Semi-analytical Techniques Research Articles

Nonlocal theory of thermoelastic diffusive materials and its application in structural dynamic thermo-elasto-diffusive responses analysis

With the rapid development of ultrafast heating technologies and its application in the micro-machining of micro/nano-electromechanical devices, where the nonlocal effects on elastic deformation and heat/mass transfer will increase and classical/generalized theory of thermoelastic diffusion coupling does not hold any more. In this work, a nonlocal theory of thermoelastic diffusive materials is developed, and the spatial nonlocal effects of heat and mass transport are both considered for the first time. With the aid of thermodynamic principles, the constitutive relations involving size-dependent characteristic lengths are derived. In the aspect of the application, the proposed theoretical model is further used to investigate the dynamic thermo-elasto-diffusive responses of a one-dimensional semi-infinite medium with one end subjected to time-dependent shock loads, and the semi-analytical techniques based on Laplace transform methods are applied to obtain the transient solutions. The effects of nonlocal parameters on structural dynamic thermo-elasto-diffusive responses are also evaluated and discussed to be expected to provide new insights on the thermoelastic diffusion coupling at the micro/nanoscale.

Analysis of MHD Fluids around a Linearly Stretching Sheet in Porous Media with Thermophoresis, Radiation, and Chemical Reaction

This paper presents the comparative analysis of MHD boundary layer fluid flow around a linearly stretching surface in the presence of radiative heat flux, heat generation/absorption, thermophoresis velocity, and chemical reaction effects in a permeable surface. The governing equations are highly nonlinear PDEs which are converted into coupled ODEs with the help of dimensionless variables and solved by using semianalytical techniques. The numerical and graphical outcomes are observed and presented via tables and graphs. Also, the Nusselt and Sherwood numbers and skin friction coefficient are illustrated by tables. On observation of heat and mass transfer, it was noticed that Maxwell fluid dominates the other fluids such as Newtonian, Williamson, and Casson fluid due to high rate of thermal conductivity, and hence, Maxwell fluid has better tendency for heat and mass transfer than other Newtonian and non-Newtonian fluids.

Dissipative heat energy on Cu and Al2O3 ethylene–glycol-based nanofluid flow over a heated semi-infinite vertical plate

The present analysis describes the effect of dissipative heat energy transfer in Ethylene–Glycol (EG) based on conducting nanofluid over a heated semi-infinite vertical plate past through a porous medium. Uniform magnetic field, heat source/sink, and the effect of particle concentration also have been discussed by incorporating in the energy and solutal transfer equations, respectively. In addition to that, the thermal properties of the nanofluid are affected by the thermal slip boundary condition since; the temperature slip is favorable for the reduction in the heat transfer. Assuming self-similar transformations, the governing PDEs are transformed into non-linear coupled ODEs. These transformed equations are solved by using semi-analytical techniques such as Adomian Decomposition Method (ADM). The characteristics of different parameters on the flow phenomena are obtained and presented via graphs. The numerical values of the thermophysical properties of both the nanoparticles and the base fluid are shown in the table. Validation of the present work is obtained by comparing our result with the earlier established result and it is found that both the results are coinciding with each other. However, the main quantified results are the following: due to heavy density of the Cu nanoparticles, increasing volume fraction in ethylene–glycol base fluid resists the fluid motion and the inclusion of dissipative heat energy is favorable to enhance the nanofluid temperature.

Mathematical Models with Buckling and Contact Phenomena for Elastic Plates: A Review

A review of mathematical models for elastic plates with buckling and contact phenomena is provided. The state of the art in this domain is presented. Buckling effects are discussed on an example of a system of nonlinear partial differential equations, describing large deflections of the plate. Unilateral contact problems with buckling, including models for plates, resting on elastic foundations, and contact models for delaminated composite plates, are formulated. Dynamic nonlinear equations for elastic plates, which possess buckling and contact effects are also presented. Most commonly used boundary and initial conditions are set up. The advantages and disadvantages of analytical, semi-analytical, and numerical techniques for the buckling and contact problems are discussed. The corresponding references are given.

Conducting Non-adaptive Experiments in a Live Setting: A Bayesian Approach to Determining Optimal Sample Size

Abstract This research studies the use of predetermined experimental plans in a live setting with a finite implementation horizon. In this context, we seek to determine the optimal experimental budget in different environments using a Bayesian framework. We derive theoretical results on the optimal allocation of resources to treatments with the objective of minimizing cumulative regret, a metric commonly used in online statistical learning. Our base case studies a setting with two treatments assuming Gaussian priors for the treatment means and noise distributions. We extend our study through analytical and semi-analytical techniques which explore worst-case bounds, the presence of unequal prior distributions, and the generalization to k treatments. We determine theoretical limits for the experimental budget across all possible scenarios. The optimal level of experimentation that is recommended by this study varies extensively and depends on the experimental environment as well as the number of available units. This highlights the importance of such an approach which incorporates these factors to determine the budget.

