Alternative Methods of Regular and Singular Perturbation Problems

Making exact approximations to solve equations distinguishes applied mathematicians from pure mathematicians, physicists, and engineers. Perturbation problems, both regular and singular, are pervasive in diverse fields of applied mathematics and engineering. This research paper provides a comprehensive overview of algebraic methods for solving perturbation problems, featuring a comparative analysis of their strengths and limitations. Serving as a valuable resource for researchers and practitioners, it offers insights and guidance for tackling perturbation problems in various disciplines, facilitating the advancement of applied mathematics and engineering.

Perturbation solutions for asymmetric laminar flow in porous channel with expanding and contracting walls

The cases of large Reynolds number and small expansion ratio for the asym- metric laminar flow through a two-dimensional porous expanding channel are considered. The Navier-Stokes equations are reduced to a nonlinear fourth-order ordinary differential equation by introducing a time and space similar transformation. A singular perturbation method is used for the large suction Reynolds case to obtain an asymptotic solution by matching outer and inner solutions. For the case of small expansion ratios, we are able to obtain asymptotic solutions by double parameter expansion in either a small Reynolds number or a small asymmetric parameter. The asymptotic solutions indicate that the Reynolds number and expansion ratio play an important role in the flow behavior. Nu- merical methods are also designed to confirm the correctness of the present asymptotic solutions.

Connection between Volterra series and perturbation method in nonlinear systems analyses

This paper is concerned with the connection between the Volterra series and the regular perturbation method in nonlinear systems analyses. It is revealed for the first time that, for a forced polynomial nonlinear system, if its derived linear system is a damped dissipative system, the steady response obtained through the regular perturbation method is exactly identical to the response given by the Volterra series. On the other hand, if the derived linear system is an undamped conservative system, then the Volterra series is incapable of modeling the forced polynomial nonlinear system. Numerical examples are further presented to illustrate these points. The results provide a new criterion for quickly judging whether the Volterra series is applicable for modeling a given polynomial nonlinear system.

On the Downshift of Wave Frequency for Bragg Resonance

For surface gravity waves propagating over a horizontal bottom that consists of a patch of sinusoidal ripples,strong wave reflection occurs under the Bragg resonance condition.The critical wave frequency,at which the peak reflection coefficient is obtained,has been observed in both physical experiments and direct numerical simulations to be downshifted from the well-known theoretical prediction.It has long been speculated that the downshift may be attributed to higher-order rippled bottom and free-surface boundary effects,but the intrinsic mechanism remains unclear.By a regular perturbation analysis,we derive the theoretical solution of frequency downshift due to third-order nonlinear effects of both bottom and free-surface boundaries.It is found that the bottom nonlinearity plays the dominant role in frequency downshift while the free-surface nonlinearity actually causes frequency upshift.The frequency downshift/upshift has a quadratic dependence in the bottom/free-surface steepness.Polychromatic bottom leads to a larger frequency downshift relative to the monochromatic bottom.In addition,direct numerical simulations based on the high-order spectral method are conducted to validate the present theory.The theoretical solution of frequency downshift compares well with the numerical simulations and available experimental data.

Hydrodynamics and Sensitivity Analysis of a Williamson Fluid in Porous-Walled Wavy Channel

In this work,a steady,incompressible Williamson fluid model is investigated in a porous wavy channel.This situation arises in the reabsorption of useful substances from the glomerular filtrate in the kidney.After 80%reabsorption,urine is left,which behaves like a thinning fluid.The laws of conservation of mass and momentum are used to model the physical problem.The analytical solution of the problem in terms of stream function is obtained by a regular perturbation expansion method.The asymptotic integration method for small wave amplitudes and the RK-Fehlberg method for pressure distribution has been used inside the channel.It is demonstrated that the forward flow becomes fast in the narrow region(at x=0.75),which dominates the upward flow inside the channel.To study the impact of model parameters on outputs,we applied normalized local sensitivity analysis and noticed that the most influential parameter for the longitudinal velocity profile is the dimensionless wave amplitude.The reabsorption parameter is sensitive for transverse velocity in the narrow region,and the Weissenberg number has a strong effect on the pressure inside the channel.Further,the least sensitive parameters for the velocity components and pressure have been identified.

The Onset of Buoyancy and Surface Tension Driven Convection in a Ferrofluid Layer by Influence of General Boundary Conditions

This paper investigated the buoyancy and surface tension-driven ferro-thermal-convection (FTC) in a ferrofluid (FF) layer due to influence of general boundary conditions. The lower surface is rigid with insulating to temperature perturbations, while the upper surface is stress-free and subjected to general thermal boundary condition. The numerically Galerkin technique (GT) and analytically regular perturbation technique (RPT) are applied for solving the problem of eigenvalue. It is analyzed that increasing Biot number, decreases the magnetic and Marangoni number is to postponement the onset. Additionally, magnetization nonlinearity parameter has no effect on FTC in the non-existence of Biot number. The results under the limiting cases are found to be in good agreement with those available in the literature.

Regularized luni-solar gravity dynamics on resident space objects

Resident space object population in highly elliptical high perigee altitude(>600 km)orbits is significantly affected by luni-solar gravity.Using regularization,an analytical orbit theory with luni-solar gravity effects as third-body perturbations in terms of Kustaanheimo-Stiefel regular elements is developed.Numerical tests with different cases resulted in good accuracy for both short-and long-term orbit propagations.It is observed that the luni-solar perturbations affect the accuracy of the analytical solution seasonally.The analytical theory is tested with the observed orbital parameters of the few objects in highly elliptical orbits.The analytical evolution of osculating perigee altitude is found to be concurrent with observed data.Solar perturbation,when compared with lunar perturbation,is established to be dominant over such orbits.