Abstract: Taking into account that many problems of natural
sciences and engineering are reduced to solving initial-value problem
for ordinary differential equations, beginning from Newton, the
scientists investigate approximate solution of ordinary differential
equations. There are papers of different authors devoted to the
solution of initial value problem for ODE. The Euler-s known
method that was developed under the guidance of the famous
scientists Adams, Runge and Kutta is the most popular one among
these methods.
Recently the scientists began to construct the methods preserving
some properties of Adams and Runge-Kutta methods and called them
hybrid methods. The constructions of such methods are investigated
from the middle of the XX century. Here we investigate one
generalization of multistep and hybrid methods and on their base we
construct specific methods of accuracy order p = 5 and p = 6 for
k = 1 ( k is the order of the difference method).
Abstract: AAM has been successfully applied to face alignment,
but its performance is very sensitive to initial values. In case the initial
values are a little far distant from the global optimum values, there
exists a pretty good possibility that AAM-based face alignment may
converge to a local minimum. In this paper, we propose a progressive
AAM-based face alignment algorithm which first finds the feature
parameter vector fitting the inner facial feature points of the face and
later localize the feature points of the whole face using the first
information. The proposed progressive AAM-based face alignment
algorithm utilizes the fact that the feature points of the inner part of the
face are less variant and less affected by the background surrounding
the face than those of the outer part (like the chin contour). The
proposed algorithm consists of two stages: modeling and relation
derivation stage and fitting stage. Modeling and relation derivation
stage first needs to construct two AAM models: the inner face AAM
model and the whole face AAM model and then derive relation matrix
between the inner face AAM parameter vector and the whole face
AAM model parameter vector. In the fitting stage, the proposed
algorithm aligns face progressively through two phases. In the first
phase, the proposed algorithm will find the feature parameter vector
fitting the inner facial AAM model into a new input face image, and
then in the second phase it localizes the whole facial feature points of
the new input face image based on the whole face AAM model using
the initial parameter vector estimated from using the inner feature
parameter vector obtained in the first phase and the relation matrix
obtained in the first stage. Through experiments, it is verified that the
proposed progressive AAM-based face alignment algorithm is more
robust with respect to pose, illumination, and face background than the
conventional basic AAM-based face alignment algorithm.
Abstract: Climate change has profound consequences for the agriculture of south-eastern Australia and its climate-induced water shortage in the Murray-Darling Basin. Post Keynesian Economics (PKE) macro-dynamics, along with Kaleckian investment and growth theory, are used to develop an ecological-economic system dynamics model of this complex nonlinear river basin system. The Murray- Darling Basin Simulation Model (MDB-SM) uses the principles of PKE to incorporate the fundamental uncertainty of economic behaviors of farmers regarding the investments they make and the climate change they face, particularly as regards water ecosystem services. MDB-SM provides a framework for macroeconomic policies, especially for long-term fiscal policy and for policy directed at the sustainability of agricultural water, as measured by socio-economic well-being considerations, which include sustainable consumption and investment in the river basin. The model can also reproduce other ecological and economic aspects and, for certain parameters and initial values, exhibit endogenous business cycles and ecological sustainability with realistic characteristics. Most importantly, MDBSM provides a platform for the analysis of alternative economic policy scenarios. These results reveal the importance of understanding water ecosystem adaptation under climate change by integrating a PKE macroeconomic analytical framework with the system dynamics modelling approach. Once parameterised and supplied with historical initial values, MDB-SM should prove to be a practical tool to provide alternative long-term policy simulations of agricultural water and socio-economic well-being.
Abstract: An evolutionary computing technique for solving initial value problems in Ordinary Differential Equations is proposed in this paper. Neural network is used as a universal approximator while the adaptive parameters of neural networks are optimized by genetic algorithm. The solution is achieved on the continuous grid of time instead of discrete as in other numerical techniques. The comparison is carried out with classical numerical techniques and the solution is found with a uniform accuracy of MSE ≈ 10-9 .
Abstract: The main objective of the present paper is to derive an easy numerical technique for the analysis of the free vibration through the stepped regions of plates. Based on the utilities of the step by step integration initial values IV and Finite differences FD methods, the present improved Initial Value Finite Differences (IVFD) technique is achieved. The first initial conditions are formulated in convenient forms for the step by step integrations while the upper and lower edge conditions are expressed in finite difference modes. Also compatibility conditions are created due to the sudden variation of plate thickness. The present method (IVFD) is applied to solve the fourth order partial differential equation of motion for stepped plate across two different panels under the sudden step compatibility in addition to different types of end conditions. The obtained results are examined and the validity of the present method is proved showing excellent efficiency and rapid convergence.
