Simple Perturbation Method Research Articles

An analytic solution of Navier–Stokes flow past a sphere in the region of intermediate Reynolds number

We study an analytical solution of steady, laminar, and incompressible flow past a sphere in the region of intermediate Reynolds number. The flow is governed by the Navier–Stokes (N–S) equation and the continuity equation. By applying a simple perturbation method to solve the equations, a second-order approximation cannot be obtained, as well-known (Whitehead’s paradox). Many analytical studies, such as Oseen approximation, matching technique, the homotopy analysis method, etc, have been conducted to resolve the paradox. The drag coefficients of these solutions are valid in the region of Reynolds number (R d is the diameter-based Reynolds number) However, the solution cannot express the flow separation behind a sphere observed in experiments. We also develop a perturbation technique to construct a solution of the N–S equation asymptotically to solve the paradox. The solution consists of power series of , where h is an arbitrary constant . By setting the value of h to avoid divergence of the solution in the all-region where steady flow exists, the solution expresses the flow separation behind a sphere, coinciding with experiments in the region of intermediate Reynolds number (), although the existing analytical solutions could not express. Also, the present solution gives the drag coefficient which agrees with experimental and numerical values in the region of .

An Analytical Solution for Radioactive MHD Flow TiO2-Fe3O4/H2O Nanofluid and Its Biological Applications

The measuring instrument for medical investigation and bio screening depending on organic labeling and nanoparticle identification is an active research in the current world. An analytical approach is used in order to investigate mass and heat transfer properties on an unstable movement of Casson nanofluid beyond the stretched exterior under the influence of induction heat. In addition to this, a simple perturbation method is used to determine the flow of dominating mathematical statements, while titanium oxide and ferrous oxide nanoparticles are drooping in H2O based nanofluid. The velocity increases with the temperature slip. The field of magnetic has the stability to raise the ending layer surface to slow down the velocity. The friction factor grows with the amplification in the nanoparticle volume fraction and it lessens with the raise of the thermal slip. The thermal convection effect of nanoparticles reveals that the temperature distribution increases, as a result, helps to damage cancer cells in the process of medicine distribution.

ANALYSIS OF APPROXIMATE SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS IN THEIR GENERAL FORM WITH GENERALIZED ORDER

In this paper, we dedicate our study on the approximate solutions of van der Pol equation in their general form. First, we prove the approximate analytic solutions to this equation by different perturbation methods, simple perturbation method (SPM), Lindstedt-Poincaré method (PLM) and Averaging method (AM). Then we compare these approximations with each other and with the exact solution. Second, we introduce a new form of generalized Van Der Pol oscillator with fractional-order derivatives. Which is analyzed through phase portraits, Poincaré maps and analytic solutions, we use numerical simulation to illustrate the behavior of the fractional order system.

Hypersingular integral equations of the first kind: A modified homotopy perturbation method and its application to vibration and active control

This paper attempts to propose and investigate a modification of the homotopy perturbation method to study hypersingular integral equations of the first kind. Along with considering this matter, of course, the novel method has been compared with the standard homotopy perturbation method. This method can be conveniently fast to get the exact solutions. The validity and reliability of the proposed scheme are discussed. Different examples are included to prove so. According to the results, we further state that new simple homotopy perturbation method is so efficient and promises the exact solution. The modification of the homotopy perturbation method has been discovered to be the significant ideal tool in dealing with the complicated function-theoretic analytical structures within an analytical method.

Mn3O4 with different morphologies tuned through one-step electrochemical method for high-performance lithium-ion batteries anode

A simple, green, efficient, and controllable electrochemical potential perturbation method has been developed to synthesize Mn3O4 nanomaterials. No toxic or explosive chemicals are used during the fabrication process. Taking pure Mn metal as the working electrode and 2 M KCl aqueous solution as the electrolyte solution, the Mn3O4 products with rod-like, spherical-like and mixed structures can be obtained through cyclic voltammetry technology under rapid scan rates of 5, 200 and 50 V s−1 within a potential range of −2.0 to 2.0 V (vs Saturated calomel electrode, SCE), respectively. Rapid and repeated redox treatment of Mn metal surface accompanying intense oxygen and hydrogen evolution will lead to the dispersion of Mn metal and the formation of the Mn3O4 nanomaterials. The lithium-ion storage behaviors of these as-fabricated Mn3O4 nanomaterials have been studied contrastively, showing that the spherical-like Mn3O4-S nanoparticles with the highest Li-ion chemical diffusion coefficient display the best cycling stability and rate capability. As a consequence, it shows an excellent specific capacity of up to 780 mAh g−1 after 250 cycles a high rate of 1000 mA g−1 and a good rate capability of 895, 798, 722, 683, 601 and 530 mAh g−1 at current densities of 200, 500, 800, 1000, 1500 and 2000 mA g−1, respectively. This work is of importance for energy storage as it provides a new and efficient electrochemical way for the fabrication of anode materials for the next-generation LIBs with outstanding electrochemical performances.

