frequency responses of the MEE structures. Nearly exact numerical solutions for three-phase physical fields of MEE structures can be generated by the proposed EFEM even if very coarse mesh patterns are used. Intensive numerical studies are conducted to confirm and verify the excellent properties of the proposed EFEM in performing dynamic analysis of the MEE structures.

Propagation of dark solitons of DNLS equations along a large-scale background

We study dynamics of dark solitons in the theory of the derivative nonlinear Schrödinger equations by the method based on imposing the condition that this dynamics must be Hamiltonian. Combining this condition with Stokes’ remark that relationships for harmonic linear waves and small-amplitude soliton tails satisfy the same linearized equations, so the corresponding solutions can be converted one into the other by replacement of the packet’s wave number k by iκ, κ being the soliton’s inverse half-width, we find the Hamiltonian and the canonical momentum of the soliton’s motion. The Hamilton equations are reduced to the Newton equation whose solutions for some typical situations are compared with exact numerical solutions of the Kaup-Newell DNLS equation.

Comparisons of Numerical and Solitary Wave Solutions for the Stochastic Reaction–Diffusion Biofilm Model including Quorum Sensing

This study deals with a stochastic reaction–diffusion biofilm model under quorum sensing. Quorum sensing is a process of communication between cells that permits bacterial communication about cell density and alterations in gene expression. This model produces two results: the bacterial concentration, which over time demonstrates the development and decomposition of the biofilm, and the biofilm bacteria collaboration, which demonstrates the potency of resistance and defense against environmental stimuli. In this study, we investigate numerical solutions and exact solitary wave solutions with the presence of randomness. The finite difference scheme is proposed for the sake of numerical solutions while the generalized Riccati equation mapping method is applied to construct exact solitary wave solutions. The numerical scheme is analyzed by checking consistency and stability. The consistency of the scheme is gained under the mean square sense while the stability condition is gained by the help of the Von Neumann criteria. Exact stochastic solitary wave solutions are constructed in the form of hyperbolic, trigonometric, and rational forms. Some solutions are plots in 3D and 2D form to show dark, bright and solitary wave solutions and the effects of noise as well. Mainly, the numerical results are compared with the exact solitary wave solutions with the help of unique physical problems. The comparison plots are dispatched in three dimensions and line representations as well as by selecting different values of parameters.

Nonisospectral water wave field: Fast and adaptive modal identification and prediction via reduced-order nonlinear solutions.

Real-world water wave fields exhibit significant nonlinear and nonisospectral characteristics, making it challenging to predict their evolution by relying solely on numerical simulation or exact solutions using integrable system theory. Hence, this paper introduces a fast and adaptive method of modal identification and prediction in nonisospectral water wave fields using the reduced-order nonlinear solution (RONS) scheme. Specifically, we discuss the coarse graining and mode extraction of wave field snapshots from the data-driven and physics-driven perspectives and utilize the RONS method for principle modal prediction of nonisospectral water wave fields. This is achieved by investigating the standard and nonisospectral Gardner system describing nonlinear water waves as a demonstration. Through detailed comparison and analysis, the fundamental solitary behaviors and dispersive effects in the Gardner system are discussed. Subsequently, a neighbor approximation is developed that combines the essences of symbolic precomputation and numerical computation in the RONS procedure, which exploits the locality of nonlinear interactions in water wave fields.

Solving Linear and Nonlinear Fuzzy Fractional Volterra-Fredholm Integro Differential Equations Using Shehu Adomian Decomposition Method

Objectives: In applied sciences and engineering, fuzzy fractional differential equations (FFDEs) and fuzzy fractional integral equations (FFIEs) are a crucial topic. The main objective of this work is to discover an analytical approximate solution for the fuzzy fractional Volterra-Fredholm integro differential equations (FFVFIDE). In the Caputo concept, fractional derivatives are regarded. Methods: The Shehu transform is challenging to exist for nonlinear problems. So, the Shehu transform is combined with the Adomian decomposition method is called the Shehu Adomian decomposition method (SHADM) and has been proposed to solve both linear and nonlinear FFVFIDEs. Findings: Both linear and nonlinear FFVIFIDEs can be solved using this technique. For nonlinear terms, Adomian polynomials have been used. The main benefit of this approach is that it converges quickly to the exact solution. Figures and numerical examples demonstrate the expertise of the suggested approach. Novelty: The comparison between the exact solution and numerical solution is shown in figures for various values of fractional order . The numerical evolution demonstrates the efficiency and reliability of the proposed SHADM. The proposed approach is rapid, exact, and simple to apply and produce excellent outcomes. Keywords: Fractional calculus, fuzzy number, Mittag ­ Leffler function, Shehu Adomian decomposition method, fuzzy fractional Volterra­-Fredholm integro differential equation

