Exact Numerical Solutions Research Articles

A third-order numerical method for solving fractional ordinary differential equations

In this paper, we developed a novel numerical method for solving general nonlinear fractional ordinary differential equations (FODEs). First, we transformed the nonlinear FODEs into the equivalent Volterra integral equations. We then developed a time-stepping algorithm for the numerical solution of the Volterra integral equations based on the third-order Taylor expansion for approximating the integrands in the Volterra integral equations on a chosen mesh with the mesh parameter $ h $. This approximation led to implicit nonlinear algebraic equations in the unknowns at each given mesh point, and an iterative algorithm based on Newton's method was developed to solve the resulting implicit equations. A convergence analysis of this numerical scheme showed that the error between the exact solution and numerical solution at each mesh point is $ \mathcal{O}(h^{3}) $, independent of the fractional order. Finally, four numerical examples were solved to verify the theoretical results and demonstrate the effectiveness of the proposed method.

Numerical way out through finite volume method for Navier-Stokes type blood flow inside stenosed curved arteries

This article explores the intricate fluid dynamics within stenosed curved arteries, focusing on Newtonian or viscous fluids' incompressible, two-dimensional, and time-independent flow. The modeled Navier-Stokes equations in curvilinear coordinates are solved numerically through the finite volume method by using open foam, and exact solutions are also computed after using the assumptions of mild stenosis. The flow inside a stenosed curved artery having computational results through FVM is the main achievement of the present work. We have included 2D and 3D graphics depicting pressure gradient, velocity distribution, and shear stress for physical visualization. This helps provide a clear understanding of the physical circumstances exhibited in the analysis. This work contributes significantly to biomechanics by unraveling the intricacies of fluid flow in stenosed curved arteries. The combination of numerical simulations, exact solutions, and visual representations enhances our understanding of the underlying physical phenomena, paving the way for further advancements in cardiovascular research and clinical applications. These visual representations serve as crucial tools for physical visualization, enhancing the interpretability of the complex flow patterns observed in the analysis.

A unified core model of double-gate and surrounding-gate MOSFETs for circuit simulation

This paper presents a new core compact model of double-gate (DGFET) and surrounding-gate (SGFET) MOSFETs for circuit simulations. The current and the terminal charges are continuous with high computation efficiency and accuracy. Despite its accuracy, it retains the same simplicity of the industry standard transistors models. The drain current is worked out without invoking the charge-sheet approximation exploiting a quadratic symmetric polynomial interpolation of the charge in the channel. Apart this clear approximation, no other simplification is used to work out the drain current, the terminal charges, the potential, and electric field in the channel. The accuracy of the model is shown by comparison with the exact numerical solution and experimental data of the literature.

Analytical Solution of the Susceptible-Infected-Recovered/Removed Model for the Not-Too-Late Temporal Evolution of Epidemics for General Time-Dependent Recovery and Infection Rates

The dynamical equations of the susceptible-infected-recovered/removed (SIR) epidemics model play an important role in predicting and/or analyzing the temporal evolution of epidemic outbreaks. Crucial input quantities are the time-dependent infection (a(t)) and recovery (μ(t)) rates regulating the transitions between the compartments S→I and I→R, respectively. Accurate analytical approximations for the temporal dependence of the rate of new infections J˚(t)=a(t)S(t)I(t) and the corresponding cumulative fraction of new infections J(t)=J(t0)+∫t0tdxJ˚(x) are available in the literature for either stationary infection and recovery rates or for a stationary value of the ratio k(t)=μ(t)/a(t). Here, a new and original accurate analytical approximation is derived for general, arbitrary, and different temporal dependencies of the infection and recovery rates, which is valid for not-too-late times after the start of the infection when the cumulative fraction J(t)≪1 is much less than unity. The comparison of the analytical approximation with the exact numerical solution of the SIR equations for different illustrative examples proves the accuracy of the analytical approach.

