In this study, we consider a linear nonhomogeneous differential equation with variable coefficients and variable delays and present a novel matrix-collocation method based on Morgan-Voyce polynomials to obtain the approximate solutions under the initial conditions. The method reduces the equation with variable delays to a matrix equation with unknown MorganVoyce coefficients. Thereby, the solution is obtained in terms of Morgan-Voyce polynomials. In addition, two test problems together with error analysis are performed to illustrate the accuracy and applicability of the method; the obtained results are scrutinized and interpreted by means of tables and figures.

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In this paper, an efficient decomposition method is constructed and used for solving system of nonlinear equations. The method based on the decomposition technique of Noor [M.A.Noor, K.I.Noor, Some iterative schemes for nonlinear equations, Appl. Math. Comput. 183(2006), 774-779]. This technique is revised to solve the system of nonlinear equations. Some illustrative examples have been presented, to demonstrate the proposed method and the results are compared with those derived from the previous methods. All test problems reveals the accuracy and fast convergence of the suggested method.

Nonlinear differential equations have many applications in different science and engineering disciplines. However, a nonlinear differential equation cannot be solved analytically and so must be solved numerically. Thus, we aim to develop a novel numerical algorithm based on Morgan-Voyce polynomials with collocation points and operational matrix method to solve nonlinear differential equations. In the our proposed method, the nonlinear differential equations including quadratic and cubic terms having the initial conditions are converted to a matrix equation. In order to obtain the matrix equations and solutions for the selected problems, code was developed in MATLAB. The solution of this method for the convergence and efficiency was compared with the equations such as Van der Pol differential equation calculated by different methods.

Recently, some authors have established a number of inequalities involving the minimum eigenvalue for the Hadamard product of M-matrices. In this paper, we improve these results and give some new lower bounds on the minimum eigenvalue for the Hadamard product of an M-matrix A and its inverse {A^{-1}}. Finally, it is shown by the numerical examples that our bounds are also better than some previous results.