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, 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: 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.
Abstract: In this paper, based on the estimation of the Cauchy matrix of linear impulsive differential equations, by using Banach fixed point theorem and Gronwall-Bellman-s inequality, some sufficient conditions are obtained for the existence and exponential stability of almost periodic solution for Cohen-Grossberg shunting inhibitory cellular neural networks (SICNNs) with continuously distributed delays and impulses. An example is given to illustrate the main results.
Abstract: In this article an evolutionary technique has been used
for the solution of nonlinear Riccati differential equations of fractional order. In this method, genetic algorithm is used as a tool for
the competent global search method hybridized with active-set algorithm for efficient local search. The proposed method has been
successfully applied to solve the different forms of Riccati
differential equations. The strength of proposed method has in its
equal applicability for the integer order case, as well as, fractional
order case. Comparison of the method has been made with standard
numerical techniques as well as the analytic solutions. It is found
that the designed method can provide the solution to the equation
with better accuracy than its counterpart deterministic approaches.
Another advantage of the given approach is to provide results on
entire finite continuous domain unlike other numerical methods
which provide solutions only on discrete grid of points.
Abstract: The problem of natural convection about a cone embedded in a porous medium at local Rayleigh numbers based on the boundary layer approximation and the Darcy-s law have been studied before. Similarity solutions for a full cone with the prescribed wall temperature or surface heat flux boundary conditions which is the power function of distance from the vertex of the inverted cone give us a third-order nonlinear differential equation. In this paper, an approximate method for solving higher-order ordinary differential equations is proposed. The approach is based on a rational Chebyshev Tau (RCT) method. The operational matrices of the derivative and product of rational Chebyshev (RC) functions are presented. These matrices together with the Tau method are utilized to reduce the solution of the higher-order ordinary differential equations to the solution of a system of algebraic equations. We also present the comparison of this work with others and show that the present method is applicable.
Abstract: In this work, we derive two numerical schemes for
solving a class of nonlinear partial differential equations. The first
method is of second order accuracy in space and time directions, the
scheme is unconditionally stable using Von Neumann stability
analysis, the scheme produced a nonlinear block system where
Newton-s method is used to solve it. The second method is of fourth
order accuracy in space and second order in time. The method is
unconditionally stable and Newton's method is used to solve the
nonlinear block system obtained. The exact single soliton solution
and the conserved quantities are used to assess the accuracy and to
show the robustness of the schemes. The interaction of two solitary
waves for different parameters are also discussed.
Abstract: It is known that an analog Hopfield neural network
with time delay can generate the outputs which are similar to the
human electroencephalogram. To gain deeper insights into the
mechanisms of rhythm generation by the Hopfield neural networks
and to study the effects of noise on their activities, we investigated
the behaviors of the networks with symmetric and asymmetric
interneuron connections. The neural network under the study consists
of 10 identical neurons. For symmetric (fully connected) networks all
interneuron connections aij = +1; the interneuron connections for
asymmetric networks form an upper triangular matrix with non-zero
entries aij = +1. The behavior of the network is described by 10
differential equations, which are solved numerically. The results of
simulations demonstrate some remarkable properties of a Hopfield
neural network, such as linear growth of outputs, dependence of
synchronization properties on the connection type, huge
amplification of oscillation by the external uniform noise, and the
capability of the neural network to transform one type of noise to
another.
Abstract: The objective of this paper is to use the Pfaffian
technique to construct different classes of exact Pfaffian solutions and
N-soliton solutions to some of the generalized integrable nonlinear
partial differential equations in (3+1) dimensions. In this paper, I will
show that the Pfaffian solutions to the nonlinear PDEs are nothing but
Pfaffian identities. Solitons are among the most beneficial solutions
for science and technology, from ocean waves to transmission of
information through optical fibers or energy transport along protein
molecules. The existence of multi-solitons, especially three-soliton
solutions, is essential for information technology: it makes possible
undisturbed simultaneous propagation of many pulses in both directions.
Abstract: In the present paper some recommendations for the
use of software package “Mathematica" in a basic numerical analysis
course are presented. The methods which are covered in the course
include solution of systems of linear equations, nonlinear equations
and systems of nonlinear equations, numerical integration,
interpolation and solution of ordinary differential equations. A set of
individual assignments developed for the course covering all the
topics is discussed in detail.
Abstract: In this paper, using a model transformation approach a system of linear delay differential equations (DDEs) with multiple delays is converted to a non-delayed initial value problem. The variational iteration method (VIM) is then applied to obtain the approximate analytical solutions. Numerical results are given for several examples involving scalar and second order systems. Comparisons with the classical fourth-order Runge-Kutta method (RK4) verify that this method is very effective and convenient.
Abstract: A stage-structured predator-prey system with two time delays is considered. By analyzing the corresponding characteristic equation, the local stability of a positive equilibrium is investigated and the existence of Hopf bifurcations is established. Formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem. Numerical simulations are carried out to illustrate the theoretical results. Based on the global Hopf bifurcation theorem for general functional differential equations, the global existence of periodic solutions is established.
Abstract: Based on the homotopy perturbation method (HPM)
and Padé approximants (PA), approximate and exact solutions are
obtained for cubic Boussinesq and modified Boussinesq equations.
The obtained solutions contain solitary waves, rational solutions.
HPM is used for analytic treatment to those equations and PA for
increasing the convergence region of the HPM analytical solution.
The results reveal that the HPM with the enhancement of PA is a
very effective, convenient and quite accurate to such types of partial
differential equations.
Abstract: We construct an exponentially weighted Legendre- Gauss Tau method for solving differential equations with oscillatory solutions. The proposed method is applied to Sturm-Liouville problems. Numerical examples illustrating the efficiency and the high accuracy of our results are presented.
Abstract: This paper at first presents approximate analytical
solutions for systems of fractional differential equations using the
differential transform method. The application of differential
transform method, developed for differential equations of integer
order, is extended to derive approximate analytical solutions of
systems of fractional differential equations. The solutions of our
model equations are calculated in the form of convergent series with
easily computable components. After that a drive-response
synchronization method with linear output error feedback is
presented for “generalized projective synchronization" for a class of
fractional-order chaotic systems via a scalar transmitted signal.
Genesio_Tesi and Duffing systems are used to illustrate the
effectiveness of the proposed synchronization method.