Applications of the Moore-Penrose Generalized Inverse to Linear Systems of Algebraic Equations
Asmaa Mohammed Kanan,
Asma Ali Elbeleze,
Afaf Abubaker
Issue:
Volume 7, Issue 6, December 2019
Pages:
152-156
Received:
4 October 2019
Accepted:
30 October 2019
Published:
18 November 2019
Abstract: In this work, we consider linear systems of algebraic equations. These systems are studied utilizing the theory of the Moore-Penrose generalized inverse or shortly (MPGI) of matrices. Some important algorithms and theorems for computation the MPGI of matrices are given. The singular value decomposition (SVD) of a matrix has a very important role in computation the MPGI, hence it is useful to study the solutions of over- and under-determined linear systems. We use the MPGI of matrices to solve linear systems of algebraic equations when the coefficients matrix is singular or rectangular. The relationship between the MPGI and the minimal least squares solutions to the linear system is expressed by theorem. The solution of the linear system using the MPGI is often an approximate unique solution, but for some cases we can get an exact unique solution. We treat the linear algebraic system as an algebraic equation with coefficients matrix A (square or rectangular) with complex entries. A closed form for solution of linear system of algebraic equations is given when the coefficients matrix is of full rank or is not of full rank, singular square matrix or non-square matrix. The results are taken from the works mentioned in the references. A few examples including linear systems with coefficients matrix of full rank and not of full rank are provided to show our studding.
Abstract: In this work, we consider linear systems of algebraic equations. These systems are studied utilizing the theory of the Moore-Penrose generalized inverse or shortly (MPGI) of matrices. Some important algorithms and theorems for computation the MPGI of matrices are given. The singular value decomposition (SVD) of a matrix has a very important role in...
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Boundary Value Problems of Nonlinear Variable Coefficient Fractional Differential Equations
Badawi Hamza Elbadawi Ibrahim,
Qixiang Dong,
Zhengdi Zhang
Issue:
Volume 7, Issue 6, December 2019
Pages:
157-163
Received:
24 October 2019
Accepted:
19 November 2019
Published:
30 December 2019
Abstract: It is recognized that the theory of boundary value problems for fractional order-differential equations is one of the rapidly developing branches of the general theory of differential equations. As far as we know, most of the papers studied the fractional Riemann-Liouville derivative with respect to boundary values that are zero. However, for the purpose of this study, we concern ourselves with Captou type derivative of the order α∈(2, 3), with respect to boundary values that are nonzero. We establish sufficient conditions for the existence of solutions for boundary value problem of nonlinear variable coefficient of fractional order. On the other hand, the boundary value problem is formulated as follows: cDαu(t) + p(t)f(t, u(t)) + q(t) = 0, u(0) = a, u'(0) = b, u(1) = d. Where a, b, d ∈ R are constants. In this paper, we investigate the existence and uniqueness of solutions for a class of boundary value problem of the nonlinear variable coefficient of fractional differential equations. The existence of solutions involving Captuo fractional derivatives is discussed under the assumption that the bounded conditions are constants. By means of the Banach contraction mapping principle and Larry- Schauder alternative, the existence of solutions are obtained. Finally, some examples are discussed to illustrate the results, which are generalized to nonlinear fractional derivatives with variable coefficients.
Abstract: It is recognized that the theory of boundary value problems for fractional order-differential equations is one of the rapidly developing branches of the general theory of differential equations. As far as we know, most of the papers studied the fractional Riemann-Liouville derivative with respect to boundary values that are zero. However, for the p...
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Stability and Hopf Bifurcation Analysis of Delayed Rosenzweig-MacArthur Model with Prey Immigration
Issue:
Volume 7, Issue 6, December 2019
Pages:
164-176
Received:
25 October 2019
Accepted:
18 November 2019
Published:
30 December 2019
Abstract: A delayed reaction Cdiffusion Rosenzweig-MacArthur model with a constant rate of prey immigration is considered. We derive the characteristic equation through partial differential equation theory, and by analyzing the distribution of the roots of the characteristic equation, the local stability of the positive equilibria is studied, and we get the conditions to determine the stability of the positive equilibria. Furthermore we find that Hopf bifurcation occurs near the positive equilibrium when the time delay passes some critical values, and we get the conditions under which the Hopf bifurcation occurs and so periodic solutions appear near the positive equilibria. By using the center manifold theory and normal form method, we derive an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. Furthermore, some numerical simulations are carried out to illustrate the analytic results of our study.
Abstract: A delayed reaction Cdiffusion Rosenzweig-MacArthur model with a constant rate of prey immigration is considered. We derive the characteristic equation through partial differential equation theory, and by analyzing the distribution of the roots of the characteristic equation, the local stability of the positive equilibria is studied, and we get the ...
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On Transmuted Type II Generalized Logistic Distribution with Application
Issue:
Volume 7, Issue 6, December 2019
Pages:
177-182
Received:
13 November 2019
Accepted:
17 December 2019
Published:
31 December 2019
Abstract: Introducing extra parameters into the baseline distribution has been a huge breakthrough in research as this enhances more flexibility of the existing models. One of the recent methods is the use of transmutation map which has attracted the interest of many researchers in the last decade. This article investigates the flexibility of transmuted type II generalized logistic distribution. The well-known type II generalized logistic distribution is transmuted using quadratic rank transmutation map to develop a transmuted type II generalized logistic distribution. The map enables the introduction of additional parameter into its parent model to make it more flexible in the analysis of data in various disciplines such as biological sciences, actuarial science, finance and insurance. Some statistical properties of the model are considered and these properties include the moment, quantiles and functions of minimum and maximum order statistics. The estimation issue of the subject model is addressed using method of maximum likelihood estimation. The model is applied to real life data to demonstrate its performance and the comparison of the result of the subject model with its parent model was done using Akaike Information criterion (AIC), Corrected Akaike Information criterion (AICC) and Bayesian Information criterion (BIC) respectively. It is believed that the results from this research work will be of immense contributions in this field and other related disciplines in modelling real data.
Abstract: Introducing extra parameters into the baseline distribution has been a huge breakthrough in research as this enhances more flexibility of the existing models. One of the recent methods is the use of transmutation map which has attracted the interest of many researchers in the last decade. This article investigates the flexibility of transmuted type...
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