Search results for: stochastic differential equations

Authors: Yuri V. Kim

Abstract:

This article presents a new approach to the Functional Testing of Space Systems (SS). It can be considered as a generic test and used for a wide class of SS that from the point of view of System Dynamics and Control may be described by the ordinary differential equations. Suggested methodology is based on using semi-natural experiment- laboratory stand that doesn’t require complicated, precise and expensive technological control-verification equipment. However, it allows for testing system as a whole totally assembled unit during Assembling, Integration and Testing (AIT) activities, involving system hardware (HW) and software (SW). The test physically activates system input (sensors) and output (actuators) and requires recording their outputs in real time. The data is then inserted in laboratory PC where it is post-experiment processed by Matlab/Simulink Identification Toolbox. It allows for estimating system dynamics in form of estimation of system differential equations by the experimental way and comparing them with expected mathematical model prematurely verified by mathematical simulation during the design process.

Keywords: system dynamics, space system ground tests and space qualification, system dynamics identification, satellite attitude control, assembling, integration and testing

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Authors: Yeliz Mert Kantar, İsmail Yeni̇lmez, Ibrahim Arik

Abstract:

In this study, the efficiency measurement of the top fifty Turkish Universities has been conducted. The top fifty Turkish Universities are listed by The Scientific and Technological Research Council of Turkey (TÜBITAK) according to the Entrepreneur and Innovative University Index every year. The index is calculated based on four components since 2018. Four components are scientific and technological research competency, intellectual property pool, cooperation and interaction, and economic and social contribution. The four components consist of twenty-three sub-components. The 2021 list announced in January 2022 is discussed in this study. Efficiency analysis have been carried out using the Stochastic Frontier Model. Statistical significance of the sub-components that make up the index with certain weights has been examined in terms of the efficiency measurement calculated through the Stochastic Frontier Model. The relationship between the efficiency ranking estimated based on the Stochastic Frontier Model and the Entrepreneur and Innovative University Index ranking is discussed in detail.

Keywords: efficiency, entrepreneur and innovative universities, turkish universities, stochastic frontier model, tübi̇tak

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Authors: J. A. Gbadeyan, R. A. Kareem

Abstract:

In this paper, we investigated the effects of unsteady internal heat generation of a two-step exothermic reactive hydromagnetic fluid flow under different chemical kinetics namely: Sensitized, Arrhenius and Bimolecular kinetics through an isothermal wall temperature channel. The resultant modeled nonlinear partial differential equations were simplified and solved using a combined Laplace-Differential Transform Method (LDTM). The solutions obtained were discussed and presented graphically to show the salient features of the fluid flow and heat transfer characteristics.

Keywords: unsteady, reactive, hydromagnetic, couette ow, exothermi creactio

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Authors: Charlene N. T. Mfangnia, Henri E. Z. Tonnang, Berge Tsanou, Jeremy Herren

Abstract:

Microsporidia MB is a recently discovered symbiont capable of blocking the transmission of Plasmodium from mosquitoes to humans. The symbiont can spread both horizontally and vertically among the mosquito population. This dual transmission gives the symbiont the ability to invade the mosquito population. The replacement of the mosquito population by the population of symbiont-infected mosquitoes then appears as a promising strategy for malaria control. In this context, the present study uses differential equations to model the transmission dynamics of Microsporidia MB in the population of female Anopheles mosquitoes. Long-term propagation scenarios of the symbiont, such as extinction, persistence or total infection, are obtained through the determination of the target and basic reproduction numbers, the equilibria, and the study of their stability. The stability is illustrated numerically, and the contribution of vertical and horizontal transmission in the spread of the symbiont is assessed. Data obtained from laboratory experiments are then used to explain the low prevalence observed in nature. The study also shows that the male death rate, the mating rate and the attractiveness of MB-positive mosquitoes are the factors that most influence the transmission of the symbiont. In addition, the introduction of temperature and the study of bifurcations show the significant influence of the environmental condition in the propagation of Microsporidia MB. This finding proves the necessity of taking into account environmental variables for the potential establishment of the symbiont in a new area.