Superconvergence error analysis of Ciarlet-Raviart mixed finite element method for damped Boussinesq equation

The primary objective of this paper is to investigate the error analysis for the damped Boussinesq equation with the Ciarlet-Raviart mixed finite element method, focusing on achieving superconvergence. By employing a skillful mean technique and a high precision estimate of the bilinear element, as well as combining with projection and interpolation technique, we first obtain a superclose error estimate between interpolation of the exact solution and numerical solution. Subsequently, by applying an interpolation post-processing approach, we efficiently derive a global superconvergence result. Finally, several numerical results are presented to support our theoretical analysis.

Modelling and simulation of MRR in electro-chemical surface grinding of Al-SiC-Gr using hybrid analytical approach

Electro-Chemical Surface Grinding (ECSG) is a hybrid machining process which utilizes synergic actions of three sub-processes viz. electrochemical dissolution, grinding action of abrasive particles and erosive action of electrolyte flow in inter electrode gap. Since, occurrence of turbulence in machining zone significantly affects the machining performance, it is required to analyze the turbulence phenomenon in ECSG process. The exact solution and Numerical solution–based hybrid analytical approach has been used to predict the Material Removal Rate (MRR) in ECSG of Al-SiC-Gr. In this model, the streaming potential of electrolytic action, abrasive action of grinding wheel and shear force of electrolyte flow between inter electrode gap have been considered. Finite Element Method based Numerical modelling has been carried out to analyze the electrolyte flow in machining zone in order to evaluate the MRR due to erosive action. It was observed that the application of shear force on oxide layer rises with the increase in flow rate of electrolyte. Confirmation experiment has been performed to validate the modelling results. Validation results show a minor error of 8.21% in prediction of MRR using analytical as well as finite element model.

Limit-state solutions for the active earth pressure behind walls rotating about the base

Determining earth pressure-induced moments in the active limit state is critical for the safety assessment of retaining structures. While traditional design methods assume a Coulomb's resultant force acting at one-third of the wall height, the literature suggests that the earth pressure distribution depends on the failure mode. This paper presents a rigorous kinematic solution for the ultimate moment acting on a wall undergoing rotation about its base (due to bending failure or overturning). In addition, an approximate static solution is considered. These solutions give a good approximation to the ultimate load. While the kinematic solution is closer to the exact numerical solution, the approximate static solution provides a reliable conservative approximation for common geometric and soil parameters. Its advantage is that it has a closed-form formulation. The kinematic solution is successfully validated against experimental data and is further used to investigate the peculiarities of the rotational failure mode and to evaluate traditional methods. It is found that traditional methods are close to the rigorous solutions and, therefore, reasonably safe. A useful by-product of this study: the formulation of the kinematic solution is versatile and can be applied to various geotechnical problems involving rotation.

The agreement between novel exact and numerical solutions of nonlinear models

Nonlinear models (NLMs), being an important topic in mathematical physics, have attracted a lot of attention in the international research community because they have numerous uses in human life. These NLMs are typically implemented to illuminate various complicated natural phenomena. NLMs have been seen in fluid mechanics, optical physics, signal processing, plasma physics, acoustic waves, traffic flow, atomic physics, mathematical physics, nuclear energy, and other applications. That's why we consider the Mikhailov-Novikov-Wang (M-N-W) integrable equation. In this project, we acquire exact solutions (ESs) and numerical solutions (NSs) of the M-N-W integrable equation. Primarily, we use the modified (G′/G)-expansion method (MDG/GEM) and adomian decomposition method (ADM) to find new ESs and NSs of the M-N-W integrable equation. This present project work is mainly concentrated on finding new ESs and NSs for a broad class of the M-N-W integrable equation using the MDG/GEM and ADM, respectively, and making a comparison between these two kinds of solutions. Additionally, using recent scientific instruments, the two-dimensional, three-dimensional, and contour plots are displayed, as well as comparison plots between graphs of analytical and numerical solutions of the M-N-W integrable equation. NLMs have appeared in a wide variety of engineering, physics, and applied mathematics; hence, it is natural to expect that this project should have broad applicability.

