Abstract: Three structure-dependent integration methods have been developed for solving equations of motion, which are second-order ordinary differential equations, for structural dynamics and earthquake engineering applications. Although they generally have the same numerical properties, such as explicit formulation, unconditional stability and second-order accuracy, a different performance is found in solving the free vibration response to either linear elastic or nonlinear systems with high frequency modes. The root cause of this different performance in the free vibration responses is analytically explored herein. As a result, it is verified that a weak instability is responsible for the different performance of the integration methods. In general, a weak instability will result in an inaccurate solution or even numerical instability in the free vibration responses of high frequency modes. As a result, a weak instability must be prohibited for time integration methods.
Abstract: In this work, a five step continuous method for the solution of third order ordinary differential equations was developed in block form using collocation and interpolation techniques of the shifted Legendre polynomial basis function. The method was found to be zero-stable, consistent and convergent. The application of the method in solving third order initial value problem of ordinary differential equations revealed that the method compared favorably with existing methods.
Abstract: This paper compared the efficiency of Simpson’s 1/3 and 3/8 rules for the numerical solution of first order Volterra integro-differential equations. In developing the solution, collocation approximation method was adopted using the shifted Legendre polynomial as basis function. A block method approach is preferred to the predictor corrector method for being self-starting. Experimental results confirmed that the Simpson’s 3/8 rule is more efficient than the Simpson’s 1/3 rule.
Abstract: This work is the modeling and simulation of fluid flow (liquid) through porous media. This type of flow occurs in many situations of interest in applied sciences and engineering, fluid (oil) consists of several individual substances in pure, single-phase flow is incompressible and isothermal. The porous medium is isotropic, homogeneous optionally, with the rectangular format and the flow is two-dimensional. Modeling of hydrodynamic phenomena incorporates Darcy's law and the equation of mass conservation. Correlations are used to model the density and viscosity of the fluid. A finite volume code is used in the discretization of differential equations. The nonlinearity is treated by Newton's method with relaxation coefficient. The results of the simulation of the pressure and the mobility of liquid flowing through porous media are presented, analyzed, and illustrated.
Abstract: Due to many applications and problems in the fields of plasma physics, geophysics, and other many topics, the interaction between the strain field and the magnetic field has to be considered. Adomian introduced the decomposition method for solving linear and nonlinear functional equations. This method leads to accurate, computable, approximately convergent solutions of linear and nonlinear partial and ordinary differential equations even the equations with variable coefficients. This paper is dealing with a mathematical model of generalized thermoelasticity of a half-space conducting medium. A magnetic field with constant intensity acts normal to the bounding plane has been assumed. Adomian’s decomposition method has been used to solve the model when the bounding plane is taken to be traction free and thermally loaded by harmonic heating. The numerical results for the temperature increment, the stress, the strain, the displacement, the induced magnetic, and the electric fields have been represented in figures. The magnetic field, the relaxation time, and the angular thermal load have significant effects on all the studied fields.
Abstract: 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.
Abstract: In this paper, the DC-link voltage control of DC-DC boost converter–inverter system is proposed. The mathematical model is developed from four different sub-circuits that depended on the switch positions. The developed differential equations are combined to develop the dynamic model. Transfer function is generated from the switched function model. Fluctuation of DC-link voltage causes connected loads malfunction. For this problem, a kind of traditional controller, the PI controller is applied to achieve constant DC-link voltage. The PI controller gains are obtained based on transfer function step response. The simulation work has been studied by using MATLAB/Simulink software and hardware prototype is implemented with a low-cost microcontroller Arduino Nano. Experimental results are collected by using ArduinoIO library package. Closed-loop DC-link voltage control system is tested with various line and load disturbances. It is found that the experimental results give equal responses with the simulation results.
Abstract: In this paper, by constructing a special set and utilizing
fixed point theory, we study the existence and multiplicity of the
positive solutions for systems of nonlinear third-order differential
equations with p-laplacian, which improve and generalize the result
of related paper.
Abstract: The water-based bioconvection of a nanofluid
containing motile gyrotactic micro-organisms over nonlinear
inclined stretching sheet has been investigated. The governing
nonlinear boundary layer equations of the model are reduced to a
system of ordinary differential equations via Oberbeck-Boussinesq
approximation and similarity transformations. Further, the modified
set of equations with associated boundary conditions are solved using
Finite Element Method. The impact of various pertinent parameters
on the velocity, temperature, nanoparticles concentration, density of
motile micro-organisms profiles are obtained and analyzed in details.
The results show that with the increase in angle of inclination δ,
velocity decreases while temperature, nanoparticles concentration,
a density of motile micro-organisms increases. Additionally, the
skin friction coefficient, Nusselt number, Sherwood number, density
number are computed for various thermophysical parameters. It
is noticed that increasing Brownian motion and thermophoresis
parameter leads to an increase in temperature of fluid which results
in a reduction in Nusselt number. On the contrary, Sherwood number
rises with an increase in Brownian motion and thermophoresis
parameter. The findings have been validated by comparing the
results of special cases with existing studies.