A semi-analytical approach for the reversible Schnakenberg reaction-diffusion system

This paper discusses how a semi-analytical approach can be used alongside the reversible Schnakenberg model in a reaction-diffusion cell. It is worth noting that semi-analytical techniques are aligned to the Galerkin method, which offers an approximation of the governing partial differential equations through a system of ordinary differential equations. The bifurcation diagrams, steady-state curves and regions of the parameter space where bifurcations take place are part of the semi-analytical solutions and the numerical solution.

Multistage cooling and freezing of a saline spherical water droplet

In this paper, the thermal behaviour of a saline water droplet during flight over a marine vessel in cold weather conditions is investigated by analytical and semi-analytical techniques. To predict and analyze the droplet cooling and freezing processes, three stages are employed: a liquid cooling stage, a solidification stage, and a solid cooling stage. The theoretical model considers heat transfer via conduction inside the droplet as well as convection, evaporation (just for the liquid cooling stage), and radiation heat transfer from the droplet's surface to the ambient air. A novel semi-analytical solution technique is developed to analyze the inward moving boundary problem for the solidification stage. The results show that the liquid cooling stage is very short, and the temperature at the droplet's center remains close to the initial droplet temperature. During the solidification stage, the velocity of the inward freezing front within the droplet is approximately constant, and the temperature variations are linear when the temperature inside the droplet reaches the freezing temperature. The solid cooling stage is much longer than the other stages, and the temperature changes are non-linear. For a case study, theoretical predictions show that the average temperature of a droplet with a diameter of 1 mm at the moment of impact on the deck is approximately −1.95°C. Moreover, there is an ice shell with a thickness of 0.06 mm on the surface of the water droplet at the moment of impact.

Increasing confidence in estimating stimulated reservoir volume by integrating RTA and microseismic analysis

Rate Transient Analysis (RTA) and microseismic monitoring are gaining momentum in modelling Stimulated Reservoir Volume (SRV) in Multi-Frac Horizontal Wells (MFHWs) in unconventional reservoirs. From a behavioural perspective, RTA uses history matching and production data analysis to estimate fracture volume and productivity, and microseismic analysis maps frack-ing-induced micro-earthquakes to calibrate the fracture network from a spatiotemporal point of view. Defining the concepts of normalized rate, material balance time and pseudo-time, dynamic drainage volume, together with convolution, deconvolution and analytical models, make RTA a powerful and computationally efficient tool for modelling MFHWs (Blasingame et al., 1991; Agarwal et al., 1998, Mattar and Anderson, 2003). Poe (2005) proposed a rate-transient analysis method for evaluating the performance of wells with limited pressure data using the superposition theory and dimensionless parameters. Soliman and Adams (2010) estimated fracture properties by applying Flow Regime Identification (FRI) plots to early production data, followed by using analytical models derived for each distinct flow regime. Kuchuk et al. (2016) calibrated reservoir models by history-matching the transient flow rate and pressure measurements. Brown (2009), Stalgorova and Mattar (2012a, 2012b), Deng et al. (2015), and Yuan et al. (2015) divided the reservoir into a series of linear flow regions and derived analytical pressure transient models from the pressure diffusion equation, not only to confirm the validity of the identification, characterization and diagnostic analyses but also to provide production forecasts and carry out optimization studies. Clarkson et al. (2015) successfully applied RTA analytical and semi-analytical modelling techniques to a gas condensate MFHW in a Western Canadian Basin, highlighting the fact that building a predictive understanding of drainage volume dynamics is best started with physics-based analytical models rather than multi-phase numerical simulations. This is particularly important in unconventional reservoirs where the complex, small-scale physics and rock-fluid interactions significantly hinder gathering enough measurements to support the numerically added complexities.

Solving FIDEs by Using Semi-Analytical Techniques

This paper mainly focuses on the recent advances in the semi-analytical approximated methods for solving Fredholm Integro-Differential Equations (FIDEs) of the second kind by using Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM) and Direct Homotopy Analysis Method (DHAM). Convergence analysis of the exact solution of the proposed methods is established. Moreover, we proved the uniqueness of the solution. To illustrate the methods, an example is presented.

A new analysis of a partial differential equation arising in biology and population genetics via semi analytical techniques

In this work, we propose a new optimal perturbation iteration method for solving the generalized Fitzhugh–Nagumo equation with time-dependent coefficients. This research reveals that the new proposed technique, with the aid of symbolic computations, provides a straightforward and impressive mathematical tool for solving nonlinear partial differential equations. Implementing this method to Fitzhugh–Nagumo equation illustrates its potency. Convergence analysis also shows that OPIM, unlike many other methods in literature, converges fast to exact analytical solutions of the nonlinear problems at lower order of approximations.

