Search results for: exact solutions of the Einstein equations

Authors: Safia Akram

Abstract:

The influence of nanofluid with different waveforms in the presence of inclined asymmetric channel on peristaltic transport of a pseudoplastic fluid is examined. The governing equations for two-dimensional and two directional flows of a pseudoplastic fluid along with nanofluid are modeled and then simplified under the assumptions of long wavelength and low Reynolds number approximation. The exact solutions for temperature and nanoparticle volume fraction are calculated. Series solution of the stream function and pressure gradient are carried out using perturbation technique. The flow quantities have been examined for various physical parameters of interest. It was found, that the magnitude value of the velocity profile decreases with an increase in volume flow rate (Q) and relaxation times (ζ) and increases in sinusoidal, multisinusoidal, trapezoidal and triangular waves. It was also observed that the size of the trapping bolus decreases with the drop of the width of the channel ‘d’ and increases with a rise of relaxation times ζ.

Keywords: nanofluid particles, peristaltic flow, pseudoplastic fluid, different waveforms, inclined asymmetric channel

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Authors: Taha Zakaraia Abdel Wahid

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In the present study, a development of the papers is introduced. The behavior of the unsteady non-equilibrium distribution functions for a rarefied gas mixture under the effect of non-linear thermal radiation field is presented. For the best of our knowledge this is done for the first time at all. The distinction and comparisons between the unsteady perturbed and the unsteady equilibrium velocity distribution functions are illustrated. The equilibrium time for the rarefied gas mixture is determined for the first time. The non-equilibrium thermodynamic properties of the system is investigated. The results are applied to the Argon-Neon binary gas mixture, for various values of both of molar fraction parameters and radiation field intensity. 3D-Graphics illustrating the calculated variables are drawn to predict their behavior and the results are discussed.

Keywords: radiation field, binary gas mixture, exact solutions, travelling wave method, unsteady BGK model, irreversible thermodynamics

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Authors: Edris Rawashdeh, I. Abu-Falahah, H. M. Jaradat

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The variable-coefficient non-linear dispersive wave equation is investigated with the aid of symbolic computation. By virtue of a newly developed simplified bilinear method, multi-soliton solutions for such an equation have been derived. Effects of the inhomogeneities of media and nonuniformities of boundaries, depicted by the variable coefficients, on the soliton behavior are discussed with the aid of the characteristic curve method and graphical analysis.

Keywords: dispersive wave equations, multiple soliton solution, Hirota Bilinear Method, symbolic computation

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Authors: Gleda Kutrolli, Maksi Kutrolli, Etjon Meco

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SARS-CoV-2 virus is currently one of the most infectious pathogens for humans. It started in China at the end of 2019 and now it is spread in all over the world. The origin and diffusion of the SARS-CoV-2 epidemic, is analysed based on the discussion of viral phylogeny theory. With the aim of understanding the spread of infection in the affected countries, it is crucial to modelize the spread of the virus and simulate its activity. In this paper, the prediction of coronavirus outbreak is done by using SIR model without vital dynamics, applying different numerical technique solving ordinary differential equations (ODEs). We find out that ABM and MRT methods perform better than other techniques and that the activity of the virus will decrease in April but it never cease (for some time the activity will remain low) and the next cycle will start in the middle July 2020 for Norway and Denmark, and October 2020 for Sweden, and September for Finland.

Keywords: forecasting, ordinary differential equations, SARS-COV-2 epidemic, SIR model

Procedia PDF Downloads 115

Authors: Mohammadamin Esmaeilzadehazimi, Morteza Shayan Arani, Mohammad Toorani, Aouni Lakis

