In this paper, we study the numerical method for solving second order Fuzzy
Differential Equations (FDEs) using Block Backward Differential Formulas (BBDF)
under generalized concept of higher-order fuzzy differentiability. Implementation of
the method using Newton iteration is discussed. Numerical results obtained by BBDF
are presented and compared with Backward Differential Formulas (BDF) and exact
solutions. Several numerical examples are provided to illustrate our methods.
Recently, oil refining industry is facing with lower profit margin due to un-
certainty. This causes oil refinery to include stochastic optimization in making a decision
to maximize the profit. In the past, deterministic linear programming approach is widely
used in oil refinery optimization problems. However, due to volatility and unpredictability
of oil prices in the past ten years, deterministic model might not be able to predict the
reality of the situation as it does not take into account the uncertainties thus, leads to
non-optimal solution. Therefore, this study will develop two-stage stochastic linear pro-
gramming for the midterm production planning of oil refinery to handle oil price volatility.
Geometric Brownian motion (GBM) is used to describe uncertainties in crude oil price,
petroleum product prices, and demand for petroleum products. This model generates the
future realization of the price and demands with scenario tree based on the statistical
specification of GBM using method of moment as input to the stochastic programming.
The model developed in this paper was tested for Malaysia oil refinery data. The result
of stochastic approach indicates that the model gives better prediction of profit margin.
In this paper, Maxwell fluid over a flat plate for convective boundary layer
flow with pressure gradient parameter is considered. The aim of this study is to compare
and analyze the effects of the presence and absence of λ (relaxation time), and also the
effects of m (pressure gradient parameter) and Pr (Prandtl number)on the momentum
and thermal boundary layer thicknesses. An approximation technique namely Homotopy
Perturbation Method (HPM) has been used with an implementation of Adam and Gear
Method’s algorithms. The obtained results have been compared for zero relaxation time
and also pressure gradient parameter with the published work of Fathizadeh and Rashidi.
The current outcomes are found to be in good agreement with the published results.
Physical interpretations have been given for the effects of the m, Pr and β (Deborah
number) with λ. This study will play an important role in industrial and engineering
The flow of water over an obstacle is a fundamental problem in fluid mechanics.
Transcritical flow means the wave phenomenon near the exact criticality. The transcriti-
cal flow cannot be handled by linear solutions as the energy is unable to propagate away
from the obstacle. Thus, it is important to carry out a study to identify suitable model
to analyse the transcritical flow. The aim of this study is to analyse the transcritical
flow over a bump as localized obstacles where the bump consequently generates upstream
and downstream flows. Nonlinear shallow water forced Korteweg-de Vries (fKdV) model
is used to analyse the flow over the bump. This theoretical model, containing forcing
functions represents bottom topography is considered as the simplified model to describe
water flows over a bump. The effect of water dispersion over the forcing region is in-
vestigated using the fKdV model. Homotopy Analysis Method (HAM) is used to solve
this theoretical fKdV model. The HAM solution which is chosen with a special choice
of }-value describes the physical flow of waves and the significance of dispersion over a
bump is elaborated.
Heat and mass transfer of MHD boundary-layer flow of a viscous incompress-
ible fluid over an exponentially stretching sheet in the presence of radiation is investi-
gated. The two-dimensional boundary-layer governing partial differential equations are
transformed into a system of nonlinear ordinary differential equations by using similarity
variables. The transformed equations of momentum, energy and concentration are solved
by Homotopy Analysis Method (HAM). The validity of HAM solution is ensured by com-
paring the HAM solution with existing solutions. The influence of physical parameters
such as magnetic parameter, Prandtl number, radiation parameter, and Schmidt num-
ber on velocity, temperature and concentration profiles are discussed. It is found that
the increasing values of magnetic parameter reduces the dimensionless velocity field but
enhances the dimensionless temperature and concentration field. The temperature dis-
tribution decreases with increasing values of Prandtl number. However, the temperature
distribution increases when radiation parameter increases. The concentration boundary
layer thickness decreases as a result of increase in Schmidt number.
A mechanistic model has been used to explain the effect of radiation. The
model consists of parameters which represent the biological process following ionizing
radiation. The parameters in the model are estimated using local and global optimiza-
tion algorithms. The aim of this study is to compare the efficiency between local and
global optimization method, which is Pattern Search and Genetic Algorithm respectively.
