Abstract: In this paper, we explore the applicability of the Sinc-
Collocation method to a three-dimensional (3D) oceanography model.
The model describes a wind-driven current with depth-dependent
eddy viscosity in the complex-velocity system. In general, the
Sinc-based methods excel over other traditional numerical methods
due to their exponentially decaying errors, rapid convergence and
handling problems in the presence of singularities in end-points.
Together with these advantages, the Sinc-Collocation approach that
we utilize exploits first derivative interpolation, whose integration
is much less sensitive to numerical errors. We bring up several
model problems to prove the accuracy, stability, and computational
efficiency of the method. The approximate solutions determined by
the Sinc-Collocation technique are compared to exact solutions and
those obtained by the Sinc-Galerkin approach in earlier studies. Our
findings indicate that the Sinc-Collocation method outperforms other
Sinc-based methods in past studies.
Abstract: Fluid flow and heat transfer of vertical full cone
embedded in porous media is studied in this paper. Nonlinear
differential equation arising from similarity solution of inverted cone
(subjected to wall temperature boundary conditions) embedded in
porous medium is solved using a hybrid neural network- particle
swarm optimization method.
To aim this purpose, a trial solution of the differential equation is
defined as sum of two parts. The first part satisfies the initial/
boundary conditions and does contain an adjustable parameter and
the second part which is constructed so as not to affect the
initial/boundary conditions and involves adjustable parameters (the
weights and biases) for a multi-layer perceptron neural network.
Particle swarm optimization (PSO) is applied to find adjustable
parameters of trial solution (in first and second part). The obtained
solution in comparison with the numerical ones represents a
remarkable accuracy.
Abstract: In the present study, a procedure was developed to
determine the optimum reaction rate constants in generalized
Arrhenius form and optimized through the Nelder-Mead method. For
this purpose, a comprehensive mathematical model of a fixed bed
reactor for dehydrogenation of heavy paraffins over Pt–Sn/Al2O3
catalyst was developed. Utilizing appropriate kinetic rate expressions
for the main dehydrogenation reaction as well as side reactions and
catalyst deactivation, a detailed model for the radial flow reactor was
obtained. The reactor model composed of a set of partial differential
equations (PDE), ordinary differential equations (ODE) as well as
algebraic equations all of which were solved numerically to
determine variations in components- concentrations in term of mole
percents as a function of time and reactor radius. It was demonstrated
that most significant variations observed at the entrance of the bed
and the initial olefin production obtained was rather high. The
aforementioned method utilized a direct-search optimization
algorithm along with the numerical solution of the governing
differential equations. The usefulness and validity of the method was
demonstrated by comparing the predicted values of the kinetic
constants using the proposed method with a series of experimental
values reported in the literature for different systems.
Abstract: Three-dimensional simulation of harmonic up
generation in free electron laser amplifier operating simultaneously
with a cold and relativistic electron beam is presented in steady-state
regime where the slippage of the electromagnetic wave with respect
to the electron beam is ignored. By using slowly varying envelope
approximation and applying the source-dependent expansion to wave
equations, electromagnetic fields are represented in terms of the
Hermit Gaussian modes which are well suited for the planar wiggler
configuration. The electron dynamics is described by the fully threedimensional
Lorentz force equation in presence of the realistic planar
magnetostatic wiggler and electromagnetic fields. A set of coupled
nonlinear first-order differential equations is derived and solved
numerically. The fundamental and third harmonic radiation of the
beam is considered. In addition to uniform beam, prebunched
electron beam has also been studied. For this effect of sinusoidal
distribution of entry times for the electron beam on the evolution of
radiation is compared with uniform distribution. It is shown that
prebunching reduces the saturation length substantially. For
efficiency enhancement the wiggler is set to decrease linearly when
the radiation of the third harmonic saturates. The optimum starting
point of tapering and the slope of radiation in the amplitude of
wiggler are found by successive run of the code.
Abstract: The governing two-dimensional equations of a heterogeneous material composed of a fluid (allowed to flow in the absence of acoustic excitations) and a crystalline piezoelectric cubic solid stacked one-dimensionally (along the z direction) are derived and special emphasis is given to the discussion of acoustic group velocity for the structure as a function of the wavenumber component perpendicular to the stacking direction (being the x axis). Variations in physical parameters with y are neglected assuming infinite material homogeneity along the y direction and the flow velocity is assumed to be directed along the x direction. In the first part of the paper, the governing set of differential equations are derived as well as the imposed boundary conditions. Solutions are provided using Hamilton-s equations for the wavenumber vs. frequency as a function of the number and thickness of solid layers and fluid layers in cases with and without flow (also the case of a position-dependent flow in the fluid layer is considered). In the first part of the paper, emphasis is given to the small-frequency case. Boundary conditions at the bottom and top parts of the full structure are left unspecified in the general solution but examples are provided for the case where these are subject to rigid-wall conditions (Neumann boundary conditions in the acoustic pressure). In the second part of the paper, emphasis is given to the general case of larger frequencies and wavenumber-frequency bandstructure formation. A wavenumber condition for an arbitrary set of consecutive solid and fluid layers, involving four propagating waves in each solid region, is obtained again using the monodromy matrix method. Case examples are finally discussed.