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This study uses a hybrid finite element method to predict the dynamic behavior of both fully and partially-filled truncated conical shells stiffened with ring stiffeners. The method combines classical shell theory and the finite element method, and employs displacement functions derived from exact solutions of Sanders' shell equilibrium equations for conical shells. The shell-fluid interface is analyzed by utilizing the velocity potential, Bernoulli's equation, and impermeability conditions to determine an explicit expression for fluid pressure. The equations of motion presented in this study apply to both conical and cylindrical shells. This study presents the first comparison of the method applied to ring-stiffened shells with other numerical and experimental findings. Vibration frequencies for conical shells with various boundary conditions and geometries in a vacuum and filled with water are compared with experimental and numerical investigations, achieving good agreement. The study thoroughly investigates the influence of geometric parameters, stiffener quantity, semi-vertex cone angle, level of water filled in the cone, and applied boundary conditions on the natural frequency of fluid-loaded ring-stiffened conical shells, and draws some useful conclusions. The primary advantage of the current method is its use of a minimal number of finite elements while achieving highly accurate results.

Keywords: finite element method, fluid–structure interaction, conical shell, natural frequency, ring-stiffener

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Authors: A. Khernane, N. Khelil, L. Djerou

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The aim of this work is to study the numerical implementation of the Hilbert uniqueness method for the exact boundary controllability of Euler-Bernoulli beam equation. This study may be difficult. This will depend on the problem under consideration (geometry, control, and dimension) and the numerical method used. Knowledge of the asymptotic behaviour of the control governing the system at time T may be useful for its calculation. This idea will be developed in this study. We have characterized as a first step the solution by a minimization principle and proposed secondly a method for its resolution to approximate the control steering the considered system to rest at time T.

Keywords: boundary control, exact controllability, finite difference methods, functional optimization

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Authors: Ti-Jun Xiao, Zhe Xu

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Controllability is one of the fundamental issues in control systems. In this paper, we study the controllability of second order evolutional control systems in Hilbert spaces with memory and boundary controls, which model dynamic behaviors of some viscoelastic materials. Transferring the control problem into a moment problem and showing the Riesz property of a family of functions related to Cauchy problems for some integrodifferential equations, we obtain a general boundary controllability theorem for these second order evolutional control systems. This controllability theorem is applicable to various concrete 1D viscoelastic systems and recovers some previous related results. It is worth noting that Riesz sequences can be used for numerical computations of the control functions and the identification of new Riesz sequence is of independent interest for the basis-function theory. Moreover, using the Riesz sequences, we obtain the existence and uniqueness of (weak) solutions to these second order evolutional control systems in Hilbert spaces. Finally, we derive the exact boundary controllability of a viscoelastic beam equation, as an application of our abstract theorem.

Keywords: evolutional control system, controllability, boundary control, existence and uniqueness

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Authors: Z. B. Ibrahim, N. Ismail, K. I. Othman

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Fuzzy Differential Equations (FDEs) play an important role in modelling many real life phenomena. The FDEs are used to model the behaviour of the problems that are subjected to uncertainty, vague or imprecise information that constantly arise in mathematical models in various branches of science and engineering. These uncertainties have to be taken into account in order to obtain a more realistic model and many of these models are often difficult and sometimes impossible to obtain the analytic solutions. Thus, many authors have attempted to extend or modified the existing numerical methods developed for solving Ordinary Differential Equations (ODEs) into fuzzy version in order to suit for solving the FDEs. Therefore, in this paper, we proposed the development of a fuzzy version of three-point block method based on Block Backward Differentiation Formulas (FBBDF) for the numerical solution of first order FDEs. The three-point block FBBDF method are implemented in uniform step size produces three new approximations simultaneously at each integration step using the same back values. Newton iteration of the FBBDF is formulated and the implementation is based on the predictor and corrector formulas in the PECE mode. For greater efficiency of the block method, the coefficients of the FBBDF are stored at the start of the program. The proposed FBBDF is validated through numerical results on some standard problems found in the literature and comparisons are made with the existing fuzzy version of the Modified Simpson and Euler methods in terms of the accuracy of the approximated solutions. The numerical results show that the FBBDF method performs better in terms of accuracy when compared to the Euler method when solving the FDEs.