Experimental data from the cell survival of irradiated HeLa cell line is used to find the
minimum value of the sum of squared error (SSE) between experimental data and sim-
ulation data from the model. The performance of both methods are compared based on
the computational time and the value of the objective function, SSE. The optimization
process is carried out by using the built-in function in MATLAB software. The parameter
estimation results show that genetic algorithm is more superior than pattern search for
Pressurized water reactor (PWR) type AP1000 is a third generation of a nuclear
power plant. The primary system of PWR using uranium dioxide to generate heat energy
via fission process. The process influences temperature, pressure and pH value of water
chemistry of the PWR. The aim of this paper is to transform the primary system of PWR
using fuzzy autocatalytic set (FACS). In this work, the background of primary system
of PWR and the properties of the model are provided. The simulation result, namely
dynamic concentration of PWR is verified against published data.
Markov map is one example of interval maps where it is a piecewise expanding
map and obeys the Markov property. One well-known example of Markov map is the
doubling map, a map which has two subintervals with equal partitions. In this paper, we
are interested to investigate another type of Markov map, the so-called skewed doubling
map. This map is a more generalized map than the doubling map. Thus, the aims of this
paper are to find the fixed points as well as the periodic points for the skewed doubling
map and to investigate the sensitive dependence on initial conditions of this map. The
method considered here is the cobweb diagram. Numerical results suggest that there exist
dense of periodic orbits for this map. The sensitivity of this map to initial conditions is
also verified where small differences in initial conditions give different behaviour of the
orbits in the map.
In this paper, we look at the propagation of internal solitary waves over three
different types of slowly varying region, i.e. a slowly increasing slope, a smooth bump and
a parabolic mound in a two-layer fluid flow. The appropriate mathematical model for this
problem is the variable-coefficient extended Korteweg-de Vries equation. The governing
equation is then solved numerically using the method of lines. Our numerical simulations
show that the internal solitary waves deforms adiabatically on the slowly increasing slope.
At the same time, a trailing shelf is generated as the internal solitary wave propagates
over the slope, which would then decompose into secondary solitary waves or a wavetrain.
On the other hand, when internal solitary waves propagate over a smooth bump or a
parabolic mound, a trailing shelf of negative polarity would be generated as the results of
the interaction of the internal solitary wave with the decreasing slope of the bump or the
parabolic mound. The secondary solitary waves is observed to be climbing the negative
The selection criteria play an important role in the portfolio optimization
using any ratio model. In this paper, the authors have considered the mean return as
profit and variance of return as risk on the asset return as selection criteria, as the first
stage to optimize the selected portfolio. Furthermore, the sharp ratio (SR) has been
considered to be the optimization ratio model. In this regard, the historical data taken
from Shanghai Stock Exchange (SSE) has been considered. A metaheuristic technique
has been developed, with financial tool box available in MATLAB and the particle swarm
optimization (PSO) algorithm. Hence, called as the hybrid particle swarm optimization
(HPSO) or can also be called as financial tool box particle swarm optimization (FTBPSO).
In this model, the budgets as constraint, where as two different models i.e. with
and without short sale, have been considered. The obtained results have been compared
with the existing literature and the proposed technique is found to be optimum and better
in terms of profit.
In this paper, we consider the system of Volterra-Fredholm integral equations
of the second kind (SVFI-2). We proposed fixed point method (FPM) to solve
SVFI-2 and improved fixed point method (IFPM) for solving the problem. In addition,
a few theorems and two new algorithms are introduced. They are supported by
numerical examples and simulations using Matlab. The results are reasonably good
when compared with the exact solutions.
An accurate forecasting of tropospheric ozone (O3) concentration is benefi-
cial for strategic planning of air quality. In this study, various forecasting techniques are
used to forecast the daily maximum O3 concentration levels at a monitoring station
in the Klang Valley, Malaysia. The Box-Jenkins autoregressive integrated movingaverage
(ARIMA) approach and three types of neural network models, namely, backpropagation
neural network, Elman recurrent neural network and radial basis function
neural network are considered. The daily maximum data, spanning from 1 January
2011 to 7 August 2011, was obtained from the Department of Environment, Malaysia.
The performance of the four methods in forecasting future values of ozone concentrations
is evaluated based on three criteria, which are root mean square error (RMSE),
mean absolute error (MAE) and mean absolute percentage error (MAPE). The findings
show that the Box-Jenkins approach outperformed the artificial neural network
A new method to construct the distinct Hamiltonian circuits in complete
graphs is called Half Butterfly Method. The Half Butterfly Method used the concept
of isomorphism in developing the distinct Hamiltonian circuits. Thus some theoretical
works are presented throughout developing this method.