Abstract: The paper discusses the subinterval-based numerical
method for fractional derivative computations. It is now referred
to by its acronym – SubIval. The basis of the method is briefly
recalled. The ability of the method to be applied in time stepping
solvers is discussed. The possibility of implementing a time step size
adaptive solver is also mentioned. The solver is tested on a transient
circuit example. In order to display the accuracy of the solver –
the results have been compared with those obtained by means of a
semi-analytical method called gcdAlpha. The time step size adaptive
solver applying SubIval has been proven to be very accurate as
the results are very close to the referential solution. The solver is
currently able to solve FDE (fractional differential equations) with
various derivative orders for each equation and any type of source
time functions.
Abstract: The impact of the storm leads to accidents even in the case of vessels that meet the computed safety criteria for stability. That is why, in order to clarify the causes of the accident and shipwreck, it is necessary to study the dynamics of the ship under the complex sudden impact of external forces. The task is to determine the movement and landing of the ship in the complex and sudden impact of external forces, i.e. when the ship's load changes over a relatively short period of time. For the solution, a technique was used to study the ship's dynamics, which is based on the compilation of a system of differential equations of motion. A coordinate system was adopted for the equation of motion of the hull and the determination of external forces. As a numerical method of integration, the 4th order Runge-Kutta method was chosen. The results of the calculation show that dynamic deviations were lower for high-altitude vessels. The study of the movement of the hull under a difficult situation is performed: receiving of cargo, impact of a flurry of wind and subsequent displacement of the cargo. The risk of overturning and flooding was assessed.
Abstract: Free vibration analysis of homogenous and axially functionally graded simply supported beams within the context of Euler-Bernoulli beam theory is presented in this paper. The material properties of the beams are assumed to obey the linear law distribution. The effective elastic modulus of the composite was predicted by using the rule of mixture. Here, the complexities which appear in solving differential equation of transverse vibration of composite beams which limit the analytical solution to some special cases are overcome using a relatively new approach called the Differential Transformation Method. This technique is applied for solving differential equation of transverse vibration of axially functionally graded beams. Natural frequencies and corresponding normalized mode shapes are calculated for different Young’s modulus ratios. MATLAB code is designed to solve the transformed differential equation of the beam. Comparison of the present results with the exact solutions proves the effectiveness, the accuracy, the simplicity, and computational stability of the differential transformation method. The effect of the Young’s modulus ratio on the normalized natural frequencies and mode shapes is found to be very important.
Abstract: In this paper, the Differential Transform Method (DTM) is employed to predict and to analysis the non-local critical buckling loads of carbon nanotubes with various end conditions and the non-local Timoshenko beam described by single differential equation. The equation differential of buckling of the nanobeams is derived via a non-local theory and the solution for non-local critical buckling loads is finding by the DTM. The DTM is introduced briefly. It can easily be applied to linear or nonlinear problems and it reduces the size of computational work. Influence of boundary conditions, the chirality of carbon nanotube and aspect ratio on non-local critical buckling loads are studied and discussed. Effects of nonlocal parameter, ratios L/d, the chirality of single-walled carbon nanotube, as well as the boundary conditions on buckling of CNT are investigated.
Abstract: In this study, we have investigated the strict stability
of fuzzy differential systems and we compare the classical notion of
strict stability criteria of ordinary differential equations and the notion
of strict stability of fuzzy differential systems. In addition that, we
present definitions of stability and strict stability of fuzzy differential
equations and also we have some theorems and comparison results.
Strict Stability is a different stability definition and this stability
type can give us an information about the rate of decay of the
solutions. Lyapunov’s second method is a standard technique used
in the study of the qualitative behavior of fuzzy differential systems
along with a comparison result that allows the prediction of behavior
of a fuzzy differential system when the behavior of the null solution
of a fuzzy comparison system is known. This method is a usefull
for investigating strict stability of fuzzy systems. First of all, we
present definitions and necessary background material. Secondly, we
discuss and compare the differences between the classical notion
of stability and the recent notion of strict stability. And then, we
have a comparison result in which the stability properties of the null
solution of the comparison system imply the corresponding stability
properties of the fuzzy differential system. Consequently, we give
the strict stability results and a comparison theorem. We have used
Lyapunov second method and we have proved a comparison result
with scalar differential equations.
Abstract: A mathematical model for knowledge acquisition in
teaching and learning is proposed. In this study we adopt the
mathematical model that is normally used for disease modelling
into teaching and learning. We derive mathematical conditions which
facilitate knowledge acquisition. This study compares the effects
of dropping out of the course at early stages with later stages of
learning. The study also investigates effect of individual interaction
and learning from other sources to facilitate learning. The study fits
actual data to a general mathematical model using Matlab ODE45
and lsqnonlin to obtain a unique mathematical model that can be
used to predict knowledge acquisition. The data used in this study
was obtained from the tutorial test results for mathematics 2 students
from the Central University of Technology, Free State, South Africa
in the department of Mathematical and Physical Sciences. The study
confirms already known results that increasing dropout rates and
forgetting taught concepts reduce the population of knowledgeable
students. Increasing teaching contacts and access to other learning
materials facilitate knowledge acquisition. The effect of increasing
dropout rates is more enhanced in the later stages of learning
than earlier stages. The study opens up a new direction in further
investigations in teaching and learning using differential equations.