Keywords: block, backward differentiation formulas, first order, fuzzy differential equations

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Authors: Kushakov Kholmurodjon, Muhammadjonov Akbarshox

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This present paper is devoted to a system of second-order nonlinear differential equations with a special right-hand side, exactly, the linear part and a third-order polynomial of a special form. It is shown that for some relations between the parameters, there is a second-order curve in which trajectories leaving the points of this curve remain in the same place. Thus, the curve is invariant with respect to the given system. Moreover, this system is invariant under a non-degenerate linear transformation of variables. The form of this curve, depending on the relations between the parameters and the eigenvalues of the matrix, is proved. All solutions of this system of differential equations are shown analytically.

Keywords: dynamic system, ellipse, hyperbola, Hess system, polar coordinate system

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Authors: A. I. Ma’ali, R. B. Adeniyi, A. Y. Badeggi, U. Mohammed

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An error estimation of the integrated formulation of the Lanczos tau method for some class of ordinary differential equations was reported. This paper is concern with the generalization of tau approximants and their corresponding error estimates for some class of ordinary differential equations (ODEs) characterized by m + s =3 (i.e for m =1, s=2; m=2, s=1; and m=3, s=0) where m and s are the order of differential equations and number of overdetermination, respectively. The general result obtained were validated with some numerical examples.

Keywords: approximant, error estimate, tau method, overdetermination

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Authors: Natarajan Tirupattur Srinivasan

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Quantum mechanics( QM) and (special) relativity (SR) have indeed revolutionized the very thinking of physicists, and the spectacular successes achieved over a century due to these two theories are mind-boggling. However, there is still a strong disquiet among some physicists. While the mathematical structure of these two theories has been established beyond any doubt, their physical interpretations are still being contested by many. Even after a hundred years of their existence, we cannot answer a very simple question, “What is an electron”? Physicists are struggling even now to come to grips with the different interpretations of quantum mechanics with all their ramifications. However, it is indeed strange that the (special) relativity theory of Einstein enjoys many orders of magnitude of “acceptance”, though both theories have their own stocks of weirdness in the results, like time dilation, mass increase with velocity, the collapse of the wave function, quantum jump, tunnelling, etc. Here, in this paper, it would be shown that by postulating an intrinsic internal motion to these enigmatic electrons, one can build a fairly consistent picture of reality, revealing a very simple picture of nature. This is also evidenced by Schrodinger’s ‘Zitterbewegung’ motion, about which so much has been written. This leads to a helical trajectory of electrons when they move in a laboratory frame. It will be shown that the helix is a three-dimensional wave having all the characteristics of our familiar 2D wave. Again, the helix, being a geodesic on an imaginary cylinder, supports ‘quantization’, and its representation is just the complex exponentials matching with the wave function of quantum mechanics. By postulating the instantaneous velocity of the electrons to be always ‘c’, the velocity of light, the entire relativity comes alive, and we can interpret the ‘time dilation’, ‘mass increase with velocity’, etc., in a very simple way. Thus, this model unifies both QM and SR without the need for a counterintuitive postulate of Einstein about the constancy of the velocity of light for all inertial observers. After all, if the motion of an inertial frame cannot affect the velocity of light, the converse that this constant also cannot affect the events in the frame must be true. But entire relativity is about how ‘c’ affects time, length, mass, etc., in different frames.

Keywords: quantum reconstruction, special theory of relativity, quantum mechanics, zitterbewegung, complex wave function, helix, geodesic, Schrodinger’s wave equations

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Authors: Adamu S. Salawu, Ibrahim O. Isah

