International Journal of Mathematical Research

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Online ISSN: 2306-2223
Print ISSN: 2311-7427
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No. 1

Methods for Estimating Missing Values in Descriptive Time Series Statistics: Novelty and Efficiency under Buys-Ballot

Pages: 72-80
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Methods for Estimating Missing Values in Descriptive Time Series Statistics: Novelty and Efficiency under Buys-Ballot

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DOI: 10.18488/journal.24.2020.91.72.80

Ugochinyere I. Nwosu , Chukwudi P. Obite

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Ugochinyere I. Nwosu , Chukwudi P. Obite (2020). Methods for Estimating Missing Values in Descriptive Time Series Statistics: Novelty and Efficiency under Buys-Ballot. International Journal of Mathematical Research, 9(1): 72-80. DOI: 10.18488/journal.24.2020.91.72.80
There is dearth of information in the field of statistics on the innovative estimation methods that can replace missing values in descriptive time series data. Therefore, this review work provides information on the existing and new methods of estimating missing values in descriptive time series data. The work provides new insight on the comparative performance of the recently-developed methods and the existing ones and discussed model structure and trending curves as important parameters in estimation of missing values. It is expected that the present contribution will assist statisticians seeking to solve the problem of missing values in descriptive time series data. The application of this work should be restricted to time series data with trend (linear, quadratic and exponential) and seasonal components combined in the additive and multiplicative forms. The contribution covers data missing at one point at a time in a row or column when data are arranged in a Buys-Ballot table. Use of the Buys-Ballot table arrangement in the estimation of missing values is new, convenient and merits scientific analysis.
Contribution/ Originality
This review work is one of the few papers that discussed the novelty and efficiency of Buys-Ballot table in the estimation of missing values in descriptive time series statistics.

Smart Parking System Using Fuzzy Logic Controller for Alien Cities

Pages: 62-71
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Smart Parking System Using Fuzzy Logic Controller for Alien Cities

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DOI: 10.18488/journal.24.2020.91.62.71

Muhammad Saqlain , Muhammad Saeed , Muhammad Haris Saeed

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Muhammad Saqlain , Muhammad Saeed , Muhammad Haris Saeed (2020). Smart Parking System Using Fuzzy Logic Controller for Alien Cities. International Journal of Mathematical Research, 9(1): 62-71. DOI: 10.18488/journal.24.2020.91.62.71
Traffic is regarded as one of the biggest issues in cities especially in metropolitan areas with high population density. It is the cause of wastage of precious resources that include fuel, money, and most importantly time. One of the reasons for traffic jams is the people finding an appropriate place to park. To address the issue, various techniques have been implemented by different traffic management authorities, and systems of modern cars have been integrated with smart parking solutions. The fuzzy logic controller is regarded as an Artificial Intelligence product that may be used to alleviate the problem. Four linguistic inputs are used in the paper such that parking car unit, distance, number of traffic signals, and parking area to provide one output that is time. This Fuzzy Logic Controller will be useful for the drivers to locate the shortest track among other tracks in the least amount of time. Moreover, the issue of finding an appropriate parking spot can be solved.
Contribution/ Originality
This study is one of very few studies which have investigated the application of Fuzzy Logic Controller in parking system. All the calculations are done using MATLAB’s fuzzy logic controller toolbox and its mobile usability is also presented.

Mathematical Model of an SIR Epidemic Switching with Zero Co-Infectives

Pages: 42-61
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Mathematical Model of an SIR Epidemic Switching with Zero Co-Infectives

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DOI: 10.18488/journal.24.2020.91.42.61

