Search results for: dispersion equations

Authors: Nasim Abuali Galehdari, Thomas J. Ryan, Ajit D. Kelkar

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In recent years, nanotechnology has played an important role in the design of an efficient radiation shielding polymeric composites. It is well known that, high loading of nanomaterials with radiation absorption properties can enhance the radiation attenuation efficiency of shielding structures. However, due to difficulties in dispersion of nanomaterials into polymer matrices, there has been a limitation in higher loading percentages of nanoparticles in the polymer matrix. Therefore, the objective of the present work is to provide a methodology to fabricate and then to characterize the functionally graded radiation shielding structures, which can provide an efficient radiation absorption property along with good structural integrity. Sandwich structures composed of Ultra High Molecular Weight Polyethylene (UHMWPE) fabric as face sheets and functionally graded epoxy nanocomposite as core material were fabricated. A method to fabricate a functionally graded core panel with controllable gradient dispersion of nanoparticles is discussed. In order to optimize the design of functionally graded sandwich composites and to analyze the stress distribution throughout the sandwich composite thickness, a finite element method was used. The sandwich panels were discretized using 3-Dimensional 8 nodded brick elements. Classical laminate analysis in conjunction with simplified micromechanics equations were used to obtain the properties of the face sheets. The presented finite element model would provide insight into deformation and damage mechanics of the functionally graded sandwich composites from the structural point of view.

Keywords: nanotechnology, functionally graded material, radiation shielding, sandwich composites, finite element method

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Authors: R. K. Apalowo, D. Chronopoulos, V. Thierry

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There exist a wide range of failure modes in composite structures due to the increased usage of the structures especially in aerospace industry. Moreover, temperature dependent wave response of composite and layered structures have been continuously studied, though still limited, in the last decade mainly due to the broad operating temperature range of aerospace structures. A wave finite element (WFE) and finite element (FE) based computational method is presented by which the temperature dependent wave dispersion characteristics and interaction phenomenon in composite structures can be predicted. Initially, the temperature dependent mechanical properties of the panel in the range of -100 ◦C to 150 ◦C are measured experimentally using the Thermal Mechanical Analysis (TMA). Temperature dependent wave dispersion characteristics of each waveguide of the structural system, which is discretized as a system of a number of waveguides coupled by a coupling element, is calculated using the WFE approach. The wave scattering properties, as a function of temperature, is determined by coupling the WFE wave characteristics models of the waveguides with the full FE modelling of the coupling element on which defect is included. Numerical case studies are exhibited for two waveguides coupled through a coupling element.

Keywords: finite element, temperature dependency, wave dispersion characteristics, wave finite element, wave scattering properties

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Authors: Ih-Ching Hsu

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Inspired by the so called Jacobi Identity (x y) z + (y z) x + (z x) y = 0, the following class of functional equations EQ I: F [F (x, y), z] + F [F (y, z), x] + F [F (z, x), y] = 0 is proposed, researched and generalized. Research methodologies begin with classical methods for functional equations, then evolve into discovering of any implicit algebraic structures. One of this paper’s major findings is that EQ I, under two additional conditions F (x, x) = 0 and F (x, y) + F (y, x) = 0, proves to be a fundamental functional equation for Lie Algebras. Existence of non-trivial solutions for EQ I can be proven by defining F (p, q) = [p q] = pq –qp, where p and q are quaternions, and pq is the quaternion product of p and q. EQ I can be generalized to the following class of functional equations EQ II: F [G (x, y), z] + F [G (y, z), x] + F [G (z, x), y] = 0. Concluding Statement: With a major finding proven, and non-trivial solutions derived, this research paper illustrates and provides a new functional equation scheme for studies in two major areas: (1) What underlying algebraic structures can be defined and/or derived from EQ I or EQ II? (2) What conditions can be imposed so that conditional general solutions to EQ I and EQ II can be found, investigated and applied?