Abstract: Current spectrums of a high power induction machine was calculated for the cases of full symmetry, static and dynamic 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. The paper presents the stator current spectrums in full symmetry, static and dynamic eccentricity cases, and demonstrates the harmonics present in each case. The effect of dynamic eccentricity is demonstrating via comparing the current spectrums related to dynamic eccentricity cases with the full symmetry one. The paper includes one case study, refers to dynamic eccentricity, to present the spectrum of the measured current and demonstrate the existence of the harmonics related to dynamic eccentricity. The zooms of current spectrums around the main slot harmonic zone are included to simplify the comparison and prove the existence of the dynamic eccentricity harmonics in both calculated and measured current spectrums.
Abstract: This paper describes a one-dimensional numerical model for natural gas production from the dissociation of methane hydrate in hydrate-capped gas reservoir under depressurization and thermal stimulation. Some of the hydrate reservoirs discovered are overlying a free-gas layer, known as hydrate-capped gas reservoirs. These reservoirs are thought to be easiest and probably the first type of hydrate reservoirs to be produced. The mathematical equations that can be described this type of reservoir include mass balance, heat balance and kinetics of hydrate decomposition. These non-linear partial differential equations are solved using finite-difference fully implicit scheme. In the model, the effect of convection and conduction heat transfer, variation change of formation porosity, the effect of using different equations of state such as PR and ER and steam or hot water injection are considered. In addition distributions of pressure, temperature, saturation of gas, hydrate and water in the reservoir are evaluated. It is shown that the gas production rate is a sensitive function of well pressure.
Abstract: The present contribution deals with the
thermophoretic deposition of nanoparticles over a rapidly rotating
permeable disk in the presence of partial slip, magnetic field, thermal
radiation, thermal-diffusion, and diffusion-thermo effects. The
governing nonlinear partial differential equations such as continuity,
momentum, energy and concentration are transformed into nonlinear
ordinary differential equations using similarity analysis, and the
solutions are obtained through the very efficient computer algebra
software MATLAB. Graphical results for non-dimensional
concentration and temperature profiles including thermophoretic
deposition velocity and Stanton number (thermophoretic deposition
flux) in tabular forms are presented for a range of values of the
parameters characterizing the flow field. It is observed that slip
mechanism, thermal-diffusion, diffusion-thermo, magnetic field and
radiation significantly control the thermophoretic particles deposition
rate. The obtained results may be useful to many industrial and
engineering applications.
Abstract: Mathematical models of dynamics employing exterior calculus are mathematical representations of the same unifying principle; namely, the description of a dynamic system with a characteristic differential one-form on an odd-dimensional differentiable manifold leads, by analysis with exterior calculus, to a set of differential equations and a characteristic tangent vector (vortex vector) which define transformations of the system. Using this principle, a mathematical model for economic growth is constructed by proposing a characteristic differential one-form for economic growth dynamics (analogous to the action in Hamiltonian dynamics), then generating a pair of characteristic differential equations and solving these equations for the rate of economic growth as a function of labor and capital. By contracting the characteristic differential one-form with the vortex vector, the Lagrangian for economic growth dynamics is obtained.
Abstract: In this paper, linear multistep technique using power
series as the basis function is used to develop the block methods
which are suitable for generating direct solution of the special second
order ordinary differential equations of the form y′′ = f(x,y), a < = x < = b with associated initial or boundary conditions. The continuaous hybrid formulations enable us to differentiate and evaluate at some
grids and off – grid points to obtain two different three discrete
schemes, each of order (4,4,4)T, which were used in block form for
parallel or sequential solutions of the problems. The computational
burden and computer time wastage involved in the usual reduction of
second order problem into system of first order equations are avoided
by this approach. Furthermore, a stability analysis and efficiency of
the block method are tested on linear and non-linear ordinary
differential equations whose solutions are oscillatory or nearly
periodic in nature, and the results obtained compared favourably with
the exact solution.
Abstract: A steady two-dimensional magnetohydrodynamics
flow and heat transfer over a stretching vertical sheet influenced by
radiation and porosity is studied. The governing boundary layer
equations of partial differential equations are reduced to a system of
ordinary differential equations using similarity transformation. The
system is solved numerically by using a finite difference scheme
known as the Keller-box method for some values of parameters,
namely the radiation parameter N, magnetic parameter M, buoyancy
parameter l , Prandtl number Pr and permeability parameter K. The
effects of the parameters on the heat transfer characteristics are
analyzed and discussed. It is found that both the skin friction
coefficient and the local Nusselt number decrease as the magnetic
parameter M and permeability parameter K increase. Heat transfer
rate at the surface decreases as the radiation parameter increases.
