Approximate Analytical Technique Research Articles

Survival probability determination of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitation

An approximate analytical technique based on a combination of statistical linearization and stochastic averaging is developed for determining the survival probability of stochastically excited nonlinear/hysteretic oscillators with fractional derivative elements. Specifically, approximate closed form expressions are derived for the oscillator non-stationary marginal, transition, and joint response amplitude probability density functions (PDF) and, ultimately, for the time-dependent oscillator survival probability. Notably, the technique can treat a wide range of nonlinear/hysteretic response behaviors and can account even for evolutionary excitation power spectra with time-dependent frequency content. Further, the corresponding computational cost is kept at a minimum level since it relates, in essence, only to the numerical integration of a deterministic nonlinear differential equation governing approximately the evolution in time of the oscillator response variance. Overall, the developed technique can be construed as an extension of earlier efforts in the literature to account for fractional derivative terms in the equation of motion. The numerical examples include a hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative terms. The accuracy degree of the technique is assessed by comparisons with pertinent Monte Carlo simulation data.

Response of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitations: A Path Integral approach based on Laplace’s method of integration

In this paper, an approximate analytical technique is developed for determining the non-stationary response amplitude probability density function (PDF) of nonlinear/hysteretic oscillators endowed with fractional element and subjected to evolutionary excitations. This is achieved by a novel formulation of the Path Integral (PI) approach. Specifically, a stochastic averaging/linearization treatment of the original fractional order governing equation of motion yields a first-order stochastic differential equation (SDE) for the oscillator response amplitude. Associated with this first-order SDE is the Chapman–Kolmogorov (CK) equation governing the evolution in time of the non-stationary response amplitude PDF. Next, the PI technique is employed, which is based on a discretized version of the CK equation solved in short time steps. This is done relying on the Laplace’s method of integration which yields an approximate analytical solution of the integral involved in the CK equation. In this manner, the repetitive integrations generally required in the classical numerical implementation of the procedure are avoided. Thus, the non-stationary response amplitude PDF is approximately determined in closed-form in a computationally efficient manner. Notably, the technique can also account for arbitrary excitation evolutionary power spectrum forms, even of the non-separable kind. Applications to oscillators with Van der Pol and Duffing type nonlinear restoring force models, and Preisach hysteretic models, are presented. Appropriate comparisons with Monte Carlo simulation data are shown, demonstrating the efficiency and accuracy of the proposed approach.

Exploration of radiative heat energy on KKL model micropolar nanofluid past a permeable moving surface

AbstractNanofluid comprises nanoparticles that have varied applications in several industries as well as science and engineering due to their heavier thermal conductivity in comparison to conventional fluids. In such applications the role of transportation is vital. However, biomedical applications such as the flow of blood in the human body, peristaltic pumping, processes, etc. the use of nanofluid are important. Therefore, the present investigation aims at the enhancement of thermophysical properties on the radiative micropolar nanofluid flow past a permeable moving surface. These physical properties are studied incorporating the concept of Koo–Kleinsteuer–Li (KKL) model. The crux of the study is the approximate analytical technique, that is, the Homotopy Perturbation Method (HPM). The illustration of the contributing parameters on the flow phenomena is obtained via graphical as well as tabular form. A comparative study is carried out for the good agreement of the present result with an earlier established result in a particular case. Some important findings of this work are a retardation in the linear velocity profile is observed due to increase in the volume fraction of the particle. Suction slows down the linear velocity profile whereas the impact is counter productive in case of injection. A hike in the angular velocity profile is seen in case of greater suction whereas the opposite effect is observed for injection. An increase in the thermal radiation boosts up the fluid temperature and as a result of this an increase in thermal layer thickness is observed.

Comparison between homotopy analysis method and homotopy renormalization method in fluid mechanics

In this paper, the homotopy renormalization method (HTR) is compared with the homotopy analysis method (HAM), an analytical approximation technique for highly nonlinear problems. Four problems in fluid mechanics, including the Blasius viscous flow, magnetohydrodynamic flow, free-convective boundary-layer flow, and Von Kármán swirling viscous flow, are used for detailed comparison. It is found that the HAM approximations are much more accurate than the HTR approximations at the same order. Furthermore, higher-order approximations with smaller errors can be obtained relatively simply by the HAM, while the HTR struggles in such scenarios. All of these confirm the superiority of the HAM for highly nonlinear problems in fluid mechanics.