Keywords: fundamental functional equation, generalized functional equations, Lie algebras, quaternions

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Authors: Victor I. Ilgisonis, Ludmila V. Konovaltseva, Vladimir P. Lakhin, Ekaterina A. Sorokina

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The analytical solutions for geodesic acoustic eigenmodes in tokamak plasmas with circular concentric magnetic surfaces are found. In the frame of ideal magnetohydrodynamics the dispersion relation taking into account the toroidal coupling between electrostatic perturbations and electromagnetic perturbations with poloidal mode number |m| = 2 is derived. In the absence of such a coupling the dispersion relation gives the standard continuous spectrum of geodesic acoustic modes. The analysis of the existence of global eigenmodes for plasma equilibria with both off-axis and on-axis maximum of the local geodesic acoustic frequency is performed.

Keywords: tokamak, MHD, geodesic acoustic mode, eigenmode

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Authors: Xijun Yu, Zhenzhen Li, Zupeng Jia

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This paper presents a new cell-centered Lagrangian scheme for two-dimensional compressible flow. The new scheme uses a semi-Lagrangian form of the Euler equations. The system of equations is discretized by Discontinuous Galerkin (DG) method using the Taylor basis in Eulerian space. The vertex velocities and the numerical fluxes through the cell interfaces are computed consistently by a nodal solver. The mesh moves with the fluid flow. The time marching is implemented by a class of the Runge-Kutta (RK) methods. A WENO reconstruction is used as a limiter for the RKDG method. The scheme is conservative for the mass, momentum and total energy. The scheme maintains second-order accuracy and has free parameters. Results of some numerical tests are presented to demonstrate the accuracy and the robustness of the scheme.

Keywords: cell-centered Lagrangian scheme, compressible Euler equations, RKDG method

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Authors: H. Triki, Y. Hamaizi, A. El-Akrmi

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We consider the higher order nonlinear Schrödinger equation model with fourth-order dispersion, cubic-quintic terms, and self-steepening. This equation governs the propagation of fem to second pulses in optical fibers. We present new bright and dark solitary wave type solutions for such a model under certain parametric conditions. This kind of solution may be useful to explain some physical phenomena related to wave propagation in a nonlinear optical fiber systems supporting high-order nonlinear and dispersive effects.

Keywords: nonlinear Schrödinger equation, high-order effects, soliton solution

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Authors: Sugeng Rianto, P.W. Arinto Yudi, Soemarno Muhammad Nurhuda

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A computation of a 3D compressible fluid flow for virtual environment with haptic interaction can be a non-trivial issue. This is especially how to reach good performances and balancing between visualization, tactile feedback interaction, and computations. In this paper, we describe our approach of computation methods based on parallel programming on a GPU. The 3D fluid flow solvers have been developed for smoke dispersion simulation by using combinations of the cubic interpolated propagation (CIP) based fluid flow solvers and the advantages of the parallelism and programmability of the GPU. The fluid flow solver is generated in the GPU-CPU message passing scheme to get rapid development of haptic feedback modes for fluid dynamic data. A rapid solution in fluid flow solvers is developed by applying cubic interpolated propagation (CIP) fluid flow solvers. From this scheme, multiphase fluid flow equations can be solved simultaneously. To get more acceleration in the computation, the Navier-Stoke Equations (NSEs) is packed into channels of texel, where computation models are performed on pixels that can be considered to be a grid of cells. Therefore, despite of the complexity of the obstacle geometry, processing on multiple vertices and pixels can be done simultaneously in parallel. The data are also shared in global memory for CPU to control the haptic in providing kinaesthetic interaction and felling. The results show that GPU based parallel computation approaches provide effective simulation of compressible fluid flow model for real-time interaction in 3D computer graphic for PC platform. This report has shown the feasibility of a new approach of solving the compressible fluid flow equations on the GPU. The experimental tests proved that the compressible fluid flowing on various obstacles with haptic interactions on the few model obstacles can be effectively and efficiently simulated on the reasonable frame rate with a realistic visualization. These results confirm that good performances and balancing between visualization, tactile feedback interaction, and computations can be applied successfully.

