Journal of Progressive Research in Mathematics Journal of Progressive Research in Mathematics Scitech Research Organisation en-US Journal of Progressive Research in Mathematics 2395-0218 Stability and Oscillation of θ-methods for Differential Equation with Piecewise Constant Arguments <p>This paper studies the numerical properties of θ-methods for the alternately advanced and retarded differential equation u′(t) = au(t)+bu(2[(t+1)/2]). Using two classes of θ-methods, namely the linear θ-method and the one-leg θ-method, the stability regions of numerical methods are determined, and the conditions of oscillation for the θ-methods are derived. Moreover, we give the conditions under which the numerical stability regions contain the analytical stability regions. It is shown that the θ-methods preserve the oscillation of the analytic solution. In addition, the relationships between stability and oscillation are presented. Several numerical examples are given.</p> Qi Wang Xueyang Liu ##submission.copyrightStatement## 2022-01-17 2022-01-17 19 1 1 16 Generalized John Numbers <p>In this paper, we define and investigate the generalized John sequences and we deal with, in detail, two special cases, namely, John and John-Lucas sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences. Furthermore, we show that there are close relations between John and John-Lucas numbers and Pell, Pell-Lucas numbers.</p> Yüksel Soykan ##submission.copyrightStatement## 2022-03-15 2022-03-15 19 1 17 34 Nonparametric Tests for the Umbrella Alternative in a Mixed Design for Location <p>This paper further investigates existing test statistics proposed by Magel et al. (2010) for detecting umbrella alternatives when the peak is known, and the underlying design consists of a completely randomized design (CRD) and randomized complete block design (RCBD). Magel et al. (2010) assumed equal variance between the CRD and the RCBD portions for the power estimates that they conducted.&nbsp; We investigate the powers of the tests compared to each other when testing for location in this design when the variance of the CRD portion is 2, 4, and 9 times larger than the variance of the RCBD portion. Underlying normal, t, and exponential distributions are considered as well as a variety of location shifts, and different ratios between the sample size in the CRD portion compared to the number of blocks in the RCBD portion.</p> Eid Alotaibi Rhonda Magel ##submission.copyrightStatement## 2022-04-11 2022-04-11 19 1 35 43 A New Family of Optimal Eighth-Order Iterative Scheme for Solving Nonlinear Equations <p>The objective of this manuscript is to introduce a new family of optimal eight-order iterative methods for computing the numerical zeros of a nonlinear univariate equation that is not dependent on the second derivative. The family was designed to enhance the order of convergence by merging Bawazir’s method and Newton’s method as a third step. To demonstrate the performance of the offered scheme, assorted numerical comparisons have been investigated. In addition, the efficiency index of the new family is 1.6818.</p> Z. Al-Turkmani Ibrahim Ahmed Al-Subaihi ##submission.copyrightStatement## 2022-04-12 2022-04-12 19 1 44 53 On the Use of Green's Functions in Solving Boundary Value Problems <p>There is no doubt that Green's functions have a long history in their use in many fields of applied mathematics and physics and especially in solving boundary value problems, hence we thought that it is worthwhile to write this article to summarize some important results in this concern emphasizing the beauty behind using them and the elegant mathematical techniques used as tools in conjunction with them. Famous problems relate to wave propagation and potential theory will be tackled in some details, giving the solutions of the partial differential equation which are connected with the problem. There remains also to<br>mention that Green's functions are used in many other applications as will be pointed out in the conclusions.</p> Aymaan S. Aboukhisheem Alham M. Al-Refai Ahlam E. Elashegh Ali M. Awin ##submission.copyrightStatement## 2022-05-06 2022-05-06 19 1 54 72 On the Numerical Solution Of Schrodinger Equation <p>In the beginning, we start with reviewing basic concepts such as inner product and Hilbert spaces ; then we introduce Schrodinger Equation focusing on the solution of time–dependent and time–independent &nbsp;with a stress on the harmonic oscillator &nbsp;problem which will be the ingredient for our subject ; namely, the numerical solution of .</p> <p>The numerical solution of is then tackledusing the so–called potential morphing method .Calculations were carried&nbsp; out for the ground state of the &nbsp;which represents the frame of reference to work with. The obtained results were compared with similar ones and found to be in very good agreement. Finally, a brief discussion related to possible future work is given ; moreover recent&nbsp; works on the subject are exposed to</p> Khadija A. Ben Mussa Amna M. Gresh Nagah A. Elbhilil Ali M. Awin ##submission.copyrightStatement## 2022-06-19 2022-06-19 19 1 82 93 An Application Of Maximal Numerical Range On Norm Of Basic Elementary Operator In Tensor Product <p>Many researchers in operator theory have attempted to determine the relationship between the norm of basic elementary operator and the norms of its coefficient operators. Various results have been obtained using varied approaches. In this paper, we attempt this problem by the use of the Stampfli’s maximal numerical range in a tensor product.</p> Benjamin Kimeu Daniel Sammy Wabomba Musundi Kinyanjui Jeremiah Ndungu ##submission.copyrightStatement## 2022-05-12 2022-05-12 19 1 73 81