Exact periodic solution of the undamped Helmholtz oscillator subject to a constant force and arbitrary initial conditions

The Helmholtz oscillator is a nonlinear mixed‐parity oscillator that models the asymmetric vibrations of many engineering and scientific systems. This paper investigated a general and completely integrable form of the Helmholtz oscillator and derived its exact periodic solution from the first integral of the governing differential equation. The considered Helmholtz oscillator has general linear and nonlinear stiffness constants, is subject to a constant force, and has arbitrary initial conditions. Its exact time period was derived in terms of the complete elliptic integral of the first kind, while the exact displacement was derived in terms of the Jacobi elliptic sine function. The validity of the exact solutions was verified for various combinations of the system parameters and initial conditions by comparing with numerical solutions. The exact solutions and numerical solutions were found to match perfectly. Furthermore, the exact solution was applied to analyze the vibration response of real‐world systems that could be modeled as Helmholtz oscillators. The present exact solutions provide benchmark solutions that could be used to determine the accuracy of new and existing approximate solutions for the Helmholtz oscillator.

Exact numerical analysis of EMEC mode instability in more realistic Cairns distributed non-thermal plasmas

Non-thermal plasma systems beyond the state of thermal equilibrium must have non-thermality dependent effective temperatures. These particle populations cannot have Maxwellian temperatures T∥,⊥e(M) which are typically considered at thermal equilibrium in the context of Maxwellian plasmas. Previously, in such dilute environments, numerous non-thermal distributions incorporating the concept of Maxwellian temperature T∥,⊥e(M) have been utilized, one of them is Cairns distribution (here termed as type-A). To ameliorate this inconsistency, we propose a more realistic form of Cairns distribution (type-B) with the vigorous definition of effective temperature and effective thermal velocity. Both Cairns distributions (A and B) are utilized to calculate the transverse dielectric response function (TDERF) of electromagnetic electron cyclotron (EMEC) mode. The exact numerical solution of TDERFs of EMEC instability reveals interesting and more promising repercussions in the case of Cairns-B i.e. an augmentation in the behavior of oscillatory and imaginary frequencies of the instability.

Effect of the Surface Charge-Dependent Boundary Slip on the Electrophoresis of a Hydrophobic Polarizable Rigid Colloid.

The electrophoresis of a hydrophobic charged rigid colloid is studied by considering the lateral movement of the adsorbed surface charge. The slip velocity condition at the hydrophobic surface is modified to take into account the impact of the frictional and electric forces created by the adsorbed laterally mobile surface charge. Though the dependency of the surface charge on the slip velocity in the context of electrophoresis has been addressed before, the effect of the laterally mobile adsorbed surface charge on the electrophoresis of hydrophobic colloids has not been studied. The dielectric colloid is considered to polarize and create an induced immobile surface charge when subjected to an imposed electric field. The impact of the mobile surface charge along with the immobile induced surface charge on electrophoresis of a hydrophobic colloid is elucidated by numerically solving the governing electrokinetic equations in their full form. We have also developed a simplified model under a weak applied field consideration, which can be further reduced to a closed-form analytic expression for the mobility under the Debye-Hückel approximation. This analytic model for mobility is in excellent agreement with the exact numerical solution for an entire range of the Debye length when the ζ-potential is in the order of the thermal potential. One of the notable features of this closed-form mobility expression is that it accounts for the mobile adsorbed surface charge on the hydrodynamic slip condition and the dielectric polarization of the particle. We find that the mobility of the surface charge decreases the electrophoretic mobility of the hydrophobic dielectric colloid. However, the mobile surface charge enhances the mobility of a conducting hydrophobic colloid.