Keywords: CIP, compressible fluid, GPU programming, parallel computation, real-time visualisation

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Authors: Payel Das, Gnaneshwar Nelakanti

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In this paper we analyse the iterated discrete Legendre Galerkin method for Fredholm-Hammerstein integral equations with smooth kernel. Using sufficiently accurate numerical quadrature rule, we obtain superconvergence rates for the iterated discrete Legendre Galerkin solutions in both infinity and $L^2$-norm. Numerical examples are given to illustrate the theoretical results.

Keywords: hammerstein integral equations, spectral method, discrete galerkin, numerical quadrature, superconvergence

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Authors: Jan Bohata, Stanislav Zvanovec, Matej Komanec, Jakub Jaros, David Hruby

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Even though there is a rapid development in new optical networks, still optical communication infrastructures remain composed of thousands of kilometers of aging optical cables. Many of them are located in a harsh environment which contributes to an increased attenuation or induced birefringence of the fibers leading to the increase of polarization mode dispersion (PMD). In this paper, we report experimental results from environmental optical cable tests and characterization in the climate chamber. We focused on the evaluation of optical network reliability in a harsh environment. For this purpose, a special thermal chamber was adopted, targeting to the large temperature changes between -60 °C and 160 C° with defined humidity. Single mode optical cable 230 meters long, having six tubes and a total number of 72 single mode optical fibers was spliced together forming one fiber link, which was afterward tested in the climate chamber. The main emphasis was put to the polarization mode dispersion (PMD) changes, which were evaluated by three different PMD measuring methods (general interferometry technique, scrambled state-of-polarization analysis and polarization optical time domain reflectometer) in order to fully validate obtained results. Moreover, attenuation and chromatic dispersion (CD), as well as the PMD, were monitored using 17 km long single mode optical cable. Results imply a strong PMD dependence on thermal changes, imposing the exceeding 200 % of its value during the exposure to extreme temperatures and experienced more than 20 dB insertion losses in the optical system. The derived statistic is provided in the paper together with an evaluation of such as optical system reliability, which could be a crucial tool for the optical network designers. The environmental tests are further taken in context to our previously published results from long-term monitoring of fundamental parameters within an optical cable placed in a harsh environment in a special outdoor testbed. Finally, we provide a correlation between short-term and long-term monitoring campaigns and statistics, which are necessary for optical network safety and reliability.

Keywords: optical fiber, polarization mode dispersion, harsh environment, aging

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Authors: Ogunrinde Roseline Bosede

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This paper presents the development, analysis and implementation of an inverse polynomial numerical method which is well suitable for solving initial value problems in first order ordinary differential equations with applications to sample problems. We also present some basic concepts and fundamental theories which are vital to the analysis of the scheme. We analyzed the consistency, convergence, and stability properties of the scheme. Numerical experiments were carried out and the results compared with the theoretical or exact solution and the algorithm was later coded using MATLAB programming language.

Keywords: differential equations, numerical, polynomial, initial value problem, differential equation

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Authors: A. Panacek, M. Kilianova, R. Prucek, V. Husickova, R. Vecerova, M. Kolar, L. Kvitek, R. Zboril

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In this work we describe the preparation of NanoSilver/methylcellulose hydrogel containing silver nanoparticles (NPs) for topical bactericidal applications. Highly concentrated dispersion of silver NPs as high as of 5g/L of silver with diameter of 10nm was prepared by reduction of AgNO3 via strong reducing agent NaBH4. Silver NPs were stabilized by addition of sodium polyacrylate in order to prevent their aggregation at such high concentration. This way synthesized silver NPs were subsequently incorporated into methylcellulose suspension at elevated temperature resulting in formation of NanoSilver/methylcellulose hydrogel when temperature cooled down to laboratory conditions. In vitro antibacterial activity assay proved high bactericidal and fungicidal efficiency of silver NPs alone in the form of dispersion as well as in the form of hydrogel against broad spectrum of bacteria and yeasts including highly multiresistant strains such as methicillin-resistant Staphylococcus aureus. A very low concentrations of silver as low as 0.84mg/L Ag in as-prepared dispersion gave antibacterial performance. NanoSilver/methylcellulose hydrogel showed antibacterial action at the lowest used silver concentration equal to 25mg/L. Such prepared NanoSilver/methylcellulose hydrogel represent promising topical antimicrobial formulation for treatment of burns and wounds.