Quasi-periodic breathers in Newton’s cradle

We consider the parameterized Newton’s cradle lattice with Hertzian interactions in this paper. The positive parameters are {βn : |n| ≤ b} with a fixed integer b ≥ 0, and the Hertzian potential is V(x)=11+α|x|1+α for a fixed real number α > α* ≔ 12b + 25. Corresponding to a large Lebesgue measure set of (βj)|j|≤b∈R+2b+1, we show the existence of a family of small amplitude, linearly stable, quasi-periodic breathers for Newton’s cradle lattice, which are quasi-periodic in time with 2b + 1 frequencies and localized in space with rate 1|n|1+α as |n| ≫ 1. To overcome obstacles in applying the Kolmogorov–Arnold–Moser (KAM) method due to the finite smoothness of V, especially when α is not an integer and to obtain a sharp estimate of the localization rate of the quasi-periodic breathers, the proof of our result uses the Jackson–Moser–Zehnder analytic approximation technique but with refined estimates on error bounds, depending on the smoothness and dimension, which provide crucial controls on the convergence of KAM iterations.

Homotopy analysis methods with applications to thin-film flow of a magnetohydrodynamic-modified second grade fluid

This study investigates thin film’s drainage and lifting of a steady, non-isothermal-modified second-grade incompressible fluid. An approximate analytical technique The homotopy analysis method (HAM) is used to solve the developed governing equations for velocity and temperature fields together with boundary conditions. The expressions for velocity profile, average velocity, temperature distribution, shear stress, and volume flow rate are provided explicitly. The flow of the thin film is found to be substantially dependent on the flow behavior index [Formula: see text], magnetic field parameter, dimensionless numbers say stock number [Formula: see text], and Brinkman number [Formula: see text]. The impact of the above-involved parameters on the fluid’s temperature and velocity is depicted numerically as well as graphically.

Analytical approach on the free convection of Buongiorno model nanofluid over a shrinking surface

This article focuses on the two-dimensional unsteady flow of an incompressible Casson nanofluid over a shrinking horizontal sheet under the influence of inclined magnetic field and in an advance dissipative heat due to Joule heating. The proposed Buongiorno model for the inclusion of Brownian and thermophoresis enriches the flow profiles. Casson model constituents a plastic fluid model that exhibits shear thinning characteristics and high shear viscosity. The proposed fluid model also approximates the rheological behavior of other liquids like physiological suspensions, cosmetics, syrups, etc. The governing partial differential equations (PDEs) that account for effect of Buongiorno model are converted in to nonlinear ordinary differential equations (ODEs) through suitable similarity variables. Further, approximate analytical technique such as Adomian Decomposition Method is beneficial to carry out the solution of the transformed governing equations and the significant nature of the contributing parameter for both the steady and unsteady case is presented via graphs. Moreover, the major contribution is; the Casson parameter along with suction/injection favors in the smooth enhancement in the velocity profiles and the fluid temperature also encouraged by the augmentation in Brownian and thermophoresis parameter.

Adomian Decomposition Method for the Radiative Micropolar Nanofluid Past a Porous Channel: An Analytical Approach

Present analysis leads to investigate the flow of conducting micropolar nanoliquid past a porous channel. The transverse magnetic field is imposed in the normal direction of the flow. The energy equation is enhanced by adopting the influence of thermal radiation. Similarity variable augmented with stream function is used to renovate PDES to ODEs. The present article's special desirability is the approximate analytical technique for these transformed governing equations. We have employed Adomian Decomposition Method (ADM). Descriptions of various profiles for the influence of different parameters are presented through graphs. However, the computation for the rate coefficients is also obtained and deliberated through the tabular form for the influence of diverse physical parameters. A significant contribution reveals that near the lower wall, heat transfer rate enhances greater thermal radiation while impact opposes at the upper wall. A comparison between the methodology newly adopted and the earlier solutions coincided with each other and was found to be in good agreement in particular cases. However, the major findings are laid down here as higher magnetic number favors enhancing angular velocity within the channel and fluid temperature increases due to an increase in thermophoresis parameter.

Analysis of entropy on the peristaltic transport of micropolar nanofluid: a simulation obtained using approximate analytical technique

Analysis of entropy on the peristaltic transport of micropolar nanofluid: a simulation obtained using approximate analytical technique

Multi-step fractional differential transform method for the solution of fractional order stiff systems☆

In this study, the multi-step fractional differential transform method (MSFDTM) is employed to obtain approximate analytical solutions of stiff systems of fractional order. The fractional derivative is described in the Caputo sense. Several numerical experiments are carried out to illustrate the feasibility of MSFDTM as an approximate analytical solution technique for stiff systems of fractional order.