Effects of magnetic field on phase-mixing of electrostatic oscillations in cold electron-positron-ion plasmas

Spatiotemporal evolution of nonlinear electron-positron oscillations around a homogeneous background of massive ions has been analyzed in cold electron-positron-ion (EPI) plasmas by employing a simple perturbation method, demonstrating phase-mixing and thus wave-breaking of excited oscillations at arbitrarily low amplitudes [C. Maity, Phys. Plasmas 21, 072317 (2014)]. In this work, we investigate effects of the magnetic field on the phase-mixing phenomena of electron-positron oscillations in cold EPI plasmas. A perturbative analysis of governing fluid-Maxwell's equations has been carried out up to third order to obtain a rough estimate of the phase-mixing time. It has been shown that the presence of an external ambient magnetic field may induce a delay in the process of phase-mixing of such oscillations.

Accurate and precise ab initio anharmonic free-energy calculations for metallic crystals: Application to hcp Fe at high temperature and pressure

A framework for computing the anharmonic free energy (FE) of metallic crystals using Born-Oppenheimer ab initio molecular dynamics (AIMD) simulation, with unprecedented efficiency, is introduced and demonstrated for the hcp phase of iron at extreme conditions (up to $\ensuremath{\approx}290$ GPa and 5000 K). The advances underlying this work are: (1) A recently introduced harmonically-mapped averaging temperature integration (HMA-TI) method that reduces the computational cost by order(s) of magnitude compared to the conventional TI approach. The TI path starts from zero Kelvin, where it assumes the behavior is given exactly by a harmonic treatment; this feature restricts application to systems that have no imaginary phonons in this limit. (2) A Langevin thermostat with the HMA-TI method that allows the use of a relatively large MD time step (4 fs, which is about eight times larger than the size needed for the Andersen thermostat) without loss of accuracy. (3) AIMD sampling is accelerated by using density functional theory (DFT) with a low-level parameter set, then the measured quantities of selected configurations are robustly reweighted to a higher level of DFT. This introduces a speedup of about 20--$30\ifmmode\times\else\texttimes\fi{}$ compared to directly simulating the accurate system. (4a) The temperature ($T$) dependence of the hcp equilibrium shape (i.e., $c/a$ axial ratio) is determined (including anharmonicity), with uncertainty less than $\ifmmode\pm\else\textpm\fi{}0.001$. (4b) Electronic excitation is included through Mermin's finite-temperature extension of the $T=0$ K DFT. A simple FE perturbation method is introduced to handle the difficulty associated with applying the TI method with a $T$-dependent geometry and (due to electronic excitation) potential-energy surface. (5) The FE in the thermodynamic limit is attained through extrapolation of only the (computationally inexpensive) quasiharmonic FE, because the anharmonic FE contribution has negligible finite-size effects. All methods introduced here do not affect the AIMD sampling---results are obtained through post-processing---so established AIMD codes can be employed without modification. Analytical formulas fitted to the results for the variation of the equilibrium $c/a$ ratio and FE components with $T$ are provided. Notably, effects of magnetic excitations are not included and may yet prove important to the overall FE; if so, it is plausible that such contributions can be added perturbatively to the FE values reported here. Notwithstanding these considerations, FE values are obtained with an estimated accuracy and precision of 2 meV/atom, suggesting that the capability to compute the phase diagram of iron at Earth's inner core conditions is within reach.

Thermal and Solutal Buoyancy Effect on MHD Boundary Layer Flow of a Visco-Elastic Fluid Past a Porous Plate with Varying Suction and Heat Source in the Presence of Thermal Diffusion

An analytical solution is investigated for a fully developed free convective flow of a visco-elastic incompressible electrically conducting fluid past a vertical porous plate bounded by a porous medium in the presence of thermal diffusion, variable suction and variable permeability. A magnetic field of uniform strength is applied perpendicular to the plate and the presence of heat source is also considered. The novelty of the study is to investigate the effect of thermal diffusion on a visco-elastic fluid in the presence of time dependent variable suction. The importance is due to the applications of this kind of visco-elastic fluids in many industries. The coupled dimensionless nonlinear partial differential equations are transformed into a set of ordinary differential equations by using multiple parameter perturbation on velocity whereas simple perturbation method on temperature and concentration. With corresponds to these, the expressions for skin friction, Nusselt number and Sherwood number are derived. The numerical computations have been studied through figures and tables. The presence of thermal diffusion increases fluid velocity, whereas the influence of the magnetic field reduces it. In the case of heavier species, it is noticed that concentration increases with an increase in Soret number.