Keywords: antimicrobial, burn, hydrogel, silver NPs

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Authors: Mohammadreza Akbari, Sara Akbari, Davood Domiri Ganji, Pooya Solimani, Reza Khalili

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In this paper, three complicated nonlinear differential equations(PDE,ODE) in the field of engineering and non-vibration have been analyzed and solved completely by new method that we have named it Akbari-Ganji's Method (AGM) . As regards the previous published papers, investigating this kind of equations is a very hard task to do and the obtained solution is not accurate and reliable. This issue will be emerged after comparing the achieved solutions by Numerical Method. Based on the comparisons which have been made between the gained solutions by AGM and Numerical Method (Runge-Kutta 4th), it is possible to indicate that AGM can be successfully applied for various differential equations particularly for difficult ones. Furthermore, It is necessary to mention that a summary of the excellence of this method in comparison with the other approaches can be considered as follows: It is noteworthy that these results have been indicated that this approach is very effective and easy therefore it can be applied for other kinds of nonlinear equations, And also the reasons of selecting the mentioned method for solving differential equations in a wide variety of fields not only in vibrations but also in different fields of sciences such as fluid mechanics, solid mechanics, chemical engineering, etc. Therefore, a solution with high precision will be acquired. With regard to the afore-mentioned explanations, the process of solving nonlinear equation(s) will be very easy and convenient in comparison with the other methods. And also one of the important position that is explored in this paper is: Trigonometric and exponential terms in the differential equation (the method AGM) , is no need to use Taylor series Expansion to enhance the precision of the result.

Keywords: new method (AGM), complex non-linear partial differential equations, damping ratio, energy lost per cycle

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Authors: M. R. Akbari, S. Akbari, L. Abdollahpour

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The present paper is concerned with calculating the 2-dimensional velocity profile of a viscous flow for an incompressible fluid along the leading edge of a flat plate by using the continuity and motion equations with a simple and innovative approach. A Comparison between Numerical method and AGM has been made and the results have been revealed that AGM is very accurate and easy and can be applied for a wide variety of nonlinear problems. It is notable that most of the differential equations can be solved in this approach which in the other approaches they do not have this capability. Moreover, there are some valuable benefits in this method of solving differential equations, for instance: Without any dimensionless procedure, we can solve many differential equation(s), that is, differential equations are directly solvable by this method. In addition, it is not necessary to convert variables into new ones. According to the afore-mentioned expressions which will be proved in this literature, the process of solving nonlinear differential equation(s) will be very simple and convenient in contrast to the other approaches.

Keywords: leading edge, new idea, flat plate, incompressible fluid

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Authors: Ishapathik Das

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The purpose of this article is to find a method of comparing designs for ordinal regression models using quantile dispersion graphs in the presence of linear predictor misspecification. The true relationship between response variable and the corresponding control variables are usually unknown. Experimenter assumes certain form of the linear predictor of the ordinal regression models. The assumed form of the linear predictor may not be correct always. Thus, the maximum likelihood estimates (MLE) of the unknown parameters of the model may be biased due to misspecification of the linear predictor. In this article, the uncertainty in the linear predictor is represented by an unknown function. An algorithm is provided to estimate the unknown function at the design points where observations are available. The unknown function is estimated at all points in the design region using multivariate parametric kriging. The comparison of the designs are based on a scalar valued function of the mean squared error of prediction (MSEP) matrix, which incorporates both variance and bias of the prediction caused by the misspecification in the linear predictor. The designs are compared using quantile dispersion graphs approach. The graphs also visually depict the robustness of the designs on the changes in the parameter values. Numerical examples are presented to illustrate the proposed methodology.