Propagation of generalized Korteweg-de Vries solitons along large-scale waves.

We consider propagation of solitons along large-scale background waves in the generalized Korteweg-de Vries (gKdV) equationtheory when the width of the soliton is much smaller than the characteristic size of the background wave. Due to this difference in scales, the soliton's motion does not affect the dispersionless evolution of the background wave. We obtained the Hamilton equationsfor soliton's motion and derived simple relationships which express the soliton's velocity in terms of a local value of the background wave. Solitons' paths obtained by integration of these relationships agree very well with the exact numerical solutions of the gKdV equation.

Mechanisms of like-charge attraction in many-body systems

The physics of like-charge attraction in a collinear binary system of N particles has been investigated via a polarizable-ion model. It was found that like-charge attraction between an array of (N−1) type-A particles on the left and a single type-B particle on the right is conditional upon a sufficient degree of dissimilarities between the two types of particles. By assuming that (1) the particles of one type are strongly polarizing (but weakly polarizable) such that the Coulomb fields they produce are dominant all over other electric fields, and that (2) the particles of the other type are strongly polarizable (but weakly polarizing) such that their Coulomb-field-induced dipoles are the only polarization effects worth considering, a good agreement between our approximate analytic theory and exact numerical solutions has been obtained. Our analytic results show that the attractive components of the electrostatic force between the two types of particles could be thought of as arising from imaginary charges induced in the polarized particle(s) by the polarizing ones, resulting in a screening of the real charge(s) in the polarized particle(s). Like-charge attraction occurs if the field contributions from the imaginary charges collectively outweigh the field contribution(s) from the real charge(s).

Asymptotical Stability Criteria for Exact Solutions and Numerical Solutions of Nonlinear Impulsive Neutral Delay Differential Equations

In this paper, the idea of two transformations is first proposed and applied. Some new different sufficient conditions for the asymptotical stability of the exact solutions of nonlinear impulsive neutral delay differential equations (INDDEs) are obtained. A new numerical scheme for INDDEs is also constructed based on the idea. The numerical methods that can preserve the stability and asymptotical stability of the exact solutions are provided. Two numerical examples are provided to demonstrate the theoretical results.

Validity of One-Dimensional Radiative Transfer in Participating Media

For radiative transfer through an absorbing, scattering medium with planar geometry, the validity of the one-dimensional assumption is examined. Using the generalized zonal with Monte Carlo method as a solver, exact numerical solutions are generated to identify conditions under which the one-dimensional assumption is valid. Specifically, for the predictions of transmissivity and reflectivity for a collimated or diffuse energy beam through a planar slab, numerical data are presented to show that a medium aspect ratio (ratio of the width to the thickness) of five or more is a necessary but not sufficient condition for the data to agree with the one-dimensional result. The implication of this condition on the utilization of the one-dimensional radiative transfer assumption in some practical engineering problems such as pipe flow and thermal insulation is discussed.

Memory efficient Fock-space recursion scheme for computing many-fermion resolvents

A fundamental roadblock to the exact numerical solution of many-fermion problems is the exponential growth of the Hilbert space with system size. It manifests as extreme dynamical memory and computation-time requirements for simulating many-fermion processes. Here we construct a novel reorganization of the Hilbert space to establish that the exponential growth of dynamical-memory requirement is suppressed inversely with system size in our approach. Consequently, the state-of-the-art resolvent computation can be performed with substantially less memory. The memory-efficiency does not rely on Hamiltonian symmetries, sparseness, or boundary conditions and requires no additional memory to handle long-range density-density interaction and hopping. We provide examples calculations of interacting fermion ground state energy, the many-fermion density of states and few-body excitations in interacting ground states in one and two dimensions.