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Direct solution to several forms of fourth-order ordinary differential equations is not easily obtained without first reducing them to a system of first-order equations. Thus, numerical methods are being developed with the underlying techniques in the literature, which seeks to approximate some classes of fourth-order initial value problems with admissible error bounds. Multistep methods present a great advantage of the ease of implementation but with a setback of several functions evaluation for every stage of implementation. However, hybrid methods conventionally show a slightly higher order of truncation for any k-step linear multistep method, with the possibility of obtaining solutions at off mesh points within the interval of solution. In the light of the foregoing, we propose the continuous form of a hybrid multistep method with Chebyshev polynomial as a basis function for the numerical integration of fourth-order initial value problems of ordinary differential equations. The basis function is interpolated and collocated at some points on the interval [0, 2] to yield a system of equations, which is solved to obtain the unknowns of the approximating polynomial. The continuous form obtained, its first and second derivatives are evaluated at carefully chosen points to obtain the proposed block method needed to directly approximate fourth-order initial value problems. The method is analyzed for convergence. Implementation of the method is done by conducting numerical experiments on some test problems. The outcome of the implementation of the method suggests that the method performs well on problems with oscillatory or trigonometric terms since the approximations at several points on the solution domain did not deviate too far from the theoretical solutions. The method also shows better performance compared with an existing hybrid method when implemented on a larger interval of solution.

Keywords: Chebyshev polynomial, collocation, hybrid multistep method, initial value problems, interpolation

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Authors: Divij Gupta

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Pendular systems have a range of both mathematical and engineering applications, ranging from modelling the behaviour of a continuous mass-density rope to utilisation as Tuned Mass Dampers (TMD). Thus, it is of interest to study the differential equations governing the motion of such systems. Here we attempt to generalise these equations of motion for the plane compound pendulum with a finite number of N point masses. A Lagrangian approach is taken, and we attempt to find the generalised form for the Euler-Lagrange equations of motion for the i-th bob of the N -bob pendulum. The co-ordinates are parameterized as angular quantities to reduce the number of degrees of freedom from 2N to N to simplify the form of the equations. We analyse the form of these equations up to N = 4 to determine the general form of the equation. We also develop a MATLAB program to compute a solution to the system for a given input value of N and a given set of initial conditions.

Keywords: classical mechanics, differential equation, lagrangian analysis, pendulum

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Authors: Korosh Khorshidi, Mohammad Khodadadi

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In this paper, exact close form solution for out of plate free flexural vibration of moderately thick rectangular nanoplates are presented based on nonlocal modified trigonometric shear deformation theory, with assumptions of the Levy's type boundary conditions, for the first time. The aim of this study is to evaluate the effect of small-scale parameters on the frequency parameters of the moderately thick rectangular nano-plates. To describe the effects of small-scale parameters on vibrations of rectangular nanoplates, the Eringen theory is used. The Levy's type boundary conditions are combination of six different boundary conditions; specifically, two opposite edges are simply supported and any of the other two edges can be simply supported, clamped or free. Governing equations of motion and boundary conditions of the plate are derived by using the Hamilton’s principle. The present analytical solution can be obtained with any required accuracy and can be used as benchmark. Numerical results are presented to illustrate the effectiveness of the proposed method compared to other methods reported in the literature. Finally, the effect of boundary conditions, aspect ratios, small scale parameter and thickness ratios on nondimensional natural frequency parameters and frequency ratios are examined and discussed in detail.

Keywords: exact solution, nonlocal modified sinusoidal shear deformation theory, out of plane vibration, moderately thick rectangular plate

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Authors: Y. M. Aiyesimi, G. T. Okedayo, O. W. Lawal

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The analysis has been carried out to study unsteady MHD thin film flow of a third grade fluid down an inclined plane with heat transfer when the slippage between the surface of plane and the lower surface of the fluid is valid. The governing nonlinear partial differential equations involved are reduced to linear partial differential equations using regular perturbation method. The resulting equations were solved analytically using method of separation of variable and eigenfunctions expansion. The solutions obtained were examined and discussed graphically. It is interesting to find that the variation of the velocity and temperature profile with the slip and magnetic field parameter depends on time.

