Homotopy continuation methods (HCMs) can be used to find the solutions
of polynomial equations. The advantages of HCMs over classical methods such as the
Newton and bisection methods are that HCMs are able to resolve divergence and starting
value problems. In this paper, we develop Super Ostrowski-HCM as a technique to
overcome the starting value problem. We compare the performance of this proposed
method with Ostrowski-HCM. The results provide evidence of the superiority of Super
Ostrowski-HCM.
Subsea cable laying process is a difficult task for an engineer due to many
uncertain situations which occur during the operation. It is very often that the cable being
laid out is not perfectly fit on the route being planned, which results in the formation of
slack. In order to control wastages during installation, the slack needs to be minimized
and the movement of a ship/vessel needs to be synchronized with the cable being laid out.
The current problem was addressed using a mathematical model by considering a number
of defining parameters such as the external forces, the cable properties and geometry. Due
to the complexity, the model is developed for a steady-state problem assuming velocity
of the vessel is constant, seabed is flat and the effect of wind and wave is insignificant.
Non-dimensional system is used to scale the engineering parameters and grouped them
into only two main parameters which are the hydrodynamic drag of the fluid and the
bending stiffness of the cable. There are two solutions generated in this article; numerical
and asymptotic solutions. The result of these solutions suggests that the percentage of
slack can be reduced by the increase of the prescribed cable tension, and also the increase
in either the drag coefficient of the sea water or the bending stiffness of the cable, similarly
will result in lower slack percentage
The box plot has been used for a very long time since 70s in checking the existence
of outliers and the asymmetrical shape of data. The existing box plot is constructed
using five values of statistics calculated from either the discrete or continous data. Many
improvement of box plots have deviated from the elegant and simplier approach of exploratory
data analysis by incorporating many other statistic values resulting the turning
back of the noble philosophy behind the creation of box plot. The modification using
range value with the minimum and maximum values are being incorporated to suit the
need of selected discrete distribution when outliers is not an important criteria anymore.
The new modification of box plot is not based on the asymmetrical shape of distribution
but more on the spreading and partitioning data into range measure. The new propose
name for the box plot with only three values of statistics is called range-box plot.
Hantaviruses are etiological agents of zoonotic diseases and certain other dis-
eases, which pose a serious threat to human health. When rodent and predator popula-
tions share in an ecology, the competitive force of the populations can lead to a reduction
or elimination of a hantavirus outbreak. The effect of the predator eliminating rodents
and predator populations that tends to reduce or eliminate hantavirus infection is investi-
gated. The existence of several equilibrium points of the model is identified and local and
global stabilities of the model at these equilibrium points are analysed in detail. Numerical
simulations are carried out to illustrate our model results.
A mathematical model is considered to determine the effectiveness of disin-
fectant solution for surface decontamination. The decontamination process involved the
diffusion of bacteria into disinfectant solution and the reaction of the disinfectant killing
effect. The mathematical model is a reaction-diffusion type. Finite difference method and
method of lines with fourth-order Runge-Kutta method are utilized to solve the model
numerically. To obtain stable solutions, von Neumann stability analysis is employed to
evaluate the stability of finite difference method. For stiff problem, Dormand-Prince
method is applied as the estimated error of fourth-order Runge-Kutta method. MATLAB
programming is selected for the computation of numerical solutions. From the results
obtained, fourth-order Runge-Kutta method has a larger stability region and better ac-
curacy of solutions compared to finite difference method when solving the disinfectant
solution model. Moreover, a numerical simulation is carried out to investigate the effect
of different thickness of disinfectant solution on bacteria reduction. Results show that
thick disinfectant solution is able to reduce the dimensionless bacteria concentration more
effectively.
Riverbank filtration (RBF) system is a surface water technology that is based
on the natural treatment of filtration instead of the use of chemicals, to pre-treat sur-
face water and provides public water supplies. Hydraulic conductivity value is one of the
significant factors affecting the water quality in RBF systems. In this article, an analyti-
cal modelling is developed to investigate the effect of this parameter on one dimensional
contaminant transport in RBF system. The model is solved by using Green’s function
approach. The model is applied for the first RBF system conducted in Malaysia. Gener-
ally, the results show that increasing the hydraulic conductivity value lead to an increase
in contaminant concentration in pumping well area.
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
this problem.