In the recent economic crises, one of the precise uniqueness that all stock
markets have in common is the uncertainty. An attempt was made to forecast future
index of the Malaysia Stock Exchange Market using artificial neural network (ANN)
model and a traditional forecasting tool – Multiple Linear Regressions (MLR). This
paper starts with a brief introduction of stock exchange of Malaysia, an overview of
artificial neural network and machine learning models used for prediction. System
design and data normalization using MINITAB software were described. Training
algorithm, MLR Model and network parameter models were presented. Best training
graphs showing the training, validation, test and all regression values were analyzed.
Medical diagnosis is the extrapolation of the future course and outcome
of a disease and a sign of the likelihood of recovery from that disease. Diagnosis
is important because it is used to guide the type and intensity of the medication
to be administered to patients. A hybrid intelligent system that combines the fuzzy
logic qualitative approach and Adaptive Neural Networks (ANNs) with the capabilities
of getting a better performance is required. In this paper, a method for modeling
the survival of diabetes patient by utilizing the application of the Adaptive NeuroFuzzy
Inference System (ANFIS) is introduced with the aim of turning data into
knowledge that can be understood by people. The ANFIS approach implements the
hybrid learning algorithm that combines the gradient descent algorithm and a recursive
least square error algorithm to update the antecedent and consequent parameters. The
combination of fuzzy inference that will represent knowledge in an interpretable manner
and the learning ability of neural network that can adjust the membership functions
of the parameters and linguistic rules from data will be considered. The proposed
framework can be applied to estimate the risk and survival curve between different
diagnostic factors and survival time with the explanation capabilities.
Replicated linear functional relationship model is often used to describe
relationships between two circular variables where both variables have error terms and
replicate observations are available. We derive the estimate of the rotation parameter
of the model using the maximum likelihood method. The performance of the proposed
method is studied through simulation, and it is found that the biasness of the estimates
is small, thus implying the suitability of the method. Practical application of the
method is illustrated by using a real data set.
Optimization is central to any problem involving decision making. The area
of optimization has received enormous attention for over 30 years and it is still popular
in research field to this day. In this paper, a global optimization method called Improved
Homotopy with 2-Step Predictor-corrector Method will be introduced. The method in-
troduced is able to identify all local solutions by converting non-convex optimization
problems into piece-wise convex optimization problems. A mechanism which only consid-
ers the convex part where minimizers existed on a function is applied. This mechanism
allows the method to filter out concave parts and some unrelated parts automatically.
The identified convex parts are called trusted intervals. The descent property and the
global convergence of the method was shown in this paper. 15 test problems have been
used to show the ability of the algorithm proposed in locating global minimizer.
In DNA splicing system, the potential effect of sets of restriction enzymes and
a ligase that allow DNA molecules to be cleaved and re-associated to produce further
molecules is modelled mathematically. This modelling is done in the framework of formal
language theory, in which the nitrogen bases, nucleotides and restriction sites are modelled
as alphabets, strings and rules respectively. The molecules resulting from a splicing system
is depicted as the splicing language. In this research, the splicing language resulting from
DNA splicing systems with one palindromic restriction enzyme for one and two (nonoverlapping)
cutting sites are generalised as regular expressions.
Real life phenomena found in various fields such as engineering, physics,
biology and communication theory can be modeled as nonlinear higher order ordinary
differential equations, particularly the Duffing oscillator. Analytical solutions for these
differential equations can be time consuming whereas, conventional numerical solutions
may lack accuracy. This research propose a block multistep method integrated with a
variable order step size (VOS) algorithm for solving these Duffing oscillators directly.
The proposed VOS Block method provides an alternative numerical solution by reducing
computational cost (time) but without loss of accuracy. Numerical simulations
are compared with known exact solutions for proof of accuracy and against current
numerical methods for proof of efficiency (steps taken).
Numerical simulation of the behaviour of blood flow through a stenosed bifurcated
artery with the presence of single mild stenosis at parent artery is investigated. The
flow analysis applies the incompressible, steady, three-dimensional Navier-Stokes equations
for non-Newtonian generalized power law fluids. Behaviour of blood flow is simulated
numerically using COMSOL Multiphysicsthat based on finite element method.The
results showthe effect of severity of stenosis on flow characteristics such as axial velocity
and its exhibit flow recirculation zone for analysis on streamlines pattern.