Keywords: non-Newtonian fluid, MHD flow, thin film flow, third grade fluid, slip boundary condition, heat transfer, separation of variable, eigenfunction expansion

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Authors: Mohamed Suleiman, Zarina Bibi Ibrahim, Nor Ain Azeany, Khairil Iskandar Othman

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In this paper, a class of variable order fully implicit multistep Block Backward Differentiation Formulas (VOBBDF) using uniform step size for the numerical solution of stiff ordinary differential equations (ODEs) is developed. The code will combine three multistep block methods of order four, five and six. The order selection is based on approximation of the local errors with specific tolerance. These methods are constructed to produce two approximate solutions simultaneously at each iteration in order to further increase the efficiency. The proposed VOBBDF is validated through numerical results on some standard problems found in the literature and comparisons are made with single order Block Backward Differentiation Formula (BBDF). Numerical results shows the advantage of using VOBBDF for solving ODEs.

Keywords: block backward differentiation formulas, uniform step size, ordinary differential equations

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Authors: Harendra Singh, Rajesh Pandey

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The paper presents a numerical method based on operational matrix of integration and Ryleigh method for the solution of a class of non-linear fractional variational problems (NLFVPs). Chebyshev first kind polynomials are used for the construction of operational matrix. Using operational matrix and Ryleigh method the NLFVP is converted into a system of non-linear algebraic equations, and solving these equations we obtained approximate solution for NLFVPs. Convergence analysis of the proposed method is provided. Numerical experiment is done to show the applicability of the proposed numerical method. The obtained numerical results are compared with exact solution and solution obtained from Chebyshev third kind. Further the results are shown graphically for different fractional order involved in the problems.

Keywords: non-linear fractional variational problems, Rayleigh-Ritz method, convergence analysis, error analysis

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Authors: Gyo Woo Lee, Man Young Kim

Abstract:

The blocked-off formulations for the analysis of radiative heat transfer are formulated and examined in order to find the solutions in a two-dimensional complex enclosure. The final discretization equations using the step scheme for spatial differencing practice are proposed with the additional source term to incorporate the blocked-off procedure. After introducing the implementation for inactive region into the general discretization equation, three different problems are examined to find the performance of the solution methods.

Keywords: radiative heat transfer, Finite Volume Method (FVM), blocked-off solution procedure, body-fitted coordinate

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Authors: Kamel Al-Khaled

Abstract:

A comparative study of the Sinc-Galerkin and Sinc-Collocation methods for solving the Kuramoto-Sivashinsky equation is given. Both approaches depend on using Sinc basis functions. Firstly, a numerical scheme using Sinc-Galerkin method is developed to approximate the solution of Kuramoto-Sivashinsky equation. Sinc approximations to both derivatives and indefinite integrals reduces the solution to an explicit system of algebraic equations. The error in the solution is shown to converge to the exact solution at an exponential. The convergence proof of the solution for the discrete system is given using fixed-point iteration. Secondly, a combination of a Crank-Nicolson formula in the time direction, with the Sinc-collocation in the space direction is presented, where the derivatives in the space variable are replaced by the necessary matrices to produce a system of algebraic equations. The methods are tested on two examples. The demonstrated results show that both of the presented methods more or less have the same accuracy.

Keywords: Sinc-Collocation, nonlinear PDEs, numerical methods, fixed-point

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Authors: Somveer Singh, Vineet Kumar Singh

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We present a new model based on viscoelasticity for the Non-Newtonian fluids.We use a matrix formulated algorithm to approximate solutions of a class of partial integro-differential equations with the given initial and boundary conditions. Some numerical results are presented to simplify application of operational matrix formulation and reduce the computational cost. Convergence analysis, error estimation and numerical stability of the method are also investigated. Finally, some test examples are given to demonstrate accuracy and efficiency of the proposed method.