Keywords: differential equations, stability analysis, malaria, microsporidia MB, horizontal transmission, vertical transmission, numerical illustration

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Authors: Brando Vagenende, Marie-Anne Guerry

Abstract:

For stochastic matrices of any order, the geometric description of the convex set of eigenvalues is completely known. The purpose of this study is to investigate the subset of the monotone matrices. This type of matrix appears in contexts such as intergenerational occupational mobility, equal-input modeling, and credit ratings-based systems. Monotone matrices are stochastic matrices in which each row stochastically dominates the previous row. The monotonicity property of a stochastic matrix can be expressed by a nonnegative lower-order matrix with the same eigenvalues as the original monotone matrix (except for the eigenvalue 1). Specifically, the aim of this research is to focus on the properties of eigenvalues of monotone matrices. For those matrices up to order 3, there already exists a complete description of the convex set of eigenvalues. For monotone matrices of order at least 4, this study gives, through simulations, more insight into the geometric description of their eigenvalues. Furthermore, this research treats in a geometric and algebraic way the properties of eigenvalues of monotone matrices of order at least 4.

Keywords: eigenvalues of matrices, finite Markov chains, monotone matrices, nonnegative matrices, stochastic matrices

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Authors: José Alberto García Fernández, Zhimin Du, Xinqiao Jin

Abstract:

Nowadays, data center industry faces strong challenges for increasing the speed and data processing capacities while at the same time is trying to keep their devices a suitable working temperature without penalizing that capacity. Consequently, the cooling systems of this kind of facilities use a large amount of energy to dissipate the heat generated inside the servers, and developing new cooling techniques or perfecting those already existing would be a great advance in this type of industry. The installation of a temperature sensor matrix distributed in the structure of each server would provide the necessary information for collecting the required data for obtaining a temperature profile instantly inside them. However, the number of temperature probes required to obtain the temperature profiles with sufficient accuracy is very high and expensive. Therefore, other less intrusive techniques are employed where each point that characterizes the server temperature profile is obtained by solving differential equations through simulation methods, simplifying data collection techniques but increasing the time to obtain results. In order to reduce these calculation times, complicated and slow computational fluid dynamics simulations are replaced by simpler and faster finite element method simulations which solve the Burgers‘ equations by backward, forward and central discretization techniques after simplifying the energy and enthalpy conservation differential equations. The discretization methods employed for solving the first and second order derivatives of the obtained Burgers‘ equation after these simplifications are the key for obtaining results with greater or lesser accuracy regardless of the characteristic truncation error.

Keywords: Burgers' equations, CFD simulation, data center, discretization methods, FEM simulation, temperature profile

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Authors: Johnson Oladele Fatokun, I. P. Akpan

Abstract:

In this paper, the one-dimensional time dependent Schrödinger equation is discretized by the method of lines using a second order finite difference approximation to replace the second order spatial derivative. The evolving system of stiff ordinary differential equation (ODE) in time is solved numerically by an L-stable trapezoidal-like integrator. Results show accuracy of relative maximum error of order 10-4 in the interval of consideration. The performance of the method as compared to an existing scheme is considered favorable.

Keywords: Schrodinger’s equation, partial differential equations, method of lines (MOL), stiff ODE, trapezoidal-like integrator

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Authors: Adamu S. Salawu, Ibrahim O. Isah

Abstract:

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: Vandana N. Purav

Abstract:

The Dynamical systems concept is a mathematical formalization for any fixed rule that describes the time dependence of a points position in its ambient space. e.g. pendulum of a clock, the number of fish each spring in a lake, the number of rabbits spring in an enclosure, etc. The Dynamical system theory used to describe the complex nature that is dynamical systems with differential equations called continuous dynamical system or dynamical system with difference equations called discrete dynamical system. The concept of dynamical system has its origin in Newtonian mechanics.