Cardiovascular disease (CVD) includes coronary heart disease, cerebrovascular disease (stroke), peripheral artery disease, and atherosclerosis of the aorta. All females face the threat of CVD. But becoming aware of symptoms and signs is a great challenge since most adults at increased risk of cardiovascular disease (CVD) have no symptoms or obvious signs especially in females. The symptoms may be identified by the assessment of their risk factors. The Bayesian approach is a specific way in dealing with this kind of problem by formalizing a priori beliefs and of combining them with the available observations. This study aimed to identify associated risk factors in CVD among female patients presenting with ST Elevation Myocardial Infarction (STEMI) using Bayesian logistic regression and obtain a feasible model to describe the data. A total of 874 STEMI female patients in the National Cardiovascular Disease Database-Acute Coronary Syndrome (NCVD-ACS) registry year 2006-2013 were analysed. Bayesian Markov Chain Monte Carlo (MCMC) simulation approach was applied in the univariate and multivariate analysis. Model performance was assessed through the model calibration and discrimination. The final multivariate model of STEMI female patients consisted of six significant variables namely smoking, dyslipidaemia, myocardial infarction (MI), renal disease, Killip class and age group. Females aged 65 years and above have higher incidence of CVD and mortality is high among female patients with Killip class IV. Also, renal disease was a strong predictor of CVD mortality. Besides, performance measures for the model was considered good. Bayesian logistic regression model provided a better understanding on the associated risk factors of CVD for female patients which may help tailor prevention or treatment plans more effectively.
This paper revisits the comrade matrix approach in finding the greatest com-
mon divisor (GCD) of two orthogonal polynomials. The present work investigates on the
applications of the QR decomposition with iterative refinement (QRIR) to solve certain
systems of linear equations which is generated from the comrade matrix. Besides iterative
refinement, an alternative approach of improving the conditioning behavior of the coeffi-
cient matrix by normalizing its columns is also considered. As expected the results reveal
that QRIR is able to improve the solutions given by QR decomposition while the nor-
malization of the matrix entries do improves the conditioning behavior of the coefficient
matrix leading to a good approximate solutions of the GCD.
Tumour cells behave differently than normal cells in the body. They grow and
divide in an uncontrolled manner (actively proliferating) and fail to respond to signal.
However, there are cells that become inactive and reside in quiescent phase (G0). These
cells are known as quiescence cells that are less sensitive to drug treatments (radiotherapy
and chemotherapy) than actively proliferation cells. This paper proposes a new mathe-
matical model that describes the interaction of tumour growth and immune response by
considering tumour population that is divided into three different phases namely inter-
phase, mitosis and G0. The model consists of a system of delay differential equations
where the delay, represents the time for tumour cell to reside interphase before entering
mitosis phase. Stability analysis of the equilibrium points of the system was performed
to determine the dynamics behaviour of system. Result showed that the tumour popu-
lation depends on number of tumour cells that enter active (interphase and mitosis) and
G0phases. This study is important for treatment planning since tumour cell can resist
treatment when they refuge in a quiescent state.
In irradiation process, instead of traverse on the targeted cells, there is side
effect happens to non-targeted cells. The targeted cells that had been irradiated with
ionizing radiation emits damaging signal molecules to the surrounding and then, dam-
age the bystander cells. The type of damage considered in this work is the number of
double-strand breaks (DSBs) of deoxyribonucleic acid (DNA) in cell’s nucleus. By us-
ing mathematical approach, a mechanistic model that can describe this phenomenon is
developed based on a structured population approach. Then, the accuracy of the model
is validated by its ability to match the experimental data. The Particle Swarm (PS)
optimization is employed for the data fitting procedure. PS optimization searches the
parameter value that minimize the errors between the model simulation data and exper-
imental data. It is obtained that the mathematical modelling proposed in this paper is
strongly in line with the experimental data.
Rainfall is an interesting phenomenon to investigate since it is directly related
to all aspects of life on earth. One of the important studies is to investigate and under-
stand the rainfall patterns that occur throughout the year. To identify the pattern, it
requires a rainfall curve to represent daily observation of rainfall received during the year.