Effects of fractional order time derivative on the solitary wave dynamics of the generalized ZK–Burgers equation

The generalized ZK-Burgers equation has nonlinear characteristics of the governing equation and helps to understand nonlinear waves in many fluid problems of plasma mediums. This work aims to study the effects of fractional order of derivative on the wave solutions of this interesting equation via exact and analytical approximate solution techniques. The analytical approximate wave solutions are retrieved using homotopy analysis method taking the fractional derivative in Caputo’s sense, while exact solutions are reaped with the help of first integral method taking the fractional derivative in Jumarie’s modified Riemann–Liouville sense. The obtained solutions will be helpful to understand the dynamical framework of many nonlinear physical phenomena.

High static low dynamic stiffness outriggers effects on vibration control on cantilever Timoshenko Beam under earthquake excitation

High Static Low Dynamic Stiffness (HSLDS) is a kind of nonlinear visco-elastic device with features of passive control systems. This device presents the main advantages of working without a need for external energy and a maintenance of low cost. Thus, the paper deals with effects of HSLDS-outriggers at a predefined location on a high rise building subjected under earthquake excitation. The partial derivative equations based on the Timoshenko theory are used to model the tall building as an elastic-continuum beam. The nonstationary random approach is used to illustrate the dynamics of earthquake excitation of repeated sequences. Known as the powerful analytical tool and currently applicable to a variety of stochastic and deterministic problems, the stochastic averaging generalized by harmonic function is developed to linearize the modal equation of the structural system. It is showed that the direct simulation is good agreement with equivalent linearization technique. In doing so, it appears that this approximate analytical technique is very convenient to quantify the threshold values of parameters of the HSLDS control device. The obtained results come out that the control device significantly improved the seismic performance of the structural system at an acceptable level.

Squeezing flow analysis of CuO–water and CuO–kerosene-based nanofluids: a comparative study

Heating and cooling are two important procedures in manufacturing as well as transportation industries. Rather than the conventional fluids, solutions of fluids with metal nanoparticles have higher thermal conductivity for effective cooling. Therefore, present paper is a comparative study of squeezing flow analysis of copper oxide–water and oil (kerosene)-based nanofluid between two parallel plates. Magnetohydrodynamic flow of kerosene-based nanofluid along with the dissipative heat energy may enhance the thermal properties of the fluid. Assuming self-similar variables, the governing equations get transformed into non-dimensional forms and approximate analytical techniques such as variation parameter method is employed for these transformed equations. With the well posed physical parameters, the computation is carried out using the mathematical package MATHEMATICA and displayed via graphs and numerical results are shown in tabular form. Favourable cases in comparison with earlier studies are also studied wherever possible. However, when the plates are away from each other, it is seen that the kerosene-based nanofluid velocity overrides the water-based nanofluid whereas the impact is reversed in the case of squeezing.

An analytical study on fluid flow characteristics for flow within a microchannel of rectangular cross section

The present work considers fluid flow analysis within a microchannel of rectangular cross section with different exact and approximate analytical techniques. Thus, velocity slip at the solid wall of the channel is considered. The slip flow is transformed to a no-slip flow with some minor modifications by applying an average slip velocity. The flow is represented by the Navier-Stokes equation exposed to slip boundary conditions. The governing equation is solved considering separation of variables method (SOV) as an exact method and, Integral and Variational methods as approximate methods. The Poiseuille number and slip coefficient are also determined for each method. The present prediction agrees well with available literature. To predict accuracy level of all the techniques, the results are presented in a comparative scale along with numerical prediction. It is observed that the accuracy level for the velocity profile obtained in each technique depends on the aspect ratio where as for the prediction of Poiseuille number and slip coefficient, all the technique show less dependency on the aspect ratio.

New approximate analytical technique for the solution of time fractional fluid flow models

In this note, we broaden the utilization of an efficient computational scheme called the approximate analytical method to obtain the solutions of fractional-order Navier–Stokes model. The approximate analytical solution is obtained within Liouville–Caputo operator. The analytical strategy generates the series form solution, with less computational work and fast convergence rate to the exact solutions. The obtained results have shown a simple and useful procedure to analyze complex problems in related areas of science and technology.