Abstract: Piezoelectric transformers are electronic devices made
from piezoelectric materials. The piezoelectric transformers as the
name implied are used for changing voltage signals from one level to another. Electrical energy carried with signals is transferred by means of mechanical vibration. Characterizing in both electrical and
mechanical properties leads to extensively use and efficiency enhancement of piezoelectric transformers in various applications. In
this paper, study and analysis of electrical and mechanical properties of multi-layer piezoelectric transformers in forms of potential and
displacement distribution throughout the volume, respectively. This
paper proposes a set of quasi-static mathematical model of electromechanical
coupling for piezoelectric transformer by using a set of
partial differential equations. Computer-based simulation utilizing the three-dimensional finite element method (3-D FEM) is exploited
as a tool for visualizing potentials and displacements distribution
within the multi-layer piezoelectric transformer. This simulation was
conducted by varying a number of layers. In this paper 3, 5 and 7 of
the circular ring type were used. The computer simulation based on
the use of the FEM has been developed in MATLAB programming environment.
Abstract: Dengue, a disease found in most tropical and
subtropical areas of the world. It has become the most common
arboviral disease of humans. This disease is caused by any of four
serotypes of dengue virus (DEN1-DEN4). In many endemic
countries, the average age of getting dengue infection is shifting
upwards, dengue in pregnancy and infancy are likely to be
encountered more frequently. The dynamics of the disease is studied
by a compartmental model involving ordinary differential equations
for the pregnant, infant human and the vector populations. The
stability of each equilibrium point is given. The epidemic dynamic is
discussed. Moreover, the numerical results are shown for difference
values of dengue antibody.
Abstract: Calcium is very important for communication among
the neurons. It is vital in a number of cell processes such as secretion,
cell movement, cell differentiation. To reduce the system of reactiondiffusion
equations of [Ca2+] into a single equation, two theories
have been proposed one is excess buffer approximation (EBA) other
is rapid buffer approximation (RBA). The RBA is more realistic than
the EBA as it considers both the mobile and stationary endogenous
buffers. It is valid near the mouth of the channel. In this work we have
studied the effects of different types of buffers on calcium diffusion
under RBA. The novel thing studied is the effect of sodium ions on
calcium diffusion. The model has been made realistic by considering
factors such as variable [Ca2+], [Na+] sources, sodium-calcium
exchange protein(NCX), Sarcolemmal Calcium ATPase pump. The
proposed mathematical leads to a system of partial differential equations
which has been solved numerically to study the relationships
between different parameters such as buffer concentration, buffer
disassociation rate, calcium permeability. We have used Forward
Time Centred Space (FTCS) approach to solve the system of partial
differential equations.
Abstract: In this paper, by constructing a special set and utilizing fixed point theory in coin, we study the existence of solution of singular two point’s boundary value problem for second-order differential equation, which improved and generalize the result of related paper.
Abstract: In this paper, some new nonlinear generalized
Gronwall-Bellman-Type integral inequalities with mixed time delays
are established. These inequalities can be used as handy tools
to research stability problems of delayed differential and integral
dynamic systems. As applications, based on these new established
inequalities, some p-stable results of a integro-differential equation
are also given. Two numerical examples are presented to illustrate
the validity of the main results.
Abstract: In this paper, we study the existence of solution of
the four-point boundary value problem for second-order differential
equations with impulses by using leray-Schauder theory:
Abstract: This paper presents the mathematical model of electric field and magnetic field in transmission system, which performs in second-order partial differential equation. This research has conducted analyzing the electromagnetic field radiating to atmosphere around the transmission line, when there is the transmission line transposition in case of long distance distribution. The six types of 500 kV transposed HV transmission line with double circuit will be considered. The computer simulation is applied finite element method that is developed by MATLAB program. The problem is considered to two dimensions, which is time harmonic system with the graphical performance of electric field and magnetic field. The impact from simulation of six types long distance distributing transposition will not effect changing of electric field and magnetic field which surround the transmission line.
Abstract: The integral form of equations of motion of composite
beams subjected to varying time loads are discretized using a
developed finite element model. The model consists of a straight five
node twenty-two degrees of freedom beam element. The stability
analysis of the beams is studied by solving the matrix form
characteristic equations of the system. The principle of virtual work
and the first order shear deformation theory are employed to analyze
the beams with large deformation and small strains. The regions of
dynamic instability of the beam are determined by solving the
obtained Mathieu form of differential equations. The effects of nonconservative
loads, shear stiffness, and damping parameters on
stability and response of the beams are examined. Several numerical
calculations are presented to compare the results with data reported
by other researchers.
Abstract: In this paper, parallelism in the solution of Ordinary
Differential Equations (ODEs) to increase the computational speed is
studied. The focus is the development of parallel algorithm of the two
point Block Backward Differentiation Formulas (PBBDF) that can
take advantage of the parallel architecture in computer technology.
Parallelism is obtained by using Message Passing Interface (MPI).
Numerical results are given to validate the efficiency of the PBBDF
implementation as compared to the sequential implementation.