Abstract: In this paper, we studied the optimal portfolio selection in a defined contribution (DC) pension scheme with multiple contributors under constant elasticity of variance (CEV) model and the impact of stochastic additional voluntary contribution on the investment strategies. We assume that the voluntary contributions are stochastic and also consider investments in a risk free asset and a risky asset to increase the expected returns of the contributing members. We derived a stochastic differential equation which consists of the members’ monthly contributions and the invested fund and obtained an optimized problem with the help of Hamilton Jacobi Bellman equation. Furthermore, we find an explicit solution for the optimal investment strategy with stochastic voluntary contribution using power transformation and change of variables method and the corresponding optimal fund size was obtained. We discussed the impact of the voluntary contribution on the optimal investment strategy with numerical simulations and observed that the voluntary contribution reduces the optimal investment strategy of the risky asset.
Abstract: Minimizing the weight in flexible structures means
reducing material and costs as well. However, these structures could
become prone to vibrations. Attenuating these vibrations has become
a pivotal engineering problem that shifted the focus of many research
endeavors. One technique to do that is to design and implement
an active control system. This system is mainly composed of a
vibrating structure, a sensor to perceive the vibrations, an actuator
to counteract the influence of disturbances, and finally a controller to
generate the appropriate control signals. In this work, two different
techniques are explored to create two different mathematical models
of an active control system. The first model is a finite element model
with a reduced number of nodes and it is called a super-element.
The second model is in the form of state-space representation, i.e.
a set of partial differential equations. The damping coefficients are
calculated and incorporated into both models. The effectiveness of
these models is demonstrated when the system is excited by its first
natural frequency and an active control strategy is developed and
implemented to attenuate the resulting vibrations. Results from both
modeling techniques are presented and compared.
Abstract: This paper focuses on the study of two dimensional magnetohydrodynamic (MHD) steady incompressible viscous Williamson nanofluid with exponential internal heat generation containing gyrotactic microorganism over a stretching sheet. The governing equations and auxiliary conditions are reduced to a set of non-linear coupled differential equations with the appropriate boundary conditions using similarity transformation. The transformed equations are solved numerically through spectral relaxation method. The influences of various parameters such as Williamson parameter γ, power constant λ, Prandtl number Pr, magnetic field parameter M, Peclet number Pe, Lewis number Le, Bioconvection Lewis number Lb, Brownian motion parameter Nb, thermophoresis parameter Nt, and bioconvection constant σ are studied to obtain the momentum, heat, mass and microorganism distributions. Moment, heat, mass and gyrotactic microorganism profiles are explored through graphs and tables. We computed the heat transfer rate, mass flux rate and the density number of the motile microorganism near the surface. Our numerical results are in better agreement in comparison with existing calculations. The Residual error of our obtained solutions is determined in order to see the convergence rate against iteration. Faster convergence is achieved when internal heat generation is absent. The effect of magnetic parameter M decreases the momentum boundary layer thickness but increases the thermal boundary layer thickness. It is apparent that bioconvection Lewis number and bioconvection parameter has a pronounced effect on microorganism boundary. Increasing brownian motion parameter and Lewis number decreases the thermal boundary layer. Furthermore, magnetic field parameter and thermophoresis parameter has an induced effect on concentration profiles.
Abstract: In this paper, a theoretical study on the forced vibration of one degree of freedom system equipped with inerter, working under load-type or displacement-type excitation, is presented. Differential equations of movement are solved under cosinusoidal excitation, and explicit relations for the magnitude, resonant magnitude, phase angle, resonant frequency, and critical frequency are obtained. Influence of the inertance and damping on these dynamic characteristics is clarified. From the obtained results, one concludes that the inerter increases the magnitude of vibration and the phase angle of the damped mechanical system. Moreover, the magnitude ratio and difference of phase angles are not depending on the actual type of excitation. Consequently, such kind of similitude allows for the comparison of various theoretical and experimental results, which can be broadly found in the literature.
Abstract: In this paper, the problem proposed by Von Karman is treated in the attendance of additional flow field effects when the liquid is spaced above the rotating rigid disk. To be more specific, a purely viscous fluid flow yield by rotating rigid disk with Navier’s condition is considered in both magnetohydrodynamic and hydrodynamic frames. The rotating flow regime is manifested with heat source/sink and chemically reactive species. Moreover, the features of thermophoresis and Brownian motion are reported by considering nanofluid model. The flow field formulation is obtained mathematically in terms of high order differential equations. The reduced system of equations is solved numerically through self-coded computational algorithm. The pertinent outcomes are discussed systematically and provided through graphical and tabular practices. A simultaneous way of study makes this attempt attractive in this sense that the article contains dual framework and validation of results with existing work confirms the execution of self-coded algorithm for fluid flow regime over a rotating rigid disk.