Keywords: model misspecification, multivariate kriging, multivariate logistic link, ordinal response models, quantile dispersion graphs

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Authors: Hassan Manouzi

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We present in this paper a useful strategy to solve stochastic partial differential equations (SPDEs) involving stochastic coefficients. Using the Wick-product of higher order and the Wiener-Itˆo chaos expansion, the SPDEs is reformulated as a large system of deterministic partial differential equations. To reduce the computational complexity of this system, we shall use a decomposition-coordination method. To obtain the chaos coefficients in the corresponding deterministic equations, we use a least square formulation. Once this approximation is performed, the statistics of the numerical solution can be easily evaluated.

Keywords: least squares, wick product, SPDEs, finite element, wiener chaos expansion, gradient method

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Authors: M.S. El-Feky, Passant Youssef, Mohamed I. Serag

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nowadays, nanotechnology is the main trend of research in different areas due to the new potential of using nanometer materials sized less than 100nm. Nanomaterials are needed in cement composites to act as bridging for Nano and micro-cracks to increase tensile strength, reduce the permeability of gases and water in concrete to solve corrosion problem, react with excess Calcium Hydroxide, produce additional C-S-H, act as filler materials to densify the cement matrix and increase its mechanical properties. The present study focuses on the effectiveness of super-plasticizers and ultrasonic processing on the dispersion of Carbon Nanotube at first in water and then in cement composites in combination with Nano silica to enhance the mechanical properties of cement composites. A qualitative analysis using a compressive strength test is conducted with a view to investigate the influence of different dispersion techniques on the mechanical properties of cement composites containing Carbon Nanotube (CNT) and Nano Silica (NS) particles with different percentages. In addition, micro-structural analysis was carried out to understand the surface morphology and microstructure of cement composites with different dosages of NS addition. The investigational study results showed that the combination of NS with a low amount of CNT had a positive effect on the hydration reaction; on the other hand, the combination of CNT and a high amount of NS had a negative effect on the hydration reaction. The compressive strength can be improved by optimum combination 0.02% CNT and 1% NS with gain in strength by 72% and 35% after 7 and 28 days compared to control samples; these results were with an agreement with the morphology structure of composites using microstructure analysis.

Keywords: nano silica, dispersion, sonication, carbon nano tubes

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Authors: Elmo Thiago Lins Cöuras Ford, Valentina Alessandra Carvalho do Vale

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The simulation in wind tunnel is used thoroughly to model real situations of drainages of air. Besides the automotive industry, a great number of applications can be numbered: dispersion of pollutant, studies of pedestrians comfort and dispersion of particles. This work had the objective of visualizing the characteristics aerodynamics of two automobiles in different ways. To accomplish that drainage of air a fan that generated a speed exists (measured with anemometer of hot thread) of 4,1m/s and 4,95m/s. To visualize the path of the air through the cars, in the wind tunnel, smoke was used, obtained with it burns of vegetable oil. For “to do smoke” vegetable oil was used, that was burned for a tension of 20 V generated by a thread of 2,5 mm. The cars were placed inside of the wind tunnel with the drainage of “air-smoke” and photographed, registering like this the path lines around them, in the 3 different speeds.