Keywords: dynamical systems, Fibonacci numbers, Newtonian mechanics, discrete dynamical system

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Authors: H. Niranjan, S. Sivasankaran

Abstract:

Heat and mass transfer near a steady stagnation point boundary layer flow of viscous incompressible fluid through porous media investigates along a vertical plate is thoroughly studied under the presence of magneto hydrodynamic (MHD) effects. The fluid flow is steady, laminar, incompressible and in two-dimensional. The nonlinear differential coupled parabolic partial differential equations of continuity, momentum, energy and specie diffusion are converted into the non-similar boundary layer equations using similarity transformation, which are then solved numerically using the Runge-Kutta method along with shooting method. The effects of the conjugate heat transfer parameter, the porous medium parameter, the permeability parameter, the mixed convection parameter, the magnetic parameter, and the thermal radiation on the velocity and temperature profiles as well as on the local skin friction and local heat transfer are presented and analyzed. The validity of the methodology and analysis is checked by comparing the results obtained for some specific cases with those available in the literature. The various parameters on local skin friction, heat and mass transfer rates are presented in tabular form.

Keywords: MHD, porous medium, slip, convective boundary condition, stagnation point

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Authors: D. Mehdizadeh, M. Rahimian, M. Eskandari-Ghadi

Abstract:

This article is concerned with the determination of the static interaction of a vertically loaded rigid circular disc embedded at the interface of a horizontal layer sandwiched in between two different transversely isotropic half-spaces called as tri-material full-space. The axes of symmetry of different regions are assumed to be normal to the horizontal interfaces and parallel to the movement direction. With the use of a potential function method, and by implementing Hankel integral transforms in the radial direction, the government partial differential equation for the solely scalar potential function is transformed to an ordinary 4th order differential equation, and the mixed boundary conditions are transformed into a pair of integral equations called dual integral equations, which can be reduced to a Fredholm integral equation of the second kind, which is solved analytically. Then, the displacements and stresses are given in the form of improper line integrals, which is due to inverse Hankel integral transforms. It is shown that the present solutions are in exact agreement with the existing solutions for a homogeneous full-space with transversely isotropic material. To confirm the accuracy of the numerical evaluation of the integrals involved, the numerical results are compared with the solutions exists for the homogeneous full-space. Then, some different cases with different degrees of material anisotropy are compared to portray the effect of degree of anisotropy.

Keywords: transversely isotropic, rigid disc, elasticity, dual integral equations, tri-material full-space

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Authors: A. B. Disu, M. S. Dada

Abstract:

This study was conducted to investigate two dimensional heat transfer of a free convective-radiative MHD (Magneto-hydrodynamics) flow with temperature dependent viscosity and heat source of a viscous incompressible fluid in a porous medium between two vertical wavy walls. The fluid viscosity is assumed to vary as an exponential function of temperature. The flow is assumed to consist of a mean part and a perturbed part. The perturbed quantities were expressed in terms of complex exponential series of plane wave equation. The resultant differential equations were solved by Differential Transform Method (DTM). The numerical computations were presented graphically to show the salient features of the fluid flow and heat transfer characteristics. The skin friction and Nusselt number were also analyzed for various governing parameters.

Keywords: differential transform method, MHD free convection, porous medium, two dimensional radiation, two wavy walls

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Authors: Sofiane Grira, Hichem Eleuch

Abstract:

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: Olfa Kaabia, Ilyes Abid, Khaled Guesmi

Abstract:

This pioneer paper studies whether and how Bitcoin shocks are transmitted to the U.S economy. We employ a new methodology: TVP FAVAR model with stochastic volatility. We use a large dataset of 111 major U.S variables from 1959:m1 to 2016:m12. The results show that Bitcoin shocks significantly impact the U.S. economy. This significant impact is pronounced in a volatile and increasing U.S economy. The Bitcoin has a positive relationship on the U.S real activity, and a negative one on U.S prices and interest rates. Effects on the Monetary Policy exist via the inter-est rates and the Money, Credit and Finance transmission channels.

