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Öğe A graph-theoretic approach to ring analysis: Dominant metric dimensions in zero-divisor graphs(Cell Press, 2024) Ali, Nasir; Siddiqui, Hafiz Muhammad Afzal; Riaz, Muhammad Bilal; Qureshi, Muhammad Imran; Akgul, AliThis article investigates the concept of dominant metric dimensions in zero divisor graphs (ZDgraphs) associated with rings. Consider a finite commutative ring with unity, denoted as R, where nonzero elements x and y are identified as zero divisors if their product results in zero (x . y = 0). The set of zero divisors in ring R is referred to as L(R). To analyze various algebraic properties of R, a graph known as the zero-divisor graph is constructed using L(R). This manuscript establishes specific general bounds for the dominant metric dimension (Ddim) concerning the ZD-graph of R. To achieve this objective, we examine the zero divisor graphs for specific rings, such as the ring of Gaussian integers modulo m, denoted as Zm[i], the ring of integers modulo n, denoted as Zn, and some quotient polynomial rings. Our research unveils new insights into the structural similarities and differences among commutative rings sharing identical metric dimensions and dominant metric dimensions. Additionally, we present a general result outlining bounds for the dominant metric dimension expressed in terms of the maximum degree, girth, clique number, and diameter of the associated ZD-graphs. Through this exploration, we aim to provide a comprehensive framework for analyzing commutative rings and their associated zero divisor graphs, thereby advancing both theoretical knowledge and practical applications in diverse domains.Öğe Exploring Ring Structures: Multiset Dimension Analysis in Compressed Zero-Divisor Graphs(Mdpi, 2024) Ali, Nasir; Siddiqui, Hafiz Muhammad Afzal; Qureshi, Muhammad Imran; Abdallah, Suhad Ali Osman; Almahri, Albandary; Asad, Jihad; Akgul, AliThis paper explores the concept of multiset dimensions (Mdim) of compressed zero-divisor graphs (CZDGs) associated with rings. The authors investigate the interplay between the ring-theoretic properties of a ring R and the associated compressed zero-divisor graph. An undirected graph consisting of a vertex set Z(RE)\{[0]}=RE\{[0],[1]}, where RE={[x] :x is an element of R} and [x]={y is an element of R : ann(x)=ann(y)} is called a compressed zero-divisor graph, denoted by Gamma ER. An edge is formed between two vertices [x] and [y] of Z(RE) if and only if [x][y]=[xy]=[0], that is, iff xy=0. For a ring R, graph G is said to be realizable as Gamma ER if G is isomorphic to Gamma ER. We classify the rings based on Mdim of their associated CZDGs and obtain the bounds for the Mdim of the compressed zero-divisor graphs. We also study the Mdim of realizable graphs of rings. Moreover, some examples are provided to support our results. Notably, we discuss the interconnection between Mdim, girth, and diameter of CZDGs, elucidating their symmetrical significance.Öğe Iterative solutions for nonlinear equations via fractional derivatives: adaptations and advances(Taylor & Francis Ltd, 2024) Ali, Nasir; Waseem, Muhammad; Safdar, Maimoona; Akgul, Ali; Tolasa, Fikadu TesgeraIn recent years, the utilization of fractional calculus has witnessed a notable surge across various scientific and engineering domains. This manuscript delves into the exploration of adapted iterative techniques tailored for solving nonlinear equations, capitalizing on the diverse range of derivatives available for addressing different problem contexts. We scrutinize previously developed iterative methods, enhancing their efficacy by introducing an auxiliary parameter to the root search process for nonlinear equations (NLE), alongside a fixed order of fractional derivatives. The selection of the auxiliary parameter is confined to the interval (0,1] for convenience. A thorough convergence analysis is conducted, employing a fractional power series expansion of f(x) in terms of fractional derivatives. Subsequently, a series of NLEs is solved to showcase and contrast the efficiency of our proposed methods with established iterative techniques. This refined abstract aims to succinctly elucidate the objectives and contributions of our study, providing readers with a clearer understanding of the manuscript's scope and significance.Öğe Numerical investigation of generalized perturbed Zakharov-Kuznetsov equation of fractional order in dusty plasma(Taylor & Francis Ltd, 2022) Ali, Nasir; Nawaz, Rashid; Zada, Laiq; Nisar, Kottakkaran Sooppy; Ali, Zahid; Jamshed, Wasim; Hussain, Syed M.In the present work, the new iterative method with a combination of the Laplace transform of the Caputo's fractional derivative has been applied to the generalized (3 + 1) dimensional fractional perturbed Zakharov-Kuznetsov equation in a dusty plasma. The proposed method is applied without any discretization and linearization. The numerical and graphical results show the accuracy of the proposed method for nonlinear differential equations. Moreover, the methods are easy to implement and give the efficient approximate solutions.Öğe On study of flow features of hybrid nanofluid subjected to oscillatory disk(World Scientific Publ Co Pte Ltd, 2024) Safdar, Maimoona; Mushtaq, Tahir; Ali, Nasir; Akguel, AliIn recent years, an entirely new class of nanofluids has been developed, namely hybrid nanofluids. Their efficiency when it comes to heat transfer has been demonstrated to be much higher than the traditional nanofluids. In this study, the flow of hybrid nanofluid is analyzed subjected to the flow geometry of oscillatory disk that also rotates with the constant angular speed. Mainly, we investigated hybrid nanofluid flow subjected to an oscillatory disk that rotates at a constant angular speed as an incompressible, laminar and time-dependent electrically conducting flow. The implication of similarity functions normalizes the flow phenomenon. Furthermore, we solved the governing system by implementing the finite differences, and then the iterative procedure is carried out with the help of Successive Over-Relaxation (SOR) technique. In the later part of the paper, graphical representation is given to discuss dimensionless factors in both scenarios of nanofluid and hybrid nanofluid.