Abstract: The RK5GL3 method is a numerical method for solving
initial value problems in ordinary differential equations, and is based
on a combination of a fifth-order Runge-Kutta method and 3-point
Gauss-Legendre quadrature. In this paper we describe the propagation
of local errors in this method, and show that the global order of
RK5GL3 is expected to be six, one better than the underlying Runge-
Kutta method.
Abstract: Beginning from the creator of integro-differential
equations Volterra, many scientists have investigated these
equations. Classic method for solving integro-differential
equations is the quadratures method that is successfully applied up
today. Unlike these methods, Makroglou applied hybrid methods
that are modified and generalized in this paper and applied to the
numerical solution of Volterra integro-differential equations. The
way for defining the coefficients of the suggested method is also
given.
Abstract: Solution of some practical problems is reduced to the
solution of the integro-differential equations. But for the numerical
solution of such equations basically quadrature methods or its
combination with multistep or one-step methods are used. The
quadrature methods basically is applied to calculation of the integral
participating in right hand side of integro-differential equations. As
this integral is of Volterra type, it is obvious that at replacement with
its integrated sum the upper limit of the sum depends on a current
point in which values of the integral are defined. Thus we receive the
integrated sum with variable boundary, to work with is hardly.
Therefore multistep method with the constant coefficients, which is
free from noted lack and gives the way for finding it-s coefficients is
present.
Abstract: The first and basic cause of the failure of concrete is repeated freezing (thawing) of moisture contained in the pores, microcracks, and cavities of the concrete. On transition to ice, water existing in the free state in cracks increases in volume, expanding the recess in which freezing occurs. A reduction in strength below the initial value is to be expected and further cycle of freezing and thawing have a further marked effect. By using some experimental parameters like nuclear magnetic resonance variation (NMR), enthalpy-temperature (or heat capacity) variation, we can resolve between the various water states and their effect on concrete properties during cooling through the freezing transition temperature range. The main objective of this paper is to describe the principal type of water responsible for the reduction in strength and structural damage (frost damage) of concrete following repeated freeze –thaw cycles. Some experimental work was carried out at the institute of cryogenics to determine what happens to water in concrete during the freezing transition.
Abstract: In this paper a new embedded Singly Diagonally
Implicit Runge-Kutta Nystrom fourth order in fifth order method for
solving special second order initial value problems is derived. A
standard set of test problems are tested upon and comparisons on the
numerical results are made when the same set of test problems are
reduced to first order systems and solved using the existing
embedded diagonally implicit Runge-Kutta method. The results
suggests the superiority of the new method.
Abstract: The RK1GL2X3 method is a numerical method for solving initial value problems in ordinary differential equations, and is based on the RK1GL2 method which, in turn, is a particular case of the general RKrGLm method. The RK1GL2X3 method is a fourth-order method, even though its underlying Runge-Kutta method RK1 is the first-order Euler method, and hence, RK1GL2X3 is considerably more efficient than RK1. This enhancement is achieved through an implementation involving triple-nested two-point Gauss- Legendre quadrature.
Abstract: In this paper usefulness of quasi-Newton iteration
procedure in parameters estimation of the conditional variance
equation within BHHH algorithm is presented. Analytical solution of
maximization of the likelihood function using first and second
derivatives is too complex when the variance is time-varying. The
advantage of BHHH algorithm in comparison to the other
optimization algorithms is that requires no third derivatives with
assured convergence. To simplify optimization procedure BHHH
algorithm uses the approximation of the matrix of second derivatives
according to information identity. However, parameters estimation in
a/symmetric GARCH(1,1) model assuming normal distribution of
returns is not that simple, i.e. it is difficult to solve it analytically.
Maximum of the likelihood function can be founded by iteration
procedure until no further increase can be found. Because the
solutions of the numerical optimization are very sensitive to the
initial values, GARCH(1,1) model starting parameters are defined.
The number of iterations can be reduced using starting values close
to the global maximum. Optimization procedure will be illustrated in
framework of modeling volatility on daily basis of the most liquid
stocks on Croatian capital market: Podravka stocks (food industry),
Petrokemija stocks (fertilizer industry) and Ericsson Nikola Tesla
stocks (information-s-communications industry).