A new multistage technique for approximate analytical solution of nonlinear differential equations

The article introduces a new multistage technique for solving a polynomial system of nonlinear initial and boundary value problems of differential equations. The radius of convergence R of the series solution to the problem is derived a-priorly in terms of the parameters of the polynomial system. Then guided by the convergence-control parameter h<R, the domain of the problem is split into subintervals. By stepping out in a multistage manner, corresponding subproblems are defined which are then subsequently solved with conventional Parker-Sochacki method to get a piecewise continuous solution with very high accuracy. The method is applied to SIR epidemic model, stiff differential equation modelling combustion, Lorenz chaotic problem, and the Troesch's boundary value problem. The results obtained showed a remarkable accuracy when compared with Runge-Kutta Method of order 4. The article showcased the proposed method as a simple, yet accurate approximate analytical technique for nonlinear differential equations.

Development in the Heat Transfer Properties of Nanofluid Due to the Interaction of Inclined Magnetic Field and Non-Uniform Heat Source

In the current scenario a new mathematical model is designed and examined for the unsteady course of nanofluid through permeable vertical surface due to the interaction of inclined magnetic field. Radiative heat transfer properties is included assuming the Cogley radiation, dissipative heat energy due to the conjunction o magnetic field i.e., Joule dissipation and the space and time-dependent heat source/sink amplifies the study as well. Depending upon todays need in various industries the implementation of nanofluid is vital. Therefore, present study involves the behavior of both metal and oxide nanoparticles in the base fluid kerosene. Involvement of transformation rules the problem is converted into nonlinear set of ODEs and further these are solved employing approximate analytical technique such as Variational Iteration Method (VIM). The characteristics of various flow parameters are analyzed via graphs and the numerical simulation along with the validation of the result is obtained through tables. The comparative study brings out the convergence criterion of the methodology adopted herein. However, the favorable results are; the fluid temperature augments with increasing nanoparticle volume fraction and suction enriches both the fluid velocity and temperature whereas injection retards it significantly.

Harmonic wavelets based response evolutionary power spectrum determination of linear and nonlinear structural systems with singular matrices

A new approximate analytical technique is proposed for determining the response evolutionary power spectrum (EPS) of stochastically excited structural multi-degree-of-freedom (MDOF) linear and nonlinear systems with singular matrices. Such systems can appear, indicatively, when a redundant coordinates modeling is adopted for forming the equations of motion of complex multi-body systems. For this case, it can be argued that this modeling approach facilitates the system’s stochastic response analysis, since employment of redundant DOFs is associated with computational cost efficient solution frameworks, and potentially provides with enhanced modeling flexibility. In this context, aiming at the joint time–frequency response analysis of MDOF systems, recently developed wavelet-based solution frameworks, which generalize classic input–output relationships of random vibration, are adopted and further generalized in this paper to account for systems with singular matrices. Specifically, resorting to the theory of generalized inverses of singular matrices, as well as to the theory of harmonic wavelets, a Moore–Penrose generalized matrix inverse excitation-response relationship is derived herein for determining the response EPS of linear MDOF systems. Further, a recently developed harmonic-wavelet-based statistical linearization technique is also generalized to account for the case of nonlinear MDOF systems. The validity of the proposed technique is demonstrated by pertinent numerical examples.

Accelerating FHS Option Pricing Under Linear GARCH

We propose an analytical approximation technique to accelerate the filtered historical simulation (FHS) option pricing method. The analytical approximation technique has at least two advantages over the FHS method: first, it does not suffer from random sampling error as it needs no simulation; second, it is fast in calculating option price as it is analytical. Simulation results indicate our technique approximates the FHS method quite well, and empirical results show that our technique has very good option pricing performance.

Practical Analytical Approach for Estimating Long-Term and Set-Up Shaft Resistance of Prebored Piles

This paper presents a practical analytical approach to evaluating the long-term and set-up shaft resistance of prebored piles. The key contributions of this study lie in the fact that not only all aspects involved, from pile installation to pile loading, but also the residual stress propagated from the vertically shearing effect during pile installation are properly considered in the proposed analytical approach. The installation effects of a prebored pile are simulated as undrained expansion of a cylindrical cavity followed by vertically shearing the soils along the boundary of the pile shaft. To develop the possible analytical approximation technique, simplified solutions for the stress and excess pore pressure changes in critical sate soils due to cavity expansion are adopted in the formulations, while a modified Terzaghi’s consolidation theory that incorporates a variable coefficient of consolidation with respect to the void ratio and mean effective stress of the soil is employed to model the consolidation process after the pile installation. The proposed analytical approach has been well validated through comparisons with a finite-element numerical model that involves the entire process from pile installation to loading. The approximate analytical model developed in this work, owing to its relative formulation simplicity, computational efficiency, and prediction reliability, is expected to be able to serve as a feasible analysis method and guidance for the design and construction of piles involving the preboring procedure.