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Item Analysis of gravitational waves in f(T) gravity.(UMT Lahore, 2025-07-15) Zubair IkramIn this work, we explore the propagation of modified gravitational wave forms arising from the inspiral of compact binary systems within the framework of f(T) gravity. We show that the key difference between the modified and general theory of relativity waveforms lies in the corrections introduced to the gravitational wave amplitude. Focusing on gravitational wave sources such as black hole (bh) and neutron star binaries, we analyze their signals as detected by future third-generation detectors like the Einstein Telescope (et) and the Advanced LIGO (aligo) interferometer. While our results indicate that sources within the aligo sensitivity range can provide constraints on the f(T) gravity model that are consistent with current cosmological data.Item Analysis of gravitational waves from axially symmetry framework.(UMT Lahore, 2025-07-16) Aneeza IjazIn this study, we explore the dynamics of relativistic fluids with axial symmetry, focusing on scenarios characterized by the presence of vorticity, the absence of dissipative processes, and shear-free but non-geodesic flow. Traditional views, particularly those stemming from Bondi conjecture, suggest that under specific ideal conditionsāsuch as zero dissipation and geodesic motionāthe emission of gravitational radiation is prohibited. However, our analysis introduces a modified framework that challenges this notion. We specifically consider fluid configurations where dissipation is absent, but the fluid possesses intrinsic rotation (vorticity) and the flow deviates from geodesic paths. Through this approach, we examine whether gravitational radiation can still emerge under these circumstances. Our findings reveal that the presence of vorticity, coupled with non-geodesic motion, facilitates the generation of gravitational waves, even in the absence of dissipative mechanisms and shear stress. This outcome implies that vorticity alone is a significant contributor to the emission of gravitational radiation, independent of the dissipative properties of the medium. Consequently, our results offer a broader understanding of the conditions that can give rise to gravitational wave phenomena in relativistic fluids. This insight is particularly relevant for astrophysical environments, such as rotating stellar interiors, neutron stars, or other compact objects, where non-dissipative, rotational fluid motion is expected to play a dominant role. Ultimately, our work highlights the essential influence of fluid rotation and non-geodesic behavior on gravitational wave production, challenging the restrictive interpretation of Bondi conjecture and contributing to the ongoing efforts to understand the complex mechanisms underlying gravitational radiation in relativistic astrophysical systems.Item Analysis of fractional second-grade nanofluid over a flat surface(UMT Lahore, 2025-07-02) Uzma KhalidIn the current study explains an impact of the Caputo derivative on the flow of second grade fluid with a suspension of ternary nanoparticles over a vertical plate with heat and mass transfer. The theoretical modeling of the flow problem is done by applying the generalized Fourier's and Fick's laws. For the generalization, the Caputo fractional derivative is applied in the constitutive relations for heat and mass fluxes. Laplace transform method is used to solve the prescribed generalized flow model and determined the closed form solutions for the temperature, concentration, and velocity fields. Moreover, the effect of appeared parameters specially the effect of fractional parameter are also explained graphically. On the basis of graphical description, it is concluded that obtained results for the generalized flow model are more accurate and efficient rather than artificial fractional model.Item Investigation of the chromatic number for the direct product of cyclic fuzzy graphs(UMT Lahore, 2025-03-19) Inayat KarimThis study investigates the chromaticity of the direct product of fuzzy graphs, specifically cyclic structures. Fuzzy graphs generalize classical graph theory by embracing fuzzy set theory, providing a rigorous mathematical platform for modeling systems with uncertainty, variability, or imprecise conditions. With the use of the direct product operation, combining two graphs through vertex multiplication, a new composite fuzzy graph is formed. The vertex set of the resulting graph is created as the Cartesian product of the vertex sets of the given graphs, and the edge set is defined on the basis of particular fuzzy adjacency relations. Finding the chromatic number: The purpose of this research is to determine the chromatic number of the direct product graphs discussed above. To solve optimization problems, finding the chromatic number is vital. Graph coloring is a fundamental concept in graph theory and, in fuzzy graphs, is an essential tool for allocating resources and reducing conflict. The