Keywords: aerodynamics, vehicle drag, vegetable oil, wind tunnel

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Authors: Y. J. Zhou, G. Lu

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In this paper, an analytical study is made for the dynamic behavior of human brain tissue under transient loading. In this analytical model the Mooney-Rivlin constitutive law is coupled with visco-elastic constitutive equations to take into account both the nonlinear and time-dependent mechanical behavior of brain tissue. Five ordinary differential equations representing the relationships of five main parameters (radial stress, circumferential stress, radial strain, circumferential strain, and particle velocity) are obtained by using the characteristic method to transform five partial differential equations (two continuity equations, one motion equation, and two constitutive equations). Analytical expressions of the attenuation properties for spherical wave in brain tissue are analytically derived. Numerical results are obtained based on the five ordinary differential equations. The mechanical responses (particle velocity and stress) of brain are compared at different radii including 5, 6, 10, 15 and 25 mm under four different input conditions. The results illustrate that loading curves types of the particle velocity significantly influences the stress in brain tissue. The understanding of the influence by the input loading cures can be used to reduce the potentially injury to brain under head impact by designing protective structures to control the loading curves types.

Keywords: analytical method, mechanical responses, spherical wave propagation, traumatic brain injury

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Authors: Abdel-Maksoud Abdel-Kader Soliman

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The General Regularized Long Wave (GRLW) equation is solved numerically by giving a new algorithm based on collocation method using quartic B-splines at the mid-knot points as element shape. Also, we use the Fourth Runge-Kutta method for solving the system of first order ordinary differential equations instead of finite difference method. Our test problems, including the migration and interaction of solitary waves, are used to validate the algorithm which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm.

Keywords: generalized RLW equation, solitons, quartic b-spline, nonlinear partial differential equations, difference equations

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Authors: Saurabh Rawat, Anushree Sah

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K Maps are generally and ideally, thought to be simplest form for obtaining solution of Boolean equations. Cubical Representation of Boolean equations is an alternate pick to incur a solution, otherwise to be meted out with Truth Tables, Boolean Laws, and different traits of Karnaugh Maps. Largest possible k- cubes that exist for a given function are equivalent to its prime implicants. A technique of minimization of Logic functions is tried to be achieved through cubical methods. The main purpose is to make aware and utilise the advantages of cubical techniques in minimization of Logic functions. All this is done with an aim to achieve minimal cost solution.r

Keywords: K-maps, don’t care conditions, Boolean equations, cubes

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Authors: Maali Kachouri, Anis Jarboui

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We focus on the causal relationship between governance and information transparency as well as interrelation among the various governance mechanisms. This paper employs a simultaneous equations approach to show this relationship in the Tunisian context. Based on an 8-year dataset, our sample covers 28 listed companies over 2006-2013. Our findings suggest that internal and external governance mechanisms are interdependent. Moreover, in order to analyze the causal effect between information transparency and governance mechanisms, we found evidence that information transparency tends to increase good corporate governance practices.

Keywords: simultaneous equations approach, transparency, causal relationship, corporate governance

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Authors: M. Shaban, A. Alibeigloo

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This paper studies free vibration behavior of single-walled carbon nanotubes (SWCNTs) embedded on elastic medium based on three-dimensional theory of elasticity. To accounting the size effect of carbon nanotubes, nonlocal theory is adopted to shell model. The nonlocal parameter is incorporated into all constitutive equations in three dimensions. The surrounding medium is modeled as two-parameter elastic foundation. By using Fourier series expansion in axial and circumferential direction, the set of coupled governing equations are reduced to the ordinary differential equations in thickness direction. Then, the state-space method as an efficient and accurate method is used to solve the resulting equations analytically. Comprehensive parametric studies are carried out to show the influences of the nonlocal parameter, radial and shear elastic stiffness, thickness-to-radius ratio and radius-to-length ratio.

Keywords: carbon nanotubes, embedded, nonlocal, free vibration

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Authors: Olayiwola Moruf Oyedunsi

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This paper discusses the Modified Variational Iteration Method (MVIM) for the solution of nonlinear Burgers’ equation arising in longitudinal dispersion phenomena in fluid flow through porous media. The method is an elegant combination of Taylor’s series and the variational iteration method (VIM). Using Maple 18 for implementation, it is observed that the procedure provides rapidly convergent approximation with less computational efforts. The result shows that the concentration C(x,t) of the contaminated water decreases as distance x increases for the given time t.

