### A general family of derivative free with and without memory root finding methods

JPRM-Vol. 16 (2020), Issue 1, pp. 64 – 83 Open Access Full-Text PDF
Saima Akram, Fiza Zafar, Moin-ud-Din Junja, Nusrat Yasmin
Abstract: In this manuscript, we construct a general family of optimal derivative free iterative methods by using rational interpolation. This family is further extended to a family of with-memory methods with increased order of convergence by employing two free parameters. At each iterative step, we use a suitable variation of the free parameters. These parameters are computed by using the information from current and previous iterations so that the convergence order of the existing family is increased from $$2^{n}$$ to $$2^{n}+2^{n-1}+2^{n-2}$$ without using any additional function evaluations. To check the performance of newly developed iterative schemes with and without memory, an extensive comparison with the existing with- and without memory methods is done by taking some real world problems and standard nonlinear functions. Numerical experiments illustrate that the proposed family of methods with-memory retain better computational efficiency and fast convergence speed as compared to existing with- and without memory methods. The performance of the methods is also analyzed visually by using complex plane. Numerical and dynamical comparisons confirm that the proposed families of with and without memory methods have better efficiency, convergence regions and speed in contrast with the existing methods of the same kind.

### On the generalized class of estimators for estimation of finite population mean in the presence of non-response problem

JPRM-Vol. 16 (2020), Issue 1, pp. 52 – 63 Open Access Full-Text PDF
Saba Riaz, Amna Nazeer, Javeria Abbasi, Sadia Qamar
Abstract: This work considers a generalized class of biased estimators for the estimation of the unknown population mean of the variable of interest accompanying the issue of non-response in the study and in the auxiliary variables. The asymptotic bias and the asymptotic variance of the suggested class are acquired, up to the first degree of approximation and, compared with the linear regression estimator. The efficiency of the suggested estimators while comparing with the linear regression estimator and some other existing estimators are studied regarding percent relative efficiency (PRE). Furthermore, a simulation study also affirms the excellence of the considered class of estimators.

### Characterizations of Chevalley groups using order of the finite groups

JPRM-Vol. 16 (2020), Issue 1, pp. 46 – 51 Open Access Full-Text PDF
Abid Mahboob, Taswer Hussain, Misbah Akram, Sajid Mahboob, Nasir Ali, Ali Raza
Abstract: In this paper, we prove $$\psi (A_{1}(4))< \psi(G)$$, $$\forall$$ groups which are not simple with order sixty, $$A_{1}(4)$$ is Chevalley group (Linear group) of order 60. Also we prove that $$\psi (A_{2}(2))< \psi(G)$$ using higher order non-simple groups of order 168.

### Construction of optimal derivative free iterative methods for nonlinear equations using Lagrange interpolation

JPRM-Vol. 16 (2020), Issue 1, pp. 30 – 45 Open Access Full-Text PDF
Moin-ud-din Junjua, Saima Akram, Tariq Afzal, Ayyaz Ali
Abstract: In this paper, we present a general family of optimal derivative free iterative methods of arbitrary high order for solving nonlinear equations by using Lagrange interpolation. The special cases of this family with optimal order of convergence two, four, eight and sixteen are obtained. These methods do not need the Newton’s or Steffensen’s iterations in the first step of their iterative schemes. The advantage of the new schemes is that they are also extendable to the iterative methods with-memory. Numerical experiments and polynomiographs are presented to confirm the theoretical results and to compare the new iterative methods with other well known methods of similar kind.

### Fractional Optimal Control for a Corruption model

JPRM-Vol. 16 (2020), Issue 1, pp. 11 – 29 Open Access Full-Text PDF
Ebenezer Bonyah
Abstract: In this work, a fractional optimal control of corruption model is investigated. The variable controls are included in the model to optimize the best strategy in reducing the corruption in the society. The fraction derivative employed in the study is in Atangana–Beleanu–Caputo (ABC) sense based on generalized Mittag–Leffler. The uniqueness and existence of solution of the corruption model is established. The necessary and sufficient condition for establishing fractional optimal control in ABC sense is determined. A numerical algorithm for obtaining fractional optimal control solution is presented. The numerical solution results show that the best strategy in controlling corruption in the society is to optimize all the thee controls simultaneously.
Abstract: For a commutative semiring $$R$$ with non-zero identity, the maximal graph of $$R$$, denoted by $$MG(R)$$, is the graph whose vertices are all elements of $$UM(R)$$ with two distinct vertices joined by an edge when there is a maximal co-ideal that contains both of them. In this paper, we study some properties of maximal graph such as planarity, radius, splitting and domination number.