Bootstrapping through discrete convolutional methods

AbstractBootstrapping was designed to randomly resample data from a fixed sample using Monte Carlo techniques. However, the original sample itself defines a discrete distribution. Convolutional methods are well suited for discrete distributions, and we show the advantages of utilizing these techniques for bootstrapping. The discrete convolutional approach can provide exact numerical solutions for bootstrap quantities, or at least mathematical error bounds. In contrast, Monte Carlo bootstrap methods can only provide confidence intervals which converge slowly. Additionally, for some problems the computation time of the convolutional approach can be dramatically less than that of Monte Carlo resampling. This article provides several examples of bootstrapping using the proposed convolutional technique and compares the results to those of the Monte Carlo bootstrap, and to those of the competing saddlepoint method.

Qualitative Analysis for Solving a Fractional Integro-Differential Equation of Hyperbolic Type with Numerical Treatment Using the Lerch Matrix Collocation Method

In this research, we present a qualitative analysis for studying a new modification of a nonlinear hyperbolic fractional integro-differential equation (NHFIDEq) in dual Banach space CEE, J. Under some suitable conditions, the existence and uniqueness of a solution are demonstrated with the use of fixed-point theorems. The verification of the offered method has been conducted by applying the Lerch matrix collocation (LMC) method as a numerical treatment. The major motivation for selecting the LMC approach is that it reduces the solution of the given NHFIDEq to a matrix representation form corresponding to a linear system of algebraic equations; additionally, to demonstrate that the proposed strategy has better precision than alternative numerical methods, we study the error and the convergence analysis. Finally, we introduce numerical examples illustrating comparisons between the exact solutions and numerical solutions for different values of the Lerch parameters λ and time t as well as how the absolute error in each example is calculated.

Instability of Langmuir-beam waves: Kappa-distributed electrons

In space plasmas, electron populations exhibit non-equilibrium velocity distributions with high-energy tails that are reproduced by the Kappa power-laws and contrast with the Maxwellian distributions often used in theoretical and numerical analyses. In this work, we investigate typical electron beam-plasma systems and show the influence of Kappa tails on the linear dispersion and stability spectra of Langmuir-beam waves. The most common scenarios invoke instabilities of Langmuir waves at the origin of radio emissions in solar flares and interplanetary shocks. However, the parametric domain of these instabilities is narrow (i.e., energetic beams but with very low density, nb/ne≲10−3), making their analytical and numerical characterization not straightforward, while the approximations used may lead to inconclusive results. Here, we provide exact numerical solutions of the Langmuir-beam mode, which distinguish from the classical ones (unaffected by the beam), and also from electron beam modes destabilized by more energetic and/or denser beams. Langmuir-beam solutions are only slightly modified by the Kappa distribution of the beam component, due to its very low density. However, if the main (core) population is Kappa distributed, the instability of the Langmuir-beam mode is strongly inhibited, if not suppressed. New analytical solutions are derived taking into account the more or less resonant involvement of the electron core and beam populations. As a result, the analytical solutions show an improved match with the exact solutions, making them applicable in advanced modeling of weak (weakly nonlinear) turbulence.

Ball Lens‐Assisted Cellphone Imaging with Submicron Resolution

AbstractOne of the most significant developments in life sciences—the discovery of bacteria and protists—was accomplished by Antoni van Leeuwenhoek in the 17th century using a single ball lens microscope. It is shown that the full potential of single lens designs can be realized in a contact mode of imaging by ball lenses with a refractive index of n ≈ 2, suitable for developing compact cellphone‐based microscopes. The quality of imaging is comparable to basic compound microscopes, but with a narrower field‐of‐view, and is demonstrated for various biomedical samples. The maximal magnification (M > 50) with the highest resolution (≈0.66 µm at λ = 589 nm) is achieved for imaging of nanoplasmonic structures by ball lenses made from LASFN35 glass, the index of which is tuned near n = 2 using chromatic dispersion. Due to limitations of geometrical optics, the imaging theory is developed based on an exact numerical solution of the Maxwell equations, including spherical aberration and the nearfield coupling of a point source. The modeling is performed using multiscale analysis: from the field propagation inside ball lenses with diameters 30 < D/λ < 4000 to the formation of the diffracted field at distances of ≈105 λ. It is shown that such imaging enables the transition from pixel‐ to diffraction‐limited resolution in cellphone microscopy.