Keywords: Legendre Wavelets, operational matrices, partial integro-differential equation, viscoelasticity

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Authors: Mohamed Ouzzane, Mahmoud Bady

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Due to energy and environment context, research is looking for the use of clean and energy efficient system in cooling industry. In this regard, the ejector represents one of the promising solutions. The thermal ejector is a passive component used for thermal compression in refrigeration and cooling systems, usually activated by heat either waste or solar. The present study introduces a theoretical analysis of the cooling system which uses a gas ejector thermal compression. A theoretical model is developed and applied for the design and simulation of the ejector, as well as the whole cooling system. Besides the conservation equations of mass, energy and momentum, the gas dynamic equations, state equations, isentropic relations as well as some appropriate assumptions are applied to simulate the flow and mixing in the ejector. This model coupled with the equations of the other components (condenser, evaporator, pump, and generator) is used to analyze profiles of pressure and velocity (Mach number), as well as evaluation of the cycle cooling capacity. A FORTRAN program is developed to carry out the investigation. Properties of refrigerant R134a are calculated using real gas equations. Among many parameters, it is thought that the generator pressure is the cornerstone in the cycle, and hence considered as the key parameter in this investigation. Results show that the generator pressure has a great effect on the ejector and on the whole cooling system. At high generator pressures, strong shock waves inside the ejector are created, which lead to significant condenser pressure at the ejector exit. Additionally, at higher generator pressures, the designed system can deliver cooling capacity for high condensing pressure (hot season).

Keywords: air cooling system, refrigeration, thermal ejector, thermal compression

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Authors: Ranjit Biswas

Abstract:

The existing subject Algebra is incomplete to support mathematical computations being done by scientists of all areas: Mathematics, Physics, Statistics, Chemistry, Space Science, Cosmology etc. even starting from the era of great Einstein. A huge hidden gap in the subject ‘Algebra’ is unearthed. All the scientists today, including mathematicians, physicists, chemists, statisticians, cosmologists, space scientists, and economists, even starting from the great Einstein, are lucky that they got results without facing any contradictions or without facing computational errors. Most surprising is that the results of all scientists, including Nobel Prize winners, were proved by them by doing experiments too. But in this paper, it is rigorously justified that they all are lucky. An algebraist can define an infinite number of new algebraic structures. The objective of the work in this paper is not just for the sake of defining a distinct algebraic structure, but to recognize and identify a major gap of the subject ‘Algebra’ lying hidden so far in the existing vast literature of it. The objective of this work is to fix the unearthed gap. Consequently, a different algebraic structure called ‘Region’ has been introduced, and its properties are studied.

Keywords: region, ROR, RORR, region algebra

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Authors: Sofiane Grira, Hichem Eleuch

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We explore the analytical soliton-pair solutions for unbalanced coupling between the two coherent lights and the atomic transitions in a dissipative three-level system in lambda configuration. The two allowed atomic transitions are interacting resonantly with two laser fields. For unbalanced coupling, it is possible to derive an explicit solution for non-linear differential equations describing the soliton-pair propagation in this three-level system with the same velocity. We suppose that the spontaneous emission rates from the excited state to both ground states are the same. In this work, we focus on such case where we consider the coupling between the transitions and the optical fields are unbalanced. The existence conditions for the soliton-pair propagations are determined. We will show that there are four possible configurations of the soliton-pair pulses. Two of them can be interpreted as a couple of solitons with same directions of polarization and the other two as soliton-pair with opposite directions of polarization. Due to the fact that solitons have stable shapes while propagating in the considered media, they are insensitive to noise and dispersion. Our results have potential applications in data transfer with the soliton-pair pulses, where a dissipative three-level medium could be a realistic model for the optical communication media.

Keywords: non-linear differential equations, solitons, wave propagations, optical fiber

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Authors: Prathmesh Vikas Patil, Yashaswini Lomte Patil

Abstract:

Laminated composite materials have considerable attention in various engineering applications due to their exceptional strength-to-weight ratio and mechanical properties. The analysis of laminated composite plates in three-dimensional (3D) elasticity is a complex problem, as it requires accounting for the orthotropic anisotropic nature of the material and the interactions between multiple layers. Conventional approaches, such as the classical plate theory, provide simplified solutions but are limited in performing exact analysis of the plate. To address such a challenge, the state space method emerges as a powerful numerical technique for modeling the behavior of laminated composites in 3D. The state-space method involves transforming the governing equations of elasticity into a state-space representation, enabling the analysis of complex structural systems in a systematic manner. Here, an effort is made to perform a 3D elasticity analysis of plates with cross-ply and angle-ply laminates using the state space approach. The state space approach is used in this study as it is a mixed formulation technique that gives the displacements and stresses simultaneously with the same level of accuracy.