Remnants of Galactic Subhalos and Their Impact on Indirect Dark-Matter Searches

Dark-matter subhalos, predicted in large numbers in the cold-dark-matter scenario, should have an impact on dark-matter-particle searches. Recent results show that tidal disruption of these objects in computer simulations is overefficient due to numerical artifacts and resolution effects. Accounting for these results, we re-estimated the subhalo abundance in the Milky Way using semianalytical techniques. In particular, we showed that the boost factor for gamma rays and cosmic-ray antiprotons is increased by roughly a factor of two.

MHD three dimensional flow of Oldroyd-B nanofluid over a bidirectional stretching sheet: DTM-Padé Solution

Abstract Current article is devoted with the study of MHD 3D flow of Oldroyd B type nanofluid induced by bi-directional stretching sheet. Expertise similarity transformation is confined to reduce the governing partial differential equations into ordinary nonlinear differential equations. These dimensionless equations are then solved by the Differential Transform Method combined with the Padé approximation (DTM-Padé). Dealings of the arising physical parameters namely the Deborah numbers β1 and β2, Prandtl number Pr, Brownian motion parameter Nb and thermophoresis parameter Nt on the fluid velocity, temperature and concentration profile are depicted through graphs. Also a comparative study between DTM and numerical method are presented by graph and other semi-analytical techniques through tables. It is envisage that the velocity profile declines with rising magnetic factor, temperature profile increases with magnetic parameter, Deborah number of first kind and Brownian motion parameter while decreases with Deborah number of second kind and Prandtl number. A comparative study also visualizes comparative study in details.

Some New Applications of Elzaki Transform for Solution of Linear Volterra Type Integral Equations

Volterra type integral equations have diverse applications in scientific and other fields. Modelling physical phenomena by employing integral equations is not a new concept. Similarly, extensive research is underway to find accurate and efficient solution methods for integral equations. Some of noteworthy methods include Adomian Decomposition Method (ADM), Variational Iteration Method (VIM), Method of Successive Approximation (MSA), Galerkin method, Laplace transform method, etc. This research is focused on demonstrating Elzaki transform application for solution of linear Volterra integral equations which include convolution type equations as well as one system of equations. The selected problems are available in literature and have been solved using various analytical, semi-analytical and numerical techniques. Results obtained after application of Elzaki transform have been compared with solutions obtained through other prominent semi-analytic methods i.e. ADM and MSA (limited to first four iterations). The results substantiate that Elzaki transform method is not only a compatible alternate approach to other analytic methods like Laplace transform method but also simple in application once compared with methods ADM and MSA.

Modeling and simulation of Electromagnetic Fields on a Floating Aluminium

The electromagnetic field calculation for a floating aluminum disc is difficult to calculate since the equation involved does not produce a closed solution. The numerical, analytical, semi-analytical techniques that are already developed to find these magnetic fields have no proper mathematical formulation when the disc is disturbed from its coaxial position. The stabilization of disc is going to be effected when the disc moves away from its coaxial position due to a change in inductance between the disc and coils, due to change in magnetic flux linkage, etc. In this paper, a 2D FEM model is developed to determine the magnetic fields on a floatingaluminum disc when it is moved away from its coaxial position. The 3D FEM model developed is simulated in both COMSOL-Multiphysics and ANSYS-Electronics. The results obtained by simulation are compared, for accuracy, with the numerical solution developed earlier using Finite Difference method (FDM) and also discussed.

Analysis of magnetic fields on a levitating coaxial and non-coaxial aluminum disc

The electromagnetic field calculations for a levitating aluminum disc involve integro-differential equations and the solution of these equations is difficult to obtain using conventional techniques especially when the disc is coaxially away from the axis of the coils. Over the years many analytical, semi-analytical and numerical techniques have been proposed to calculate the magnetic fields on the disc when it is coaxial with the coils. In this paper, a mathematical formulation has been developed to obtain the magnetic fields on a conducting disc using a numerical technique at different positions of the levitation and for different disc discretization’s. The numerical technique developed here is based on Finite Difference Method. Since the magnetic fields on the disc are due to the coil currents and eddy currents in the disc, first a mathematical formulation is done to calculate fields due to exciting coil currents and then a numerical technique is used to calculate fields due to eddy currents on the disc. Also, the magnetic fields on the disc are calculated when the disc moves away from the axis of the coil. A MATLAB program is developed to calculate these fields.