Keywords: bitcoin, US economy, FAVAR models, stochastic volatility

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Authors: K. Jafar, R. Nazar, A. Ishak, I. Pop

Abstract:

The present analysis considers the steady stagnation point flow and heat transfer towards a permeable sheet in an upper-convected Maxwell (UCM) electrically conducting fluid, with a constant magnetic field applied in the transverse direction to flow, and a local heat generation within the boundary layer with a heat generation rate proportional to (T-T_inf)^p. Using a similarity transformation, the governing system of partial differential equations is first transformed into a system of ordinary differential equations, which is then solved numerically using a finite-difference scheme known as the Keller-box method. Numerical results are obtained for the flow and thermal fields for various values of the shrinking/stretching parameter lambda, the magnetic parameter M, the elastic parameter K, the Prandtl number Pr, the suction parameter s, the heat generation parameter Q, and the exponent p. The results indicate the existence of dual solutions for the shrinking sheet up to a critical value lambda_c whose value depends on the value of M, K, and s. In the presence of internal heat absorbtion (Q<0), the surface heat transfer rate decreases with increasing p but increases with parameter Q and s, when the sheet is either stretched or shrunk.

Keywords: magnetohydrodynamic (MHD), boundary layer flow, UCM fluid, stagnation point, shrinking sheet

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Authors: Deva Kanta Phukan

Abstract:

An attempt is made where an electrically conducting fluid is injected from a fixed outer cylindrical casing onto an inner moving cylindrical rod. A magnetic field is applied parallel to the axis of the cylindrical rod. The basic governing set of partial differential equations for conservation of mass and momentum are reduced to a set of non-linear ordinary differential equation by introducing similarity transformation, which are integrated numerically. A perturbation solution for the case of large magnetic parameter is derived for constant Reynolds number.

Keywords: annular axi-symmetric stagnation flow, conducting fluid, magnetic field, moving cylinder

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Authors: Yutae Lee

Abstract:

In this paper, we consider the performance and availability of a redundancy model. The redundancy model is a form of resilience that ensures service availability in the event of component failure. This paper considers a 2N redundancy model. In the model there are at most one active service unit and at most one standby service unit. The active one is providing the service while the standby is prepared to take over the active role when the active fails. We design our analysis model using Stochastic Reward Nets, and then evaluate the performance and availability of 2N redundancy model using Stochastic Petri Net Package (SPNP).

Keywords: availability, performance, stochastic reward net, 2N redundancy

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Authors: Damir Latypov

Abstract:

A closed-form solution for 4-D double surface integrals arising in boundary integrals equations of a potential theory is obtained for arbitrary coplanar polygonal surfaces. The solution method is based on the construction of exact differential forms followed by the application of Stokes' theorem for each surface integral. As a result, the 4-D double surface integral is reduced to a 2-D double line integral. By an appropriate change of variables, the integrand is transformed into a separable function of integration variables. The closed-form solutions to the corresponding 1-D integrals are readily available in the integration tables. Previously closed-form solutions were known only for the case of coincident triangle surfaces and coplanar rectangles. Solutions for these cases were obtained by surface-specific ad-hoc methods, while the present method is general. The method also works for non-polygonal surfaces. As an example, we compute in closed form the 4-D integral for the case of coincident surfaces in the shape of a circular disk. For an arbitrarily shaped surface, the proposed method provides an efficient quadrature rule. Extensions of the method for non-coplanar surfaces and other than 1/R integral kernels are also discussed.