Keywords: cross ply laminates, angle ply laminates, state space method, three-dimensional elasticity analysis

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Authors: Jyh-Yang Wu, Sheng-Gwo Chen

Abstract:

In this paper, we present a simple effective numerical geometric method to estimate the divergence of a vector field over a curved surface. The conservation law is an important principle in physics and mathematics. However, many well-known numerical methods for solving diffusion equations do not obey conservation laws. Our presented method in this paper combines the divergence theorem with a generalized finite difference method and obeys the conservation law on discrete closed surfaces. We use the similar method to solve the Cahn-Hilliard equations on evolving spherical surfaces and observe stability results in our numerical simulations.

Keywords: conservation laws, diffusion equations, Cahn-Hilliard equations, evolving surfaces

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Authors: Cagri Memis, Muzaffer Kapanoglu

Abstract:

The school bus routing problem (SBRP) is a variant of the Vehicle Routing Problem (VRP) classified as a location-allocation-routing problem. In this study, the SBRP is decomposed into two sub-problems: (1) bus route generation and (2) bus stop selection to solve large instances of the SBRP in reasonable computational times. To solve the first sub-problem, we propose a genetic algorithm to generate bus routes. Once the routes have been fixed, a sub-problem remains of allocating students to stops considering the capacity of the buses and the walkability constraints of the students. While the exact method solves small-scale problems, treating large-scale problems with the exact method becomes complex due to computational problems, a deficiency that the genetic algorithm can overcome. Results obtained from the proposed approach on 150 instances up to 250 stops show that the matheuristic algorithm provides better solutions in reasonable computational times with respect to benchmark algorithms.

Keywords: genetic algorithm, matheuristic, school bus routing problem, vehicle routing problem

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Authors: F. Maass, P. Martin, J. Olivares

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The aim of this work is the application of the program GeoGebra in teaching the study of nonhomogeneous linear second order differential equations with constant coefficients. Different kind of functions or forces will be considered in the right hand side of the differential equations, in particular, the emphasis will be placed in the case of trigonometrical functions producing the resonance phenomena. In order to obtain this, the frequencies of the trigonometrical functions will be changed. Once the resonances appear, these have to be correlationated with the roots of the second order algebraic equation determined by the coefficients of the differential equation. In this way, the physics and engineering students will understand resonance effects and its consequences in the simplest way. A large variety of examples will be shown, using different kind of functions for the nonhomogeneous part of the differential equations.

Keywords: education, geogebra, ordinary differential equations, resonance

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Authors: Ali Ateş, Ansar B. Mwimbo, Ali H. Abdulkarim

Abstract:

General transport equation has a wide range of application in Fluid Mechanics and Heat Transfer problems. In this equation, generally when φ variable which represents a flow property is used to represent fluid velocity component, general transport equation turns into momentum equations or with its well known name Navier-Stokes equations. In these non-linear differential equations instead of seeking for analytic solutions, preferring numerical solutions is a more frequently used procedure. Finite difference method is a commonly used numerical solution method. In these equations using velocity and pressure gradients instead of stress tensors decreases the number of unknowns. Also, continuity equation, by integrating the system, number of equations is obtained as number of unknowns. In this situation, velocity and pressure components emerge as two important parameters. In the solution of differential equation system, velocities and pressures must be solved together. However, in the considered grid system, when pressure and velocity values are jointly solved for the same nodal points some problems confront us. To overcome this problem, using staggered grid system is a referred solution method. For the computerized solutions of the staggered grid system various algorithms were developed. From these, two most commonly used are SIMPLE and SIMPLER algorithms. In this study Navier-Stokes equations were numerically solved for Newtonian flow, whose mass or gravitational forces were neglected, for incompressible and laminar fluid, as a hydro dynamically fully developed region and in two dimensional cartesian coordinate system. Finite difference method was chosen as the solution method. This is a parametric study in which varying values of velocity components, pressure and Reynolds numbers were used. Differential equations were discritized using central difference and hybrid scheme. The discritized equation system was solved by Gauss-Siedel iteration method. SIMPLE and SIMPLER were used as solution algorithms. The obtained results, were compared for central difference and hybrid as discritization methods. Also, as solution algorithm, SIMPLE algorithm and SIMPLER algorithm were compared to each other. As a result, it was observed that hybrid discritization method gave better results over a larger area. Furthermore, as computer solution algorithm, besides some disadvantages, it can be said that SIMPLER algorithm is more practical and gave result in short time. For this study, a code was developed in DELPHI programming language. The values obtained in a computer program were converted into graphs and discussed. During sketching, the quality of the graph was increased by adding intermediate values to the obtained result values using Lagrange interpolation formula. For the solution of the system, number of grid and node was found as an estimated. At the same time, to indicate that the obtained results are satisfactory enough, by doing independent analysis from the grid (GCI analysis) for coarse, medium and fine grid system solution domain was obtained. It was observed that when graphs and program outputs were compared with similar studies highly satisfactory results were achieved.

Keywords: finite difference method, GCI analysis, numerical solution of the Navier-Stokes equations, SIMPLE and SIMPLER algoritms

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Authors: Getachew T. Sedebo, Stephan V. Joubert, Michael Y. Shatalov

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Conventional configuration of the vibratory disc gyroscope is based on in-plane non-axisymmetric vibrations of the disc with a prescribed circumferential wave number. Due to the Bryan's effect, the vibrating pattern of the disc becomes sensitive to the axial component of inertial rotation of the disc. Rotation of the vibrating pattern relative to the disc is proportional to the inertial angular rate and is measured by sensors. In the present paper, the authors investigate a possibility of making a 3D sensor on the basis of both in-plane and bending vibrations of the disc resonator. We derive equations of motion for the disc vibratory gyroscope, where both in-plane and bending vibrations are considered. Hamiltonian variational principle is used in setting up equations of motion and the corresponding boundary conditions. The theory of thin shells with the linear elasticity principles is used in formulating the problem and also the disc is assumed to be isotropic and obeys Hooke's Law. The governing equation for a specific mode is converted to an ODE to determine the eigenfunction. The resulting ODE has exact solution as a linear combination of Bessel and Neumann functions. We demonstrate how to obtain an explicit solution and hence the eigenvalues and corresponding eigenfunctions for annular disc with fixed inner boundary and free outer boundary. Finally, the characteristics equations are obtained and the corresponding eigenvalues are calculated. The eigenvalues are used for the calculation of tuning conditions of the 3D disc vibratory gyroscope.

Keywords: Bryan’s effect, bending vibrations, disc gyroscope, eigenfunctions, eigenvalues, tuning conditions

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Authors: DucTho Le, Duy Kien Dao, Quoc Tinh Bui, Haidang Phan

Abstract:

The article is concerned with the motion of ultrasonic guided waves in a unidirectional fiber-reinforced composite plate under acoustic sources. Such unidirectional composite material has orthotropic elastic properties as it is very stiff along the fibers and rather compliant across the fibers. The dispersion equations of free Lamb waves propagating in an orthotropic layer are derived that results in the dispersion curves. The connection of these equations to the Rayleigh-Lamb frequency relations of isotropic plates is discussed. By the use of reciprocity in elastodynamics, closed-form solutions of elastic wave motions subjected to time-harmonic loads in the layer are computed in a simple manner. We also consider the problem of Lamb waves generated by a set of time-harmonic sources. The obtained computations can be very useful for developing ultrasound-based methods for nondestructive evaluation of composite structures.

Keywords: lamb waves, fiber-reinforced composite plates, dispersion equations, nondestructive evaluation, reciprocity theorems

Procedia PDF Downloads 107