Improved Homoclinic Predictor for Bogdanov–Takens Bifurcation

An improved homoclinic predictor at a generic codim 2 Bogdanov–Takens (BT) bifucation is derived. We use the classical "blow-up" technique to reduce the canonical smooth normal form near a generic BT bifurcation to a perturbed Hamiltonian system. With a simple perturbation method, we derive explicit first- and second-order corrections of the unperturbed homoclinic orbit and parameter value. To obtain the normal form on the center manifold, we apply the standard parameter-dependent center manifold reduction combined with the normalization, that is based on the Fredholm solvability of the homological equation. By systematically solving all linear systems appearing from the homological equation, we remove an ambiguity in the parameter transformation existing in the literature. The actual implementation of the improved predictor in MatCont and numerical examples illustrating its efficiency are discussed.

Magnetohydrodynamic effects on a charged colloidal sphere with arbitrary double-layer thickness

An analytical study is presented for the magnetohydrodynamic (MHD) effects on a translating and rotating colloidal sphere in an arbitrary electrolyte solution prescribed with a general flow field and a uniform magnetic field at a steady state. The electric double layer surrounding the charged particle may have an arbitrary thickness relative to the particle radius. Through the use of a simple perturbation method, the Stokes equations modified with an electric force term, including the Lorentz force contribution, are dealt by using a generalized reciprocal theorem. Using the equilibrium double-layer potential distribution from solving the linearized Poisson-Boltzmann equation, we obtain closed-form formulas for the translational and angular velocities of the spherical particle induced by the MHD effects to the leading order. It is found that the MHD effects on the particle movement associated with the translation and rotation of the particle and the ambient fluid are monotonically increasing functions of κa, where κ is the Debye screening parameter and a is the particle radius. Any pure rotational Stokes flow of the electrolyte solution in the presence of the magnetic field exerts no MHD effect on the particle directly in the case of a very thick double layer (κa→0). The MHD effect caused by the pure straining flow of the electrolyte solution can drive the particle to rotate, but it makes no contribution to the translation of the particle.

On Approximating DSGE Models by Series Expansions

We show how to use a simple perturbation method to solve non-linear rational expectation models. Drawing from the applied mathematics literature we propose a method consisting of series expansions of the non-linear system around a known solution. The variables are represented in terms of their orders of approximation with respect to a perturbation parameter. The final solution, therefore, is the sum of the different orders. This approach links to formal arguments the idea that each order of approximation is solved recursively taking as given the lower order of approximation. Therefore, this method is not subject to the ambiguity concerning the order of the variables in the resulting state-space representation as, for example, has been discussed by Kim et al. (2008). Provided that the model is locally stable, the approximation technique discussed in this paper delivers stable solutions at any order of approximation. JEL Classification: C63, E0

Rank-based regression for analysis of repeated measures

We consider rank-based regression models for repeated measures. To account for possible withinsubject correlations, we decompose the total ranks into between- and within-subject ranks and obtain two different estimators based on between- and within-subject ranks. A simple perturbation method is then introduced to generate bootstrap replicates of the estimating functions and the parameter estimates. This provides a convenient way for combining the corresponding two types of estimating function for more efficient estimation.

Phase compactons

We study the phase dynamics of a chain of autonomous, self-sustained, dispersively coupled oscillators. In the quasicontinuum limit the basic discrete model reduces to a Korteveg–de Vries-like equation, but with a nonlinear dispersion. The system supports compactons–solitary waves with a compact support–and kovatons–compact formations of glued together kink–antikink pairs that propagate with a unique speed, but may assume an arbitrary width. We demonstrate that lattice solitary waves, though not exactly compact, have tails which decay at a superexponential rate. They are robust and collide nearly elastically and together with wave sources are the building blocks of the dynamics that emerges from typical initial conditions. In finite lattices, after a long time, the dynamics becomes chaotic. Numerical studies of the complex Ginzburg–Landau lattice show that the non-dispersive coupling causes a damping and deceleration, or growth and acceleration, of compactons. A simple perturbation method is applied to study these effects.