Keywords: boundary integral equations, differential forms, integration, stokes' theorem

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Authors: Homa Ghave, Parmis Shahmaleki

Abstract:

This paper presents a new multi-criteria stochastic mathematical model for a single machine scheduling with outsourcing allowed. There are multiple jobs processing in batch. For each batch, all of job or a quantity of it can be outsourced. The jobs have stochastic processing time and lead time and deterministic due dates arrive randomly. Because of the stochastic inherent of processing time and lead time, we use the chance constrained programming for modeling the problem. First, the problem is formulated in form of stochastic programming and then prepared in a form of deterministic mixed integer linear programming. The objectives are considered in the model to minimize the maximum tardiness and outsourcing cost simultaneously. Several procedures have been developed to deal with the multi-criteria problem. In this paper, we utilize the concept of satisfaction functions to increases the manager’s preference. The proposed approach is tested on instances where the random variables are normally distributed.

Keywords: single machine scheduling, multi-criteria mathematical model, outsourcing strategy, uncertain lead times and processing times, chance constrained programming, satisfaction function

Procedia PDF Downloads 234

Authors: Wedad Albalawi

Abstract:

The mathematical inequalities have been the core of mathematical study and used in almost all branches of mathematics as well in various areas of science and engineering. The inequalities by Hardy, Littlewood and Polya were the first significant composition of several science. This work presents fundamental ideas, results and techniques, and it has had much influence on research in various branches of analysis. Since 1934, various inequalities have been produced and studied in the literature. Furthermore, some inequalities have been formulated by some operators; in 1989, weighted Hardy inequalities have been obtained for integration operators. Then, they obtained weighted estimates for Steklov operators that were used in the solution of the Cauchy problem for the wave equation. They were improved upon in 2011 to include the boundedness of integral operators from the weighted Sobolev space to the weighted Lebesgue space. Some inequalities have been demonstrated and improved using the Hardy–Steklov operator. Recently, a lot of integral inequalities have been improved by differential operators. Hardy inequality has been one of the tools that is used to consider integrity solutions of differential equations. Then, dynamic inequalities of Hardy and Coposon have been extended and improved by various integral operators. These inequalities would be interesting to apply in different fields of mathematics (functional spaces, partial differential equations, mathematical modeling). Some inequalities have been appeared involving Copson and Hardy inequalities on time scales to obtain new special version of them. A time scale is an arbitrary nonempty closed subset of the real numbers. Then, the dynamic inequalities on time scales have received a lot of attention in the literature and has become a major field in pure and applied mathematics. There are many applications of dynamic equations on time scales to quantum mechanics, electrical engineering, neural networks, heat transfer, combinatorics, and population dynamics. This study focuses on Hardy and Coposon inequalities, using Steklov operator on time scale in double integrals to obtain special cases of time-scale inequalities of Hardy and Copson on high dimensions. The advantage of this study is that it uses the one-dimensional classical Hardy inequality to obtain higher dimensional on time scale versions that will be applied in the solution of the Cauchy problem for the wave equation. In addition, the obtained inequalities have various applications involving discontinuous domains such as bug populations, phytoremediation of metals, wound healing, maximization problems. The proof can be done by introducing restriction on the operator in several cases. The concepts in time scale version such as time scales calculus will be used that allows to unify and extend many problems from the theories of differential and of difference equations. In addition, using chain rule, and some properties of multiple integrals on time scales, some theorems of Fubini and the inequality of H¨older.

Keywords: time scales, inequality of hardy, inequality of coposon, steklov operator

Procedia PDF Downloads 53

Authors: M. Khoshab, M. J. Sedigh

Abstract:

Dynamic systems, which in mathematical point of view are those governed by differential equations, are much more difficult to study and to predict their behavior in comparison with static systems which are governed by algebraic equations. Economical systems such as market are among complicated dynamic systems. This paper tries to adopt a very simple mathematical model for market and to study effect of supply and demand function on behavior of the market while the supply side experiences a lag due to production restrictions.

Keywords: dynamic system, lag on supply demand, market stability, supply demand model

Procedia PDF Downloads 271

Authors: Saleh Elawgali

Abstract:

Current spectrums of a four pole-pair, 400 kW induction machine were calculated for the cases of full symmetry and static eccentricity. The calculations involve integration of 93 electrical plus four mechanical ordinary differential equations. Electrical equations account for variable inductances affected by slotting and eccentricities. The calculations were followed by Fourier analysis of the stator currents in steady state operation. Zooms of the current spectrums, around the 50 Hz fundamental harmonic as well as of the main slot harmonic zone, were included. The spectrums included refer to both calculated and measured currents.

