# Matrix Science Mathematic (MSMK)

## ABSTRACT

### A NOTE ON β-DERIVATIONS IN PRIME NEAR RING

Journal: Matrix Science Mathematic (MSMK)

This is an open access article distributed under the Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited In this paper, we prove commutativity of prime near rings by using the notion of β-derivations. Let M be a prime near ring. If there exist p,q ϵ M and two sided nonzero β-derivation f on M, where β:M→M is a homomorphism, satisfying the following conditions:

f([s,t])=s^p [β(s),β(t)]s^q ∀ s,t ϵ M
f([s,t])=-s^p [β(s),β(t)]s^q ∀ s,t ϵ M

 Pages 20-23 Year 2021 Issue 1 Volume 5

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## ABSTRACT

### A NOTE ON COMMUTATIVITY OF PRIME NEAR RING WITH GENERALIZED β-DERIVATION

Journal: Matrix Science Mathematic (MSMK)

This is an open access article distributed under the Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited In this paper, we prove commutativity of prime near rings by using the notion of β-derivations. Let M be a prime near ring. If there exist  and two sided generalized β-derivation G associated with the non-zero two sided β-derivation  on M, where  is a homomorphism, satisfying the following conditions:

1. G([p_1,q_1 ])=〖p_1〗^(u_1 ) [β(p_1 ),β(q_1)]〖p_1〗^(v_1 ) ∀ p_1,q_1 ϵ M
2. G([p_1,q_1 ])=〖p_1〗^(u_1 ) [β(p_1 ),β(q_1)]〖p_1〗^(v_1 ) ∀ p_1,q_1 ϵ M

Then M is a commutative ring.

 Pages 16-19 Year 2021 Issue 1 Volume 5

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## ABSTRACT

### ON COMMUTATIVITY OF PRIME NEAR RINGS

Journal: Matrix Science Mathematic (MSMK)

This is an open access article distributed under the Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited In this paper, we prove commutativity of prime near rings by using the notion of β-derivations. Let M be a zero symmetric prime near ring. If there exist p ≥ 0, q ≥ 0 and a nonzero two sided β-derivation d on M, where β : M M is a homomorphism, such that d satisfy one of the following conditions:

• [β(s),d(t)] = sp(β(s)(t))sq s, t M
• [β(s),d(t)] = −sp(β(s)(t))sq s, t M
• [d(s)(t)] = tp(β(s)(t))tq s, t M
• [d(s)(t)] = −tp(β(s)(t)tq s, t M
 Pages 06-15 Year 2021 Issue 1 Volume 5

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## ABSTRACT

### DIFFERENTIAL TRANSFORMATION METHOD FOR SOLVING MALARIA –HYGIENE MATHEMATICAL MODEL

Journal: Matrix Science Mathematic (MSMK)
Author: Oluwafemi, Temidayo J., Azuaba, Emmaunel, Sulemain Amina S

This is an open access article distributed under the Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited In this study, we proposed a malaria-hygiene mathematical model using non-linear differential equation. The model equations are divided into seven compartments consisting of five human compartments (Hygienic Susceptible, Unhygienic Susceptible, Hygienic Infected, Unhygienic Infected, and Recovered) and two vector compartments (Non-Disease Carrier vector and Disease carrier vector). Differential Transformation Method (DTM) is applied to solve the mathematical model. The solutions obtained by DTM are compared with Runge-Kutta order 4th method (RK4). The graphical solutions illustrate similarity between DTM and RK4. It therefore imply that DTM can be consider a reliable alternative solution method.
 Pages 01-05 Year 2021 Issue 1 Volume 5

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## ABSTRACT

### EXISTENCE RESULTS TO A CLASS OF FIRST-ORDER FUNCTIONAL INTEGRO-DIFFERENTIAL INCLUSIONS

Journal: Matrix Science Mathematic (MSMK)
Author: A.M.A. El-Sayed, Sh. M Al-Issa, M.H. Hijazi

This is an open access article distributed under the Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We establish the existence of continuous solutions to initial value problem for first order functional integro-differential inclusion. The study holds in the case when the set-valued function has Lipschitz selection, also we discuss the existence of maximal and minimal solutions. The continuous dependence and uniqueness of the solution will be proved. As an application, the initial value problem for the arbitrary-order differential inclusion will be studied.
 Pages 44-50 Year 2020 Issue 2 Volume 4

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## ABSTRACT

### SOLUTION OF THE HIV INFECTION MODEL WITH FULL LOGISTIC PROLIFERATION AND VARIABLE SOURCE TERM USING GALERKIN SCHEME

Journal: Matrix Science Mathematic (MSMK)
Author: Attaullah, Rashid Jan, A. Jabeen