two primary areas of application under investigation in this work are workforce scheduling and wireless communication networks' frequency assignment. In workforce scheduling, cyclic fuzzy graphs represent periodic work shifts, and fuzzy graphs represent assignments of tasks. This helps optimize workforce allocation so that tasks are allocated to shifts in a way that reduces conflicts and overlaps. The chromatic number here refers to the least number of shifts required to complete all tasks given constraints such as worker availability and shift patterns. Using the direct product of cyclic fuzzy graphs and fuzzy task graphs, a schedule can be optimized to enhance productivity and minimize inefficiencies in shift planning. Another important application covered in this study is frequency assignment in wireless communication networks. In wireless communication networks, interference between communication cells must be minimized to ensure effective and clear transmission of signals. Fuzzy graphs represent the network cells and interference patterns between them, whereas cyclic fuzzy graphs model periodic over-time overlap of these interferences. The direct product of the graphs provides an overall framework for analyzing interferences over both spatial and temporal domains. In this case, the chromatic number represents the minimum number of frequency bands to which network cells should be assigned in order to minimize interference. This approach allows effective frequency allocation schemes to enhance overall network performance and reliability. In support of the theoretical background, this study presents some case studies illustrating the real-world practical implications of chromatic optimization. The case studies show how direct products of cyclic fuzzy graphs and they can efficiently be applied in solving scheduling and frequency assignment problems under uncertain circumstances. With the use of chromatic properties, dramatic improvements in efficiency and conflict resolution are achievable in workforce optimization and network improvement. Moreover, the research provides new information on the chromatic behavior of direct product of fuzzy graphs, particularly those with cyclic structures. The findings identify new trends and mathematical characteristics that can be applied to develop more effective approaches to managing uncertainty in compound systems. The methods proposed not only enhance the theoretical understanding of fuzzy graph products but also provide real-world applications for industries where variability and unpredictability are significant factors in decision-making. This study relates graph theory to practical optimization problems using the chromatic properties of direct product fuzzy graphs. By focusing on cyclic structures, the research offers a better approach to scheduling and network control, demonstrating the versatility and applicability of fuzzy graph models. The findings show that chromatic optimization through cyclic fuzzy graphs provides an effective and viable solution to real-world problems in wireless communications and scheduling, hence enhancing the fields of operational research and mathematical modeling.Item Comparative analysis of proper edge coloring of direct product of path and star fuzzy graphs(UMT Lahore, 2025-02-12) Bacha KhanThis paper discovers the study of adjacent vertex distinguishing proper edge coloring (AVDPEC) in the context of direct product of fuzzy graphs, with a particular prominence on fuzzy path and fuzzy star graph combinations. In this study, we apply the classical coloring theory of crisp graphs to fuzzy graphs, as membership values add a subtle layer of intricacy to them. In circumstances where system elements are not unconditionally binary, this nuanced method allows us to describe and examine complex systems that call for a certain amount of vagueness or partial membership. We start with a detailed explanation of the essential concepts of graph coloring for both crisp and fuzzy graphs, along with an introduction to the novel fuzzy graph classes that result from this research. Our key contribution comprises an algorithm for the direct product of fuzzy path and fuzzy star networks, which acquires appropriate edge coloring that separates adjacent vertices utilizing fuzzy membership values. There are four main steps in the proposed coloring algorithm. The first step is to pick the color that matches the edge with the lowest membership value in either šš or šš. Step two comprises relocating consideration on the edges that are incident to vertices from both šš and šš. The color of the edge that has the lowest membership value that arises on these vertices is elected in this case. Reallocating colors to edges that have previously been utilized in the direct product is the focus of the third step. Any remaining edges in the direct product that do not yet have a color allocated to them are given new colors in the fourth and final step. In applications where recognizing between entities or tasks is crucial, this algorithm pursues to guarantee that each adjacent vertex pair in the fuzzy graph product is recognized by the particular colors of their incident edges. In order to highlight how changes in membership values affect the coloring procedure, we suggested expressive examples that show how the algorithm works with both crisp and fuzzy graphs. We determine a number of new characteristics and behaviors particular to fuzzy graph structures as an outcome of our exploration. The proper edge coloring in parallel computing guarantees negligible interference and effective source use by enhancing the assignment of computational jobs.Item A decision framework for ranking electronic voting machines using (m, n)-rung orthopair fuzzy soft sets.