Abstract: The RK5GL3 method is a numerical method for solving
initial value problems in ordinary differential equations, and is
based on a combination of a fifth-order Runge-Kutta method and
3-point Gauss-Legendre quadrature. In this paper we describe an
effective local error control algorithm for RK5GL3, which uses local
extrapolation with an eighth-order Runge-Kutta method in tandem
with RK5GL3, and a Hermite interpolating polynomial for solution
estimation at the Gauss-Legendre quadrature nodes.
Abstract: This paper describes Independent Component Analysis (ICA) based fixed-point algorithm for the blind separation of the convolutive mixture of speech, picked-up by a linear microphone array. The proposed algorithm extracts independent sources by non- Gaussianizing the Time-Frequency Series of Speech (TFSS) in a deflationary way. The degree of non-Gaussianization is measured by negentropy. The relative performances of algorithm under random initialization and Null beamformer (NBF) based initialization are studied. It has been found that an NBF based initial value gives speedy convergence as well as better separation performance
Abstract: In this paper zero-dissipative explicit Runge-Kutta
method is derived for solving second-order ordinary differential
equations with periodical solutions. The phase-lag and dissipation
properties for Runge-Kutta (RK) method are also discussed. The new
method has algebraic order three with dissipation of order infinity.
The numerical results for the new method are compared with existing
method when solving the second-order differential equations with
periodic solutions using constant step size.
Abstract: The implicit block methods based on the backward
differentiation formulae (BDF) for the solution of stiff initial value
problems (IVPs) using variable step size is derived. We construct a
variable step size block methods which will store all the coefficients
of the method with a simplified strategy in controlling the step size
with the intention of optimizing the performance in terms of
precision and computation time. The strategy involves constant,
halving or increasing the step size by 1.9 times the previous step size.
Decision of changing the step size is determined by the local
truncation error (LTE). Numerical results are provided to support the
enhancement of method applied.
Abstract: In present work are considered the scheme of
evaluation the transition probability in quantum system. It is based on
path integral representation of transition probability amplitude and its
evaluation by means of a saddle point method, applied to the part of
integration variables. The whole integration process is reduced to
initial value problem solutions of Hamilton equations with a random
initial phase point. The scheme is related to the semiclassical initial
value representation approaches using great number of trajectories. In
contrast to them from total set of generated phase paths only one path
for each initial coordinate value is selected in Monte Karlo process.
Abstract: In the recent works related with mixture discriminant
analysis (MDA), expectation and maximization (EM) algorithm is
used to estimate parameters of Gaussian mixtures. But, initial values
of EM algorithm affect the final parameters- estimates. Also, when
EM algorithm is applied two times, for the same data set, it can be
give different results for the estimate of parameters and this affect the
classification accuracy of MDA. Forthcoming this problem, we use
Self Organizing Mixture Network (SOMN) algorithm to estimate
parameters of Gaussians mixtures in MDA that SOMN is more robust
when random the initial values of the parameters are used [5]. We
show effectiveness of this method on popular simulated waveform
datasets and real glass data set.
Abstract: The optimal control problem of a linear distributed
parameter system is studied via shifted Legendre polynomials (SLPs)
in this paper. The partial differential equation, representing the
linear distributed parameter system, is decomposed into an n - set
of ordinary differential equations, the optimal control problem is
transformed into a two-point boundary value problem, and the twopoint
boundary value problem is reduced to an initial value problem
by using SLPs. A recursive algorithm for evaluating optimal control
input and output trajectory is developed. The proposed algorithm is
computationally simple. An illustrative example is given to show the
simplicity of the proposed approach.
Abstract: In the present paper, an improved initial value
numerical technique is presented to analyze the free vibration of
symmetrically laminated rectangular plate. A combination of the
initial value method (IV) and the finite differences (FD) devices is
utilized to develop the present (IVFD) technique. The achieved
technique is applied to the equation of motion of vibrating laminated
rectangular plate under various types of boundary conditions. Three
common types of laminated symmetrically cross-ply, orthotropic and
isotropic plates are analyzed here. The convergence and accuracy of
the presented Initial Value-Finite Differences (IVFD) technique have
been examined. Also, the merits and validity of improved technique
are satisfied via comparing the obtained results with those available
in literature indicating good agreements.