Keywords: modified variational iteration method, Burger’s equation, porous media, partial differential equation

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Authors: Amit Kumar

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Accurate identification of deteriorated air quality regions is very helpful in devising better environmental practices and mitigation efforts. In the present study, an attempt has been made to identify the air pollutant dispersion patterns especially NOX due to vehicular and industrial sources over a rapidly developing urban city, Visakhapatnam (17°42’ N, 83°20’ E), India, during April 2009. Using the emission factors of different vehicles as well as the industry, a high resolution 1 km x 1 km gridded emission inventory has been developed for Visakhapatnam city. A dispersion model AERMOD with explicit representation of planetary boundary layer (PBL) dynamics and offline coupled through a developed coupler mechanism with a high resolution mesoscale model WRF-ARW resolution for simulating the dispersion patterns of NOX is used in the work. The meteorological as well as PBL parameters obtained by employing two PBL schemes viz., non-local Yonsei University (YSU) and local Mellor-Yamada-Janjic (MYJ) of WRF-ARW model, which are reasonably representing the boundary layer parameters are considered for integrating AERMOD. Significantly different dispersion patterns of NOX have been noticed between summer and winter months. The simulated NOX concentration is validated with available six monitoring stations of Central Pollution Control Board, India. Statistical analysis of model evaluated concentrations with the observations reveals that WRF-ARW of YSU scheme with AERMOD has shown better performance. The deteriorated air quality locations are identified over Visakhapatnam based on the validated model simulations of NOX concentrations. The present study advocates the utility of tNumerical Simulation of Air Pollutant Using Coupled AERMOD-WRF Modeling System over Visakhapatnam: A Case Studyhe developed gridded emission inventory of NOX with coupled WRF-AERMOD modeling system for air quality assessment over the study region.

Keywords: WRF-ARW, AERMOD, planetary boundary layer, air quality

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Authors: Arash Jafari, Mehdi Taghaddosi, Azin Parvin

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In the paper, our purpose is to enhance the ability to solve a nonlinear differential equation which is about the motion of an incompressible fluid flow going down of an inclined plane without thermal effect with a simple and innovative approach which we have named it new method. Comparisons are made amongst the Numerical, new method, and HPM methods, and the results reveal that this method is very effective and simple and can be applied to other nonlinear problems. It is noteworthy that there are some valuable advantages in this way of solving differential equations, and also most of the sets of differential equations can be answered in this manner which in the other methods they do not have acceptable solutions up to now. A summary of the excellence of this method in comparison to the other manners is as follows: 1) Differential equations are directly solvable by this method. 2) Without any dimensionless procedure, we can solve equation(s). 3) It is not necessary to convert variables into new ones. According to the afore-mentioned assertions which will be proved in this case study, the process of solving nonlinear equation(s) will be very easy and convenient in comparison to the other methods.

Keywords: viscos fluid, incompressible fluid flow, inclined plane, nonlinear phenomena

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Authors: S. Srednyak

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We study a class of functional partial differential equations(FPDEs). This class is suggested by Quantum Field Theory. We derive general properties of solutions to such equations. In particular, we demonstrate that they lead to systems of coupled integral equations with singular kernels. We show that solutions to such hierarchies can be sought among functions with regular singularities at a countable set of subvarieties of the physical space. We also develop a formal analogy of basic constructions of differential geometry on functional manifolds, as this is necessary for in depth study of FPDEs. We also consider the case of linear overdetermined systems of functional differential equations and show that it can be completely solved in terms of formal solutions of a functional equation that is a functional analogy of a system of determined algebraic equations. This development leads us to formally define the functional analogy of algebraic geometry, which we call functional algebraic geometry. We study basic properties of functional algebraic varieties. In particular, we investigate the case of a formally discrete set of solutions. We also define and study functional analogy of discriminants. In the case of fully determined systems such that the defining functionals have regular singularities, we demonstrate that formal solutions can be sought in the class of functions with regular singularities. This case provides a practical way to apply our results to physics problems.