Functional data analysis methods are capable to convert discrete data intoa function that
can represent the rainfall curve and as a result, try to describe the hidden patterns of the
rainfall. This study focused on the distribution of daily rainfall amount using functional
data analysis. Fourier basis functions are used for periodic rainfall data. Generalized
cross-validation showed 123 basis functions were sufficient to describe the pattern of daily
rainfall amount. North and west areas of the peninsula show a significant bimodal pattern
with the curve decline between two peaks at the mid-year. Meanwhile,the east shows uni-
modal patterns that reached a peak in the last three months. Southern areas show more
uniform trends throughout the year. Finally, the functional spatial method is introduced
to overcome the problem of estimating the rainfall curve in the locations with no data
recorded. We use a leave one out cross-validation as a verification method to compare
between the real curve and the predicted curve. We used coefficient of basis functions
to get the predicted curve. It was foundthatthe methods ofspatial prediction can match
up with theexistingspatialpredictionmethodsin terms of accuracy,but it isbetterasthe new
approach provides a simpler calculation.
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.
Coxmodel is popular in survival analysis. In the case of time-varying covariate;
several subject-specific attributes possibly to change more frequently than others. This
paper deals with that issue. This study aims to analyze survival data with time-varying
covariate using a time-dependent covariate Cox model. The two case studies employed in
this work are (1) delisting time of companies from IDX and (2) delisting time of company
from LQ45 (liquidity index). The survival time is the time until a company is delisted
from IDX or LQ45. The determinants are eighteen quarterly financial ratios and two
macroeconomics indicators, i.e., the Jakarta Composite Index (JCI) and BI interest rate
that changes more frequent. The empirical results show that JCI is significant for both
delisting and liquidity whereas BI rate is significant only for liquidity. The significant
firm-specific financial ratios vary for delisting and liquidity.
The incorporation of non-linear pattern of early ages has led to new research
directions on improving the existing stochastic mortalitymodel structure. Several authors
have outlined the importance of encompassing the full age range in dealing with longevity
risk exposure, by not ignoring the dependence between young and old ages. In this study,
we consider the two extensions of the Cairns, Blake and Dowd model that incorporate the
irregularity profile seen at the mortality of lower ages, which are the Plat, and the O’Hare
and Li models respectively. The models’ performances in terms of in-sample fitting and
out-sample forecasts were examined and compared. The results indicated that the O’Hare
and Li model performs better as compared to the Plat model.
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.
This study presents a mathematical model examining wastewater pollutant removal through
an oxidation pond treatment system. This model was developed to describe the reaction
between microbe-based product mPHO (comprising Phototrophic bacteria (PSB)), dissolved
oxygen (DO) and pollutant namely chemical oxygen demand (COD). It consists
of coupled advection-diffusion-reaction equations for the microorganism (PSB), DO and
pollutant (COD) concentrations, respectively. The coupling of these equations occurred
due to the reactions between PSB, DO and COD to produce harmless compounds. Since
the model is nonlinear partial differential equations (PDEs), coupled, and dynamic, computational
algorithm with a specific numerical method, which is implicit Crank-Nicolson
method, was employed to simulate the dynamical behaviour of the system. Furthermore,
numerical results revealed that the proposed model demonstrated high accuracy when
compared to the experimental data.
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
trailing shelf.
Recent studies have shown that independent identical distributed Gaussian
random variables is not suitable for modelling extreme values observed during extremal
events. However, many real life data on extreme values are dependent and stationary
rather than the conventional independent identically distributed data. We propose a stationary
autoregressive (AR) process with Gumbel distributed innovation and characterise
the short-term dependence among maxima of an (AR) process over a range of sample
sizes with varying degrees of dependence. We estimate the maximum likelihood of the
parameters of the Gumbel AR process and its residuals, and evaluate the performance
of the parameter estimates. The AR process is fitted to the Gumbel-generalised Pareto
(GPD) distribution and we evaluate the performance of the parameter estimates fitted
to the cluster maxima and the original series. Ignoring the effect of dependence leads to
overestimation of the location parameter of the Gumbel-AR (1) process. The estimate
of the location parameter of the AR process using the residuals gives a better estimate.
Estimate of the scale parameter perform marginally better for the original series than the
residual estimate. The degree of clustering increases as dependence is enhance for the AR
process. The Gumbel-AR(1) fitted to the threshold exceedances shows that the estimates
of the scale and shape parameters fitted to the cluster maxima perform better as sample
size increases, however, ignoring the effect of dependence lead to an underestimation of
the parameter estimates of the scale parameter. The shape parameter of the original
series gives a superior estimate compare to the threshold excesses fitted to the Gumbel
distributed Generalised Pareto ditribution.
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.