Keywords: diagnostic, harmonic, induction machine, spectrum

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Authors: Liliana O. Martínez, Juan E González, Manuel Ramírez-Aranda, Ana Cervantes-Herrera

Abstract:

A serious game category mobile application called Math Dominoes is presented. The main objective of this applications is to strengthen the teaching-learning process of solving algebraic equations and is based on the board game "Double 6" dominoes. Math Dominoes allows the practice of solving first, second-, and third-degree algebraic equations. This application is aimed to students who seek to strengthen their skills in solving algebraic equations in a dynamic, interactive, and fun way, to reduce the risk of failure in subsequent courses that require mastery of this algebraic tool.

Keywords: algebra, equations, dominoes, serious games

Procedia PDF Downloads 95

Authors: Maryam Ebrahimpour, Amir Ebrahimi

Abstract:

This paper presents the steady-state amplitude analysis of MOS Colpitts oscillator with short channel device. The proposed method is based on a large signal analysis and the nonlinear differential equations that govern the oscillator circuit behaviour. Also, the short channel effects are considered in the proposed analysis and analytical equations for finding the steady-state oscillation amplitude are derived. The output voltage calculated from this analysis is in excellent agreement with simulations for a wide range of circuit parameters.

Keywords: colpitts oscillator, CMOS, electronics, circuits

Procedia PDF Downloads 318

Authors: Raphael Naryongo, Philip Ngare, Anthony Waititu

Abstract:

This paper deals with numerical simulation of Wishart processes for a single asset risky pricing model whose volatility is described by Wishart affine diffusion processes. The multi-factor specification of volatility will make the model more flexible enough to fit the stock market data for short or long maturities for better returns. The Wishart process is a stochastic process which is a positive semi-definite matrix-valued generalization of the square root process. The aim of the study is to model the log asset stock returns under the double Wishart stochastic volatility model. The solution of the log-asset return dynamics for Bi-Wishart processes will be obtained through Euler-Maruyama discretization schemes. The numerical results on the asset returns are compared to the existing models returns such as Heston stochastic volatility model and double Heston stochastic volatility model

Keywords: euler schemes, log-asset return, infinitesimal generator, wishart diffusion affine processes

Procedia PDF Downloads 339

Authors: Chahid K. Ghaddar

Abstract:

The spreadsheet engine is exploited via a non-conventional mechanism to enable novel worksheet solver functions for computational calculus. The solver functions bypass inherent restrictions on built-in math and user defined functions by taking variable formulas as a new type of argument while retaining purity and recursion properties. The enabling mechanism permits integration of numerical algorithms into worksheet functions for solving virtually any computational problem that can be modelled by formulas and variables. Several examples are presented for computing integrals, derivatives, and systems of deferential-algebraic equations. Incorporation of the worksheet solver functions with the ubiquitous spreadsheet extend the utility of the latter as a powerful tool for computational mathematics.

Keywords: calculus, differential algebraic equations, solvers, spreadsheet

Procedia PDF Downloads 308

Authors: Wedad Albalawi

Abstract:

The mathematical inequalities have been the core of mathematical study and used in almost all branches of mathematics as well in various areas of science and engineering. The inequalities by Hardy, Littlewood and Polya were the first significant composition of several science. This work presents fundamental ideas, results and techniques and it has had much influence on research in various branches of analysis. Since 1934, various inequalities have been produced and studied in the literature. Furthermore, some inequalities have been formulated by some operators; in 1989, weighted Hardy inequalities have been obtained for integration operators. Then, they obtained weighted estimates for Steklov operators that were used in the solution of the Cauchy problem for the wave equation. They were improved upon in 2011 to include the boundedness of integral operators from the weighted Sobolev space to the weighted Lebesgue space. Some inequalities have been demonstrated and improved using the Hardy–Steklov operator. Recently, a lot of integral inequalities have been improved by differential operators. Hardy inequality has been one of the tools that is used to consider integrity solutions of differential equations. Then dynamic inequalities of Hardy and Coposon have been extended and improved by various integral operators. These inequalities would be interesting to apply in different fields of mathematics (functional spaces, partial differential equations, mathematical modeling). Some inequalities have been appeared involving Copson and Hardy inequalities on time scales to obtain new special version of them. A time scale is defined as a closed subset contains real numbers. Then the inequalities of time scales version have received a lot of attention and has had a major field in both pure and applied mathematics. There are many applications of dynamic equations on time scales to quantum mechanics, electrical engineering, neural networks, heat transfer, combinatorics, and population dynamics. This study focuses on double integrals to obtain new time-scale inequalities of Copson driven by Steklov operator. They will be applied in the solution of the Cauchy problem for the wave equation. The proof can be done by introducing restriction on the operator in several cases. In addition, the obtained inequalities done by using some concepts in time scale version such as time scales calculus, theorem of Fubini and the inequality of H¨older.

Keywords: time scales, inequality of Hardy, inequality of Coposon, Steklov operator

Procedia PDF Downloads 42

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: Jaan Lellep, Shahid Mubasshar

Abstract:

A numerical solution is developed for simply supported nanoarches based on the non-local theory of elasticity. The nanoarch under consideration has a step-wise variable cross-section and is weakened by crack-like defects. It is assumed that the cracks are stationary and the mechanical behaviour of the nanoarch can be modeled by Eringen’s non-local theory of elasticity. The physical and thermal properties are sensitive with respect to changes of dimensions in the nano level. The classical theory of elasticity is unable to describe such changes in material properties. This is because, during the development of the classical theory of elasticity, the speculation of molecular objects was avoided. Therefore, the non-local theory of elasticity is applied to study the vibration of nanostructures and it has been accepted by many researchers. In the non-local theory of elasticity, it is assumed that the stress state of the body at a given point depends on the stress state of each point of the structure. However, within the classical theory of elasticity, the stress state of the body depends only on the given point. The system of main equations consists of equilibrium equations, geometrical relations and constitutive equations with boundary and intermediate conditions. The system of equations is solved by using the method of separation of variables. Consequently, the governing differential equations are converted into a system of algebraic equations whose solution exists if the determinant of the coefficients of the matrix vanishes. The influence of cracks and steps on the natural vibration of the nanoarches is prescribed with the aid of additional local compliance at the weakened cross-section. An algorithm to determine the eigenfrequencies of the nanoarches is developed with the help of computer software. The effects of various physical and geometrical parameters are recorded and drawn graphically.

Keywords: crack, nanoarches, natural frequency, step

Procedia PDF Downloads 93

Authors: Lina Wu, Ye Li

Abstract:

An equivalent relation between a harmonic form and a closed co-closed form is established on a complete non-compact manifold. This equivalence has been generalized for a differential k-form ω from Lq spaces to non-Lq spaces when q=2 in the context of p-balanced growth where p=2. Especially for a simple differential k-form on a complete non-compact manifold, the equivalent relation has been verified with the extended scope of q for from finite q-energy in Lq spaces to infinite q-energy in non-Lq spaces when with 2-balanced growth. Generalized Hadamard Theorem, Cauchy-Schwarz Inequality, and Calculus skills including Integration by Parts as well as Convergent Series have been applied as estimation techniques to evaluate growth rates for a differential form. In particular, energy growth rates as indicated by an appropriate power range in a selected test function lead to a balance between a harmonic differential form and a closed co-closed differential form. Research ideas and computational methods in this paper could provide an innovative way in the study of broadening Lq spaces to non-Lq spaces with a wide variety of infinite energy growth for a differential form.

Keywords: closed forms, co-closed forms, harmonic forms, L^q spaces, p-balanced growth, simple differential k-forms

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