(UMT Lahore, 2025-07-17) Rukhsar HamidThe purpose of this thesis is to develop a new concept of possibility (m,n)-rung orthopair fuzzy soft set ((m,n)-rofs), which is an expansion of traditional fuzzy set theory. Electronic voting machines related to elections (evms) are becoming more and more reliant on voting; therefore, it is important to have a strong and trustworthy methodology to evaluate them. The results of the study can improve the overall quality of electronic voting machines (evms) in the future and provide valuable information to voters and election officials. This research proposes an effective decision-making approach for evaluating electronic voting machines (evms) using the (m, n)-rofs with a possibility setting based on aggregation operators. Here, the incorporation of possibility settings and aggregation operators enhance the decision-making process by systematically integrating various criteria and expert opinions. To make the possibility (m,n)-rofss acceptable for multi argument scenarios, this study modifies its current structure. Thus, we first create the basic idea of a possibility (m,n)-rofss. To demonstrate the possibilities of (m,n) rung orthopair fuzzy soft sets, an example is given that shows how several models of electronic voting machines can be assessed using the approach according to established standards. Moreover, using the suggested aggregation operators as a basis, we suggest a method for p-(m,n)-ropfss to address decision-making problems, including known possibility data. These aggregation operators are useful to make research more subjective, and they can also provide a comprehensive evaluation. The foundation of our madm approach is the p (m,n)-rofssga or p (m,n)-rofssaa operators, which take contention relationships into account. Furthermore, sensitivity and comparative analyses are provided in the end, along with a real-life example to illustrate the feasibility of the proposed methods. Their inclusion not only strengthens the method but allows experts to make a cohesive decision-making process. These analyses are helpful in assessing the reliability and stability of the process of evaluation when faced with varying levels of uncertainty. The evaluation of these machines is conducted based on several critical criteria: security, usability, auditability, initial purchase cost and maintenance cost. This algorithm works well to figure out whether a specific electronic voting machine (evm) is appropriate to use in an election, which will ultimately be a great innovation in the election system of any government. Overall, this methodās actual use shows how adaptable and successful it is at handling challenging decision-making situations. It emphasizes how crucial it is to use contemporary methods to enhance decision-making procedures and guarantee the smooth and dependable operation of vital systems, such as voting mechanisms. This research allows both election officials and voters to understand the performance and sustainability of electronic voting machines during the election. The insights will be beneficial for stakeholders in the electoral process, policymakers, and election officials. It adds to the ongoing discussion about electoral technology and provides information for decision-making processes targeted at improving the credibility, accessibility, and integrity of us elections by analyzing the effectiveness and factors related to these electronic voting systems.Item Numerical solution of fractal fractional MHD Casson nano-fluid between two walls with finite difference method(UMT Lahore, 2025-07-16) Noureen AnwarThe research identically analyses that the unsteady magnetohydrodynamic (mhd) flow of a casson nanofluid between two parallel plates can be accurately modelled by means of the fractal fractional caputo derivative, which allows to incorporate memories and hereditary effects that are common in non-newtonian and nano-scale fluid systems. In the present work, a mathematical model is proposed where joule heating, viscous dissipation and diffusive nanoparticles are accounted. The governing nonlinear partial differential equations are simplified using similarity transformations and solved numerically in dimensionless form using the finite difference method. The results show