Keywords: functional equations, quantum field theory, holomorphic functions, Yang Mills mass gap problem, quantum chaos

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Authors: M. J. Gibbons, A. J. Robinson

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Increasing expectations on small form factor electronics to be more compact while increasing performance has driven conventional cooling technologies to a thermal management threshold. An emerging solution to this problem is electrospray (ES) cooling. ES cooling enables two phase cooling by utilising Coulomb forces for energy efficient fluid atomization. Generated charged droplets are accelerated to the grounded target surface by the applied electric field and surrounding gravitational force. While in transit the like charged droplets enable plume dispersion and inhibit droplet coalescence. If the electric field is increased in the cone-jet regime, a subsequent increase in the plume spray angle has been shown. Droplet segregation in the spray plume has been observed, with primary droplets in the plume core and satellite droplets positioned on the periphery of the plume. This segregation is facilitated by inertial and electrostatic effects. This result has been corroborated by numerous authors. These satellite droplets are usually more densely charged and move at a lower relative velocity to that of the spray core due to the radial decay of the electric field. Previous experimental research by Gomez and Tang has shown that the number of droplets deposited on the periphery can be up to twice that of the spray core. This result has been substantiated by a numerical models derived by Wilhelm et al., Oh et al. and Yang et al. Yang et al. showed from their numerical model, that by varying the extractor potential the dispersion radius of the plume also varies proportionally. This research aims to investigate this dispersion density and the role it plays in the local heat transfer coefficient profile (h) of ES cooling. This will be carried out for different extractor – target separation heights (H2), working fluid flow rates (Q), and extractor applied potential (V2). The plume dispersion will be recorded by spraying a 25 µm thick, joule heated steel foil and by recording the thermal footprint of the ES plume using a Flir A-40 thermal imaging camera. The recorded results will then be analysed by in-house developed MATLAB code.

Keywords: electronic cooling, electrospray, electrospray plume dispersion, spray cooling

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Authors: Y. M. Aiyesimi, S. O. Abah, G. T. Okedayo

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A general analysis has been developed to study the chemical reaction effects on unsteady MHD double-diffusive free convective flow over a vertical stretching plate. The governing nonlinear partial differential equations have been reduced to the coupled nonlinear ordinary differential equations by the similarity transformations. The resulting equations are solved numerically by using Runge-Kutta shooting technique. The effects of the chemical parameters are examined on the velocity, temperature and concentration profiles.

Keywords: chemical reaction, MHD, double-diffusive, stretching plate

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Authors: Haniye Dehestani, Yadollah Ordokhani

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In this work, we present an efficient approach for solving variable-order time-fractional partial differential equations, which are based on Legendre and Laguerre polynomials. First, we introduced the pseudo-operational matrices of integer and variable fractional order of integration by use of some properties of Riemann-Liouville fractional integral. Then, applied together with collocation method and Legendre-Laguerre functions for solving variable-order time-fractional partial differential equations. Also, an estimation of the error is presented. At last, we investigate numerical examples which arise in physics to demonstrate the accuracy of the present method. In comparison results obtained by the present method with the exact solution and the other methods reveals that the method is very effective.

Keywords: collocation method, fractional partial differential equations, legendre-laguerre functions, pseudo-operational matrix of integration

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Authors: Daniil Karzanov

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This paper studies the effect of quarterly earnings reports on the stock price. The profitability of the stock is modeled by geometric Brownian diffusion and the Constant Elasticity of Variance model. We fit several variations of stochastic differential equations to the pre-and after-report period using the Maximum Likelihood Estimation and Grid Search of parameters method. By examining the change in the model parameters after reports’ publication, the study reveals that the reports have enough evidence to be a structural breakpoint, meaning that all the forecast models exploited are not applicable for forecasting and should be refitted shortly.

Keywords: stock market, earnings reports, financial time series, structural breaks, stochastic differential equations

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