General linearized equations of wave propagation in strained elastic solids of cylindrical shapes using a simple perturbation method

The equations of particle motion in an elastic isotropic stressed medium are first derived in Cartesian coordinates and then transformed into cylindrical coordinates. The three components of the equations of motion are non-linear partial differential equations and cannot be of use in practical applications. However, noting that the particle displacement is composed of a small dynamic part superimposed on a large static part, these equations are linearized via a simple perturbation method. The linearized equations are presented in closed form. They contain variables, which may be measured and experimented upon in practice, in the field of acoustoelasticity.

Boiling Water Reactor Loading Pattern Optimization Using Simple Linear Perturbation and Modified Tabu Search Methods

An automated system for designing a loading pattern (LP) for boiling water reactors (BWRs) given a reference LP and control rod (CR) sequence has been developed. This system employs the advanced nodal code SIMULATE-3 and a BWR LP optimization code FINELOAD-3, which uses a simple linear perturbation method and a modified Tabu search method to select potential optimized LP candidates. Both of these unique methods of FINELOAD-3 were developed to achieve an effective BWR LP optimization strategy and to have high computational efficiency. FINELOAD-3 also adjusts deep CR positions to compensate for the core reactivity deviation caused by fuel shuffling. The objective function is to maximize the end-of-cycle core reactivity while satisfying the specified thermal margins and cold shutdown margin constraints. This optimization system realized the practical application for real BWR LP design. Computer time needed to obtain an optimized LP for a typical BWR/5 octant core with 15 depletion steps is ~4 h using an engineering workstation. This system was extensively tested for real BWR reload core designs and showed that the developed LPs using this system are equivalent or better than the manually optimized LPs.

An approximate solution for spherical and cylindrical piston problem

A new theory of shock dynamics (NTSD) has been derived in the form of a finite number of compatibility conditions along shock rays. It has been used to study the growth and decay of shock strengths for spherical and cylindrical pistons starting from a non-zero velocity. Further a weak shock theory has been derived using a simple perturbation method which admits an exact solution and also agrees with the classical decay laws for weak spherical and cylindrical shocks.

Nonlinear Two-Wave Interaction in a Medium Endowed with a Quasi-Periodic Structure

The nonlinear two-wave interaction in a medium endowed with a quasi-periodic structure is theoretically studied using a simple perturbation method, and the reflection coefficient of wave’s energy is numerically calculated in order to confirm the theoretical result that the reflection coefficient of one wave depends not only on its own amplitude but also on that of the other wave. We apply a method employed in the study of the propagation of nonlinear waves through a medium endowed with a periodic structure 1) to the present problem.

On the Propagation of Nonlinear Waves through a Medium Endowed with a Periodic Structure

The propagation of nonlinear waves through a medium endowed with a periodic structure is theoretically studied using a simple perturbation method, and in order to confirm the result, the reflection coefficient of the wave's energy is numerically calculated. It is shown that the reflection coefficient depends on the wave amplitude, and this fact can be explained from the characteristic behavior of the pseudo-unstable state solution and the doubly-periodic stable state solution. These concepts were previously proposed by the present author in a study of the nonlinear Mathieu equation.

Voltage-dependent STM image of a charge density wave

In the present work we write a general expression for the local density of states (LDOS) due to a commensurate charge density wave (CDW). The main goal is to investigate the voltage dependence of the contrast in the scanning tunneling microscope (STM) images of materials showing CDW's. For layered materials, having nearly two-dimensional electronic structures, the problem is complicated by the many-band situation near the Fermi level, and by the incomplete band gapping. Nevertheless, a simple perturbation method allows one to relate the amplitude and phase of the CDW to features of the band structure. We emphasize the role of particular characteristic energies, at which the CDW has a large contribution from special k points of the surface Brillouin zone, leading to different modulations in the STM image. In a second part of the paper, we consider the voltage-dependent contrast of ${\mathrm{NbSe}}_{2}.$ For this material, we find that the amplitude and the phase of the CDW change significantly as a function of energy (or voltage), resulting in a number of different possible motifs. For example, the direct comparison between occupied and empty states reveals that new states on the order of ${E}_{F}\ifmmode\pm\else\textpm\fi{}\ensuremath{\Delta},$ giving a dominant contribution to the LDOS, have different phases. As a result, in the corresponding STM images, the maxima of the corrugation have shifted positions along the diagonal of the conventional unit cell.

Construction of invariants by perturbation theory

Abstract We propose a simple perturbation method for the construction of invariants. A straightforward recurrence relation yields all the perturbation corrections hierarchically. We consider the Mathieu equation as an illustrative example.