that an increase in the magnetic field acts to suppress the velocity profile in the presence of lorentz force, while eckert number increases the thermal field as a result of increased viscous dissipation. In addition, the temperature profile is affected by heat generation and fluctuations in thermal conductivity. A decrease of the mass diffusivity leads to a decrease in the concentration of nanoparticles. The fractional order is important for control of time-dependent behavior of the system and allows modelling of the transient processes with greater realism and flexibility. This research follows the trend of the application of fractal fractional derivatives for the complex mhd nanofluid flows modeling with a high significance of nonlocal and memory effects. The discussed approach increases the capability of simulating physical systems, which are in the real world coupled ones, with the thermal and fluid transport mechanisms affected by electromagnetic fields. The results of this research could be used to develop and optimize advanced thermal control systems, such as microfluidic cooling devices, biomedical flow systems and industrial heat exchangers where accurate reconstruction of heat and mass transfer is important.Item Comparison of the Sanskruti Index in bicyclic graphs(UMT Lahore, 2025-06-24) Fatima AneesThrough this study, the authors identify the most appropriate way to characterize bicyclic graphs using the advanced Sanskruti index which describes the complexities found in graph- and molecule-based systems. The objective is to design a strong mathematical structure to determine and distinguish bicyclic graphs that achieve either the maximum or minimum value of the Sanskruti index. With the help of effective graph approaches and the key structure of bicyclic graphs, this work finds important patterns affecting the Sanskruti index. By combining analytical techniques and computer simulations, the study identifies best configurations and helps better explain how graph invariants show up and matter in chemical graphs, network structures, and mathematical optimizations.Item Computing local fractional metric dimension of path-wheel graphs(UMT Lahore, 2025-06-17) Sana ZamanMetric dimension is a useful tool for studying many distance-based issues in the field of electrical networking, robotics, computer networking, integral programming, telecommunication and robotics. The most recently created variant of the metric dimension, known as the fractional metric dimension, is widely applied to non-integral linear programming issues. In this thesis we will compute the local fractional metric dimension (LFMD) of path-wheel graphs (P_W(m,h)), in which h is the number of isomorphic copies of the wheel graph and m+1 is the number of vertices in one wheel W_m. All the obtained results are discussed by the examples of particular graphs belonging to the understudied families of graphs. Graph theory is a field of mathematics that focuses on the study of graphsāstructures made up of vertices (or nodes) connected by edges (or links). Graphs can be directed (where edges have a direction) or undirected (where edges have no direction). Graph theory explores the properties, patterns and behaviors of these graphs and is widely used to solve problems involving networks, relationships and connections in areas such as computer science, transportation, biology and social sciences. The metric dimension of a graph is the smallest number of selected vertices such that the distances from any other vertex to these selected vertices uniquely identify it. These selected vertices form whatās called a resolving set. For every pair of different vertices in the graph, there must be at least one vertex in the resolving set that has a different distance to each of them. The size of the smallest such set is the metric dimension of the graph. The local fractional metric dimension of a graph is a variation of the metric dimension that focuses only on adjacent pairs of vertices and allows the use of fractional weights instead of whole numbers when choosing resolving vertices. In this concept, each vertex is assigned a weight between 0 and 1. The sum of weights for vertices that can distinguish every pair of neighboring (adjacent) vertices must be at least 1 for each such pair. The local fractional metric dimension is the smallest possible total weight of all the vertices under these conditions.Item Existence of solution įµ©-fractional differential equation via fixed point(UMT Lahore, 2025-06-18) Fariha IrshadThis research provides an in-depth exploration of complex systems governed by fractional-order dynamics, focusing on boundary value problems (BVPs) and the modeling of computer virus spread through memory-dependent operators. The first part of the study addresses nonlinear BVPs involving Ļ-fractional derivatives, where the existence of solutions is established using advanced mathematical tools such as the ReimannāLiouville and Caputo integro-differential operators. To validate the solvability