International Journal of Mathematical Research

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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.

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 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.

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.