Reuben Iortyer Gweryina , Chinwendu Emilian Madubueze , Peter Arome Sani

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Reuben Iortyer Gweryina , Chinwendu Emilian Madubueze , Peter Arome Sani (2020). Mathematical Model of an SIR Epidemic Switching with Zero Co-Infectives. International Journal of Mathematical Research, 9(1): 42-61. DOI: 10.18488/journal.24.2020.91.42.61
This paper studies the global dynamics of an SIR epidemic switching model with zero co-infectives and intervention programmes. The model considers two epidemics of non-specific nomenclature in which the first epidemic is a precondition to the outbreak of the second epidemic. Analytical study of the model exposed the two epidemic steady states, namely, epidemic-free equilibrium (EFE) and epidemic endemic equilibrium (EEE). Both equilibrium states are shown to be globally attractive points with respect to the criteria of the basic reproduction number using Lyapunov stability theory. Some sufficient conditions on the model parameters are obtained to show the existence of the forward bifurcation. Finally, numerical simulations are done to exemplify the qualitative results and the impact of switching and intervention programmes. The numerical results shown that switching reduces the susceptibility and infectivity of the first epidemic and increases that of the second epidemic. Also, depending on the severity of the both epidemics, the different levels of intervention programmes are needed to reduce the number of infectives in both epidemics. However, equal intervention programmes are recommended for both epidemics to avoid neglecting one epidemic during outbreaks of the two epidemics.
Contribution/ Originality
This study is one of the few studies in mathematical epidemiology which have investigated the role of switching in an SIR model of two epidemics with zero co-infectives. In addition, Lyapunov functions theory and Center Manifold method is applied to the model for the global stability analysis and existence of forward bifurcation respectively.

An Order Four Numerical Scheme for Fourth-Order Initial Value Problems Using Lucas Polynomial with Application in Ship Dynamics

Pages: 28-41
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An Order Four Numerical Scheme for Fourth-Order Initial Value Problems Using Lucas Polynomial with Application in Ship Dynamics

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DOI: 10.18488/journal.24.2020.91.28.41

Luke Azeta Ukpebor , Ezekiel Olaoluwa Omole , Lawrence Osa Adoghe

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Luke Azeta Ukpebor , Ezekiel Olaoluwa Omole , Lawrence Osa Adoghe (2020). An Order Four Numerical Scheme for Fourth-Order Initial Value Problems Using Lucas Polynomial with Application in Ship Dynamics. International Journal of Mathematical Research, 9(1): 28-41. DOI: 10.18488/journal.24.2020.91.28.41
From the time immemorial, researchers have been beaming their search lights round the numerical solution of ordinary differential equation of initial value problems. This was as a result of its large applications in the area of Sciences, Engineering, Medicine, Control System, Electrical Electronics Engineering, Modeled Equations of Higher order, Thin flow, Fluid Mechanics just to mention few. There are a lot of differential equations which do not have theoretical solution; hence the use of numerical solution is very imperative. This paper presents the derivation, analysis and implementation of a class of new numerical schemes using Lucas polynomial as the approximate solution for direct solution of fourth order ODEs. The new schemes will bridge the gaps of the conventional methods such as reduction of order, Runge-kutta’s and Euler’s methods which has been reported to have a lot of setbacks. The schemes are chosen at the integration interval of seven-step being a perfection interval. The even grid-points are interpolated while the odd grid-points are collocated. The discrete scheme, additional schemes and derivatives are combined together in block mode for the solution of fourth order problems including special, linear as well as application problems from Ship Dynamics. The analysis of the schemes shows that the schemes are Reliable, P-stable and Efficient. The basic properties of the schemes were examined. Numerical results were presented to demonstrate the accuracy, the convergence rate and the speed advantage of the schemes. The schemes perform better in terms of accuracy when compared with other methods in the literature.
Contribution/ Originality
The study uses Lucas polynomial for the derivation of a new class of numerical schemes. The schemes were implemented in block mode for approximating fourth order ODEs directly without reduction. It solves variety of problems including problem in Electrical Engineering. The schemes performs excellently better than other schemes in the literatures.

Numerical Computation of Fitzhugh-Nagumo Equation: A Novel Galerkin Finite Element Approach

Pages: 20-27
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Numerical Computation of Fitzhugh-Nagumo Equation: A Novel Galerkin Finite Element Approach

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DOI: 10.18488/journal.24.2020.91.20.27