On the Analytical Solution of the SIRV-Model for the Temporal Evolution of Epidemics for General Time-Dependent Recovery, Infection and Vaccination Rates

The susceptible–infected–recovered/removed–vaccinated (SIRV) epidemic model is an important generalization of the SIR epidemic model, as it accounts quantitatively for the effects of vaccination campaigns on the temporal evolution of epidemic outbreaks. Additional to the time-dependent infection (a(t)) and recovery (μ(t)) rates, regulating the transitions between the compartments S→I and I→R, respectively, the time-dependent vaccination rate v(t) accounts for the transition between the compartments S→V of susceptible to vaccinated fractions. An accurate analytical approximation is derived for arbitrary and different temporal dependencies of the rates, which is valid for all times after the start of the epidemics for which the cumulative fraction of new infections J(t)≪1. As vaccination campaigns automatically reduce the rate of new infections by transferring persons from susceptible to vaccinated, the limit J(t)≪1 is even better fulfilled than in the SIR-epidemic model. The comparison of the analytical approximation for the temporal dependence of the rate of new infections J˚(t)=a(t)S(t)I(t), the corresponding cumulative fraction J(t), and V(t), respectively, with the exact numerical solution of the SIRV-equations for different illustrative examples proves the accuracy of our approach. The considered illustrative examples include the cases of stationary ratios with a delayed start of vaccinations, and an oscillating ratio of recovery to infection rate with a delayed vaccination at constant rate. The proposed analytical approximation is self-regulating as the final analytical expression for the cumulative fraction J∞ after infinite time allows us to check the validity of the original assumption J(t)≤J∞≪1.

Numerical solution of coupled system of Emden-Fowler equations using artificial neural network technique

In this paper, a deep artificial neural network technique is proposed to solve the coupled system of Emden-Fowler equations. A vectorized form of algorithm is developed. Implementation and simulation of this technique is performed using Python code. This technique is implemented in various numerical examples, and simulations are conducted. We have shown graphically how accurately this method works. We have shown the comparison of numerical solution and exact solution using error tables. We have also conducted a comparative analysis of our solution with alternative methods, including the Bernstein collocation method and the Homotopy analysis method. The comparative results are presented in error tables. The efficiency and accuracy of this method are demonstrated by these graphs and tables.

Minimum-Time Control for the Test Mass Release Phase of Drag-Free Spacecraft

The capture control of test mass by means of the electrostatic suspensions is crucial for drag-free spacecraft. The test mass must be released to the cage center of the inertial sensor accurately and quickly. This paper proposes a minimum-time capture control method for the test mass release phase of drag-free spacecraft. An analytical solution of optimal control is derived based on Pontryagin’s minimum principle and the linearized dynamics model of the test mass during the release phase. The parameters of the analytical solution are initially guessed with an approximate linear solution of the test mass dynamics model and are slightly modified by using differential correction. Compared with the exact numerical solution by the hp -adaptive pseudospectral method, the analytical solution is proved to be minimum-time. Numerical simulation shows that the proposed control method quickly captures the test mass to the cage center of the inertial sensor. The capture time to stabilization is only half that of the traditional controller.

On the Numerical Solutions Based On Exponential Finite Difference Method for Kuramoto-Sivashinsky Equation and Numerical Stability Analysis

In this paper, we solve the Kuramoto-Sivashinsky Equation numerically by finite-difference methods, using two different schemes which are the Fully Implicit scheme and Exponential finite difference scheme, because of the existence of the fourth derivative in the equation we suggested a treatment for the numerical solution of the two previous scheme by parting the mesh grid into five regions, the first region represents the first boundary condition, the second at the grid point x1, while the third represents the grid points x2,x3,…xn-2, the fourth represents the grid point xn-1 and the fifth is the second boundary condition. We also, study the numerical stability by Fourier (Von-Neumann) method for the two scheme which used in the solution on all mesh points to ensure the stability of the point which had been treated in the suggested style, we using two interval with two initial condition and the numerical results obtained by using these schemes are compare with Exact Solution of Equation Excellent approximate is found between the Exact Solution and numerical Solutions of these methods.