conditions, we apply fixed point frameworks including endpoint theory and alphaāpsi contraction mappings. These theoretical findings are reinforced with illustrative numerical examples, supported by graphical visualizations that demonstrate the applicability and behavior of the proposed models. In the second part, we focus on the population dynamics of computer viruses under the influence of memory effects introduced via the fractalāfractional derivative operator. A rigorous numerical investigation is conducted to examine how different values of the fractal dimension and fractional order affect the virus transmission process. To facilitate this, we develop and analyze two mathematical formulationsāone using the Caputo derivative and the other based on the fractalāfractional operator. For numerical simulation, we utilize two distinct methods: the AdamsāBashforth scheme and the Taylor operational matrix method (TOMM). Both techniques yield consistent and reliable results despite the use of different derivative definitions, highlighting the robustness of the approach. Furthermore, the study evaluates the generalized stability of the proposed models by applying UlamāHyers and UlamāHyersāRassias stability criteria. These assessments confirm the stability and sensitivity of the models with respect to small perturbations. Overall, the work emphasizes the critical role of memory effects in both boundary value formulations and virus dynamics, showing the potential of fractional and fractalāfractional derivatives in capturing long-term dependencies and providing deeper insights into complex systems.Item Abundant families of solitary wave structures for the reaction-diffusion glycolysis model(UMT Lahore, 2025-11-21) Ali AkbarIn this thesis, biochemical reactions are mediated by enzymes, which are biological catalysts that can alter the rate and specificity of chemical reactions inside cells. This paper investigates the analytical structures for a two-species glycolysis model of biochemical reaction. These species are described by substrate and activator. Thus, it is important to study it from a mathematical point of view. The new MEDA technique is utilized for investigating the analytical structures. The diverse families of trigonometric, hyperbolic, and rational form solutions are observed. Furthermore, the dynamical behavior of the obtained structures is also considered in 3D view, line graphs, and their corresponding contours.Item Closed form solitary wave solutions for the Jaulent-Miodek hierarchy model(UMT Lahore, 2025-11-23) Kashif ShahzadThe jaulent-Miodek hierarchy systemās dynamical analysis is presented in this work. The jaulent-Miodek hierarchy equation incorporates an energy-dependent Schrƶdinger potential that finds use in fluid dynamics, condensed matter physics, engineering systems, and optics. Investigating this dynamic problem from a mathematical standpoint is therefore crucial. For the jaulent-Miodek model, the closed-form invariant solution known as solitons is investigated using the Sardar subequation approach. Under the noise, the various solitons are created as bright soliton, dark soliton, dark-bright soliton, periodic, and another form. It is seen that the model contains kink wave profiles, different soliton profiles, and periodic oscillating nonlinear waves. Some of the recently developed soliton solutions are validated by reintegrating them into the appropriate framework for soft computation using Wolfram Mathematica. By selecting appropriate parameter values, the various figures are plotted in both 2D and 3D along with the related contours.Item Stability of a compact star and exotic matter(UMT Lahore, 2025-05-02) Waqar HussainWe explore how density inhomogeneities and dissipation influence the final outcome of a self-gravitating sphere's collapse. Using a perturbative approach to the thermodynamic variables and gravitational potentials, which is initially a type of Krori-Barua approach we track the collapse process starting from an initially static, shear-free perfect fluid sphere. As the core collapses, it dissipates energy through a radial heat flux, while the surrounding spacetime is filled with a mix of null energy and an anisotropic string distribution. Over time, this dynamical process transitions into a shear-like regime, influenced by both the heat flux and density fluctuations. Our findings show that the anisotropy caused by the strings drives the stellar fluid toward instability, with this effect further amplified by the density inhomogeneities. A notable and new aspect of this collapse scenario is the delay in the formation of the event horizon.Item Core-envelope model of compact objects in š(š ) gravity(UMT Lahore, 2025-02-18) Amera AzamIn this thesis, a relativistic core and envelope model is being developed for spherical symmetry anisotropic compact stars in š(š ) gravity. It is necessary to understand the complex interior structure of these stars to do the study of general relativity and astrophysics. The core area is