Hazrat Ali , Md. Kamrujjaman , Md. Shafiqul Islam

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Hazrat Ali , Md. Kamrujjaman , Md. Shafiqul Islam (2020). Numerical Computation of Fitzhugh-Nagumo Equation: A Novel Galerkin Finite Element Approach. International Journal of Mathematical Research, 9(1): 20-27. DOI: 10.18488/journal.24.2020.91.20.27
The key objective of this research paper is to find the numerical solution of the famous FitzHugh-Nagumo equation. The numerical scheme used here is the Galerkin finite element method (GFEM) in a simple and convenient way. Because the advantages of using GFEM are that it can be used directly without any linearization or any other restrictive assumption, it uses shape functions instead of trial functions, and it gives a polynomial at each point instead of value, so it can be used to find value at any point within the domain. First we derive the detail formulation of GFEM for this nonlinear parabolic partial differential equation. Then we solve the FitzHugh-Nagumo equation for various values of. Later, we solve another renowned Newell-Whitehead equation for the verification of the consistency of this algorithm. The results are depicted both graphically and numerically. All results are compared with the analytical solutions to show the convergence of the proposed algorithm. Those results demonstrate that our proposed algorithm works efficiently and gives a very good agreement with the exact solution. This can be applied for solving any nonlinear parabolic partial differential equation (PDE).
Contribution/ Originality
The study uses the new estimation methodology for approximating the numerical solution of the well-known FitzHug-Nagumo equation by GFEM. It is also remarked that this study is one of very few studies which have approximated the famous F-N partial differential equation using a new technique.

Application of Interval Valued Fuzzy Soft Max-Min Decision Making Method

Pages: 11-19
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Application of Interval Valued Fuzzy Soft Max-Min Decision Making Method

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DOI: 10.18488/journal.24.2020.91.11.19

Rana Muhammad Zulqarnain , Muhammad Saeed , Bagh Ali , Nadeem Ahmad , Liaqat Ali , Sohaib Abdal

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Rana Muhammad Zulqarnain , Muhammad Saeed , Bagh Ali , Nadeem Ahmad , Liaqat Ali , Sohaib Abdal (2020). Application of Interval Valued Fuzzy Soft Max-Min Decision Making Method. International Journal of Mathematical Research, 9(1): 11-19. DOI: 10.18488/journal.24.2020.91.11.19
In this paper, we study some basic concepts of fuzzy sets, soft sets, fuzzy soft sets, and interval-valued fuzzy soft sets. Secondly, we study the interval-valued fuzzy soft max-min decision-making function and developed the graphical model for the Interval-valued fuzzy soft max-min decision making (IVFSMmDM) method by using Interval-valued fuzzy soft max-min decision-making function. Finally, we used IVFSMmDM for faculty selection in the education department and observed that T4   is the best teacher for teaching by using hypothetical data.
Contribution/ Originality
This paper contributes a new graphical model for IVFSMmDM by using interval-valued fuzzy soft max-min decision-making function and used for faculty selection in the education department.

Numerical Solutions of Black-Scholes Model by Du Fort-Frankel FDM and Galerkin WRM

Pages: 1-10
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Numerical Solutions of Black-Scholes Model by Du Fort-Frankel FDM and Galerkin WRM

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DOI: 10.18488/journal.24.2020.91.1.10

Md. Shorif Hossan , A B M Shahadat Hossain , Md. Shafiqul Islam

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Md. Shorif Hossan , A B M Shahadat Hossain , Md. Shafiqul Islam (2020). Numerical Solutions of Black-Scholes Model by Du Fort-Frankel FDM and Galerkin WRM. International Journal of Mathematical Research, 9(1): 1-10. DOI: 10.18488/journal.24.2020.91.1.10
The main objective of this paper is to find the approximate solutions of the Black-Scholes (BS) model by two numerical techniques, namely, Du Fort-Frankel finite difference method (DF3DM), and Galerkin weighted residual method (GWRM) for both (call and put) type of European options. Since both DF3DM and GWRM are the most familiar numerical techniques for solving partial differential equations (PDE) of parabolic type, we estimate options prices by using these techniques. For this, we first convert the Black-Scholes model into a modified parabolic PDE, more specifically, in DF3DM, the first temporal vector is calculated by the Crank-Nicolson method using the initial boundary conditions and then the option price is evaluated. On the other hand, in GWRM, we use piecewise modified Legendre polynomials as the basis functions of GWRM which satisfy the homogeneous form of the boundary conditions. We may observe that the results obtained by the present methods converge fast to the exact solutions. In some cases, the present methods give more accurate results than the earlier results obtained by the adomian decomposition method [14]. Finally, all approximate solutions are shown by the graphical and tabular representations.
Contribution/ Originality
The paper’s primary contribution is finding that the approximate results of Black-Scholes model by DF3DM, and GWRM with modified Legendre polynomials as basis functions.