Semi-Analytical Methods for Calculation of Leakage Inductance and Frequency-Dependent Resistance of Windings in Transformers

Leakage inductance and frequency-dependent resistance of windings are the key parameters for the design of magnetic devices. This paper focuses on analytical or semi-analytical calculation techniques to be integrated into a medium frequency transformer optimization process. To this extend, a review of the currently most used method to calculate the inductance and resistance of windings transformer is established. Based on this review, new calculation methods are developed. Magnetic field cartography is evaluated on the basis of 2-D structures thanks to a combination of Ampere’s law and the image method. From this magnetic field map, leakage inductance can be calculated. The obtained accuracy with this method is equivalent to 3-D finite-element modeling (FEM). For frequency-dependent resistance evaluation, new methods based on 2-D or 1-D approach are compared to FEM and reviewed models. An analysis of the behavior of each model is done, showing that newly developed models are more suitable in the case of a medium frequency transformer design in terms of accuracy, robustness, and calculation time. This paper could be useful to the designers of magnetic devices because newly developed models could easily be adapted to various geometries, such as toroidal structures or even inductors.

Belief Dynamics in Social Networks: A Fluid-Based Analysis

The advent and proliferation of social media have led to the development of mathematical models describing the evolution of beliefs/opinions in an ecosystem composed of socially interacting users. The goal is to gain insights into collective dominant social beliefs and into the impact of different components of the system, such as users' interactions, while being able to predict users' opinions. Following this thread, in this paper we consider a fairly general dynamical model of social interactions, which captures all the main features exhibited by a social system. For such model, by embracing a mean-field approach, we derive a diffusion differential equation that represents asymptotic belief dynamics, as the number of users grows large. We then analyze the steady-state behavior as well as the time dependent (transient) behavior of the system. In particular, for the steady-state distribution, we obtain simple closed-form expressions for a relevant class of systems, while we propose efficient semi-analytical techniques in the most general cases. At last, we develop an efficient semi-analytical method to analyze the dynamics of the users' belief over time, which can be applied to a remarkably large class of systems.

On the solution of linear hydrodynamic stability of dean flow by using three semi-analytical approaches

In the present paper, three semi-analytical techniques are examined for solving eigenvalue problem arising from linear hydrodynamics stability of Dean Flow. To this accomplishment, hybrid of Fourier transform and Adomian decomposition method (FTADM), differential transform method (DTM) and Homotopy perturbation method (HPM) is selected and applied on the eigenvalue problem. Semi-analytical results are validated against the existing data with high accuracy. The comparison between FTADM, DTM and HPM reveals that for the same number of truncated terms, the accuracy of the FTADM is more pronounced. This may be attributed to the incorporation of all boundary conditions into our solution when using the FTADM. The results also indicate that the value of wave number (i.e., parameter engaged in our eigenvalue problem) is remarkably impressive on the convergence trend and effectiveness (i.e., the occurrence of becoming nearer to the numerical results) of our solution. In addition, critical wave number and Dean number for the onset of Dean flow instability are successfully reported.

Spike timing precision of neuronal circuits.

Spike timing is believed to be a key factor in sensory information encoding and computations performed by the neurons and neuronal circuits. However, the considerable noise and variability, arising from the inherently stochastic mechanisms that exist in the neurons and the synapses, degrade spike timing precision. Computational modeling can help decipher the mechanisms utilized by the neuronal circuits in order to regulate timing precision. In this paper, we utilize semi-analytical techniques, which were adapted from previously developed methods for electronic circuits, for the stochastic characterization of neuronal circuits. These techniques, which are orders of magnitude faster than traditional Monte Carlo type simulations, can be used to directly compute the spike timing jitter variance, power spectral densities, correlation functions, and other stochastic characterizations of neuronal circuit operation. We consider three distinct neuronal circuit motifs: Feedback inhibition, synaptic integration, and synaptic coupling. First, we show that both the spike timing precision and the energy efficiency of a spiking neuron are improved with feedback inhibition. We unveil the underlying mechanism through which this is achieved. Then, we demonstrate that a neuron can improve on the timing precision of its synaptic inputs, coming from multiple sources, via synaptic integration: The phase of the output spikes of the integrator neuron has the same variance as that of the sample average of the phases of its inputs. Finally, we reveal that weak synaptic coupling among neurons, in a fully connected network, enables them to behave like a single neuron with a larger membrane area, resulting in an improvement in the timing precision through cooperation.

Finding structure in the dark: Coupled dark energy, weak lensing, and the mildly nonlinear regime

We reexamine interactions between the dark sectors of cosmology, with a focus on robust constraints that can be obtained using only mildly nonlinear scales. While it is well known that couplings between dark matter and dark energy can be constrained to the percent level when including the full range of scales probed by future optical surveys, calibrating matter power spectrum emulators to all possible choices of potentials and couplings requires many computationally expensive n-body simulations. Here we show that lensing and clustering of galaxies in combination with the cosmic microwave background (CMB) are capable of probing the dark sector coupling to the few percent level for a given class of models, using only linear and quasilinear Fourier modes. These scales can, in principle, be described by semianalytical techniques such as the effective field theory of large-scale structure.