This is an open access article distributed under the Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited The present study is related to the numerical solution of the human immunodeficiency virus (HIV) infection model with full logistic proliferation and variable source term (depending on the viral load) used for the supply of new CD4+ T-cells from thymus instead of using simple logistic proliferation and constant source term. In simple logistic proliferation term only the healthy or infected CD4+ T-cells proliferation are considered while in full logistic proliferation term both the proliferation of healthy and infected are considered. Consequently, the variable source term is used for the supply of new healthy CD4+ T-cells from thymus, which is a decreasing function depending on the concentration of viral load. Continuous Galerkin-Petrov method, in particular cGP(2)-method has been invoked for finding the approximate solution of the model. For cGP(2)-method, we have two unknowns on each time interval which have to be calculated by solving 2 × 2 block system. This method is an accurate of order three in the whole time interval and shows the convergence of order four in the discrete time points. We examined the impact of various clinical parameters and study the existence of the infected state. Additionally, the Runge Kutta method of order four briefly RK4-method has also been used to verify and strengthen the results obtained by cGP(2)-method. Obtained results are displayed both graphically and in tabular form. The results obtained in this study confirm the idea that the cGP(2)-method is a powerful technique which can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.
 Pages 37-43 Year 2020 Issue 2 Volume 4

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## ABSTRACT

### THE EFFECT OF NON-LOCALITY IN BOUNDARY CONDITIONS ON THE EIGENVALUES OF SOME FINITE DIFFERENCE SCHEMES

Journal: Matrix Science Mathematic (MSMK)
Author: A. Elmekawy, S. M. Helal, M. El-Aza

This is an open access article distributed under the Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited In this paper, we analyze a new form of non-local boundary conditions for a two-dimensional elliptic partial differential equation model. Some relations for the existence of the different types of eigenvalues and their corresponding eigenfunctions are proved. The figures of the relations are also dragged to show the effect of the non-locality in boundary conditions on the eigenvalue problem.
 Pages 32-36 Year 2020 Issue 2 Volume 4

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## ABSTRACT

### LAGUERRE POLYNOMIALS SOLUTION FOR SOLVING HIGH-ORDER DELAY LINEAR DIFFERENTIAL EQUATIONS

Journal: Matrix Science Mathematic (MSMK)
Author: Mohamed El-Gamel, Walid Tharwat, Magdy El-Azab

This is an open access article distributed under the Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited The aim of this article is to present an efficient numerical procedure for solving higher-order linear delay differential equations with variable coefficients under the mixed conditions in terms of Laguerre polynomials. Four problems are solved and results are compared with the existing results to show the accuracy and applicability of Laguerre polynomials.
 Pages 27-31 Year 2020 Issue 1 Volume 4

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## ABSTRACT

### CUBIC B-SPLINE SOLUTION FOR A SECOND-ORDER SINGULAR LINEAR PARTIAL DIFFERENTIAL EQUATIONS

Journal: Matrix Science Mathematic (MSMK)
Author: Mohamed El-Gamel, Neveen El-Shamy

This is an open access article distributed under the Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited A new approach implementing the cubic B-spline technique is introduced for the numerical solution of a class of singular partial differential equation. Properties of these cubic B-spline functions are first presented. These properties are then used to reduce singular partial differential equation to systems of linear algebraic equations. Numerical examples illustrate the pertinent features of the method and its applicability to a large variety of problems. The examples include regular and singular problems.
 Pages 20-26 Year 2020 Issue 1 Volume 4

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## ABSTRACT

### COMPUTATION OF THE POWER OF BASE OF TWO DIGITS NUMBER USING KIFILIDEEN (MATRIX, COMBINATION AND DISTRIBUTIVE (MCD)) APPROACH

Journal: Matrix Science Mathematic (MSMK)
Author: Kifilideen L. Osanyinpeju

This is an open access article distributed under the Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited The different methods used in the multiplication of numbers are long method (column method) of multiplication, grid methods and addition methods of multiplication. The utilization of the methods mentioned to solve power (index) of base number (number that multiply itself in a number of times) are horrendous, outrageous, tedious, time consuming and too long to be carried out. This study develops computation of the power of base of number using Kifilideen (matrix, combination and distributive (MCD)) approach. The Kif matrix method of multiplication is a shorter version of the long method (column method) of multiplication. The matrix method provides a straight forward, direct and systematic means of multiplication of digit number that multiply itself repetitively.
 Pages 14-19 Year 2020 Issue 1 Volume 4