Exact numerical solutions and high order nonstandard difference schemes for a second order delay differential equation

In this work, we obtain an expression, given in terms of some special functions, for the exact numerical solution of the second order delay differential equation with general initial condition for . Based on this exact solution, we propose a family of nonstandard finite difference methods that combine high order accuracy and efficient computational properties. We also show that the numerical solutions obtained with the new schemes are consistent with asymptotic stability properties of the exact solutions.

Numerical model supplemented by thin-layer analysis for diffusiophoresis of a particle incorporating finite ion size effects

The impact of finite-sized ions on the diffusiophoresis of a charged colloid subjected to a concentration gradient of electrolyte solution consisting monovalent or multivalent ionic species, is studied. In diffusiophoresis, the ion concentration is of O(1M). In this non-dilute electrolyte solutions, the ion–ion steric interaction is important. We have adopted the Boublik–Mansoori–Carnahan–Starling–Leland (BMCSL) model to account for the ion steric interactions and the Batchelor–Green expression for the relative viscosity of suspension. We have solved the standard model numerically considering ions as point charge (PNP-model), the modified Nernst–Planck equations incorporating the ion steric interaction with constant viscosity (MNP-model), and modification of the MNP-model by incorporating the viscosity variation with the ionic volume fraction (MNPV-model). Semi-analytical expressions for mobility based on a linear perturbation technique under a thinner Debye length is presented for PNP- and MNP-models. In the MNP-model, counterion saturation in the Debye layer due to the ion steric interaction enhances the surface potential by attenuating the shielding effect, diminishes the surface conduction, and magnifies the induced electric field. These in combination create a larger mobility at a thinner Debye length compared with the PNP-model. This increment in mobility attenuates when the MNPV-model is considered. The MNPV-model is more appropriate to analyze the finite ion size effects, and it is found to yield the mobility values more close to the experimental data compared with the MNP- and PNP-model. The semi-analytical expressions for mobility based on the PNP- and MNP-models agree with the corresponding exact numerical solutions when the surface potential is in the order of thermal potential. However, a large discrepancy between the simplified expression and the exact numerical results is found for a concentrated electrolyte in which the induced electric field is large.

Diffusiophoresis of a soft particle incorporating ion partitioning and hydrophobic core

The diffusiophoresis of a soft particle governed by an imposed ionic concentration gradient is studied based on a modified ion transport equation. The surface of the rigid core of the soft particle is considered to be hydrophobic, which is coated with a charged polyelectrolyte layer (PEL) whose dielectric constant is different from the dispersion medium. This step-change in the dielectric constant of the PEL and the dispersion medium creates an ion partitioning effect. The governing equations based on the coupled Navier–Stokes–Nernst–Planck–Poisson equations are modified to incorporate the ion partitioning effect. A simplified model based on the linear perturbation technique under a small applied concentration gradient in the scale of the particle radius is developed. The simplified model agrees well with the exact numerical solutions for a Debye length smaller than the particle size. We have analyzed the impact of the core slip length and PEL permittivity on the mobility of the soft particle. The interfacial slip enhances the impact of the core charge density on the diffusiophoresis. In this case, a reversal of mobility occurs at a thinner Debye length compared to a hydrophilic core. Ion partitioning attenuates the concentration of mobile counterions in PEL, leading to a reduction in screening of PEL fixed charge. We find that the diffusiophoresis of the soft particle in a non-z:z asymmetric electrolyte strongly depends on the valence of the counterions. Diffusiophoresis of the soft particle in a KCl-NaCl mixture is considered to achieve positive mobility.