represented by polytropic equation of state (EOS). This point specifically describes the density-pressure relation. In accordance with physical situation frequently realized for a star interiors. The mathematical handling of the outer layers, where circumstances are often less harsh. It is made simpler using a linear EOS for the envelope area. The smooth matching of the stars core, envelope, and Schwarzschild outer regions is an essential part of the findings. For physical characteristics like density and pressure to be never-changing and clearly-defined in different locations. This smooth changeover is necessary to avoid singularities that would endanger such continuity crucial to the models validity. All of the physical parameters behave well in the core and envelope regions of the specific compact stars SAX J1808.4 3658 and 4U1608-52. The physical properties that characterize the internal structure of these stars, while maintaining non-singularity and continuity at the interface where the core and envelope meet. This comprehensive assessment enhances the robustness of the model. The core and envelope model of compact objects, significantly influence the mass, radius, and compactification factor of a star. The graphical representation is helpful to illustrate these relations and provides a better understanding of the relationship between the inner and the outer features of the star and other such trivial properties that the star might have. These illustrations are useful for the comprehension of structure and equilibrium of stars formed in the limit and all are important for their dynamics. Abstract the adiabatic index and the radial sound speed are important tests to determine the stability of the model. The speed of radial sound is essential in ensuring that sound waves propagate inside the star according to limitations of causality, which state that they cannot travel faster than light. Abstract xiii on the contrary, the adiabatic index explains a relationship between the variations of density and pressure, which is essentially required to evaluate the overall stability arrangements. I observed the core and envelope model stability requirements to verify that the model correctly expresses the physical reality of the compact stars. Moreover, to understand the structures of stars better, this study advances modified gravity theories by shedding light on the underlying reasons that govern the behavior of these interesting celestial bodies.Item Topological analysis and quantitative structure- property relationship (QSPR) analysis of antiāasthmatic drugs(UMT Lahore, 2025-01-18) Fatima ZahraAnti-asthmatic drugs play a crucial role in modern medicine by managing and relieving the symptoms of asthma. this thesis investigates the properties of anti-asthmatic drugs using topological indices and quantitative structureāproperty relationship (qspr) modeling. the research begins with a comprehensive review of anti-asthmatic agents, utilizing topological indices to examine their molecular structures and potential pharmacological effects. by applying qspr modeling based on these indices, we aim to predict key physicochemical properties of anti-asthmatic drugs, contributing to the development of safer and more effective treatments. motivated by earlier work on the structural analysis of pharmaceuticals, we analyze various anti-asthmatic compounds using selected topological indices. a topological index quantitatively captures theoretical characteristics of chemical structures. these drugs, including bronchodilators, corticosteroids, and leukotriene modifiers, are commonly used to reduce airway inflammation and improve respiratory function. by representing a drug molecule as a graphāwhere atoms are vertices and bonds are edgesāwe apply mathematical tools for structural characterization. our qspr analysis demonstrates a strong correlation between specific topological indices and the physicochemical properties of compounds used in anti-asthmatic therapy. these findings highlight the potential of topological and graph-theoretical methods in optimizing drug design, improving therapeutic efficacy, and minimizing side effects.Item Charged stellar filament and dark matter(UMT Lahore, 2025-01-02) Sundas ShahzadiThis study delves into the dynamical behavior and structural instability of filamentary configurations in the presence of exotic matter, employing the palatini formulation of modified gravity, specifically the f(r,t) theory. In this framework the exotic matter is represented through a non-minimal coupling between geometry and the trace of the energy-momentum tensor, which introduces novel features not present in standard general relativity. To analyze the collapse dynamics of these structures, we implement the darmois junction conditions at the boundary surface, ensuring a smooth matching between the interior and exterior spacetime geometries. Through this approach, we derive a collapse equation that governs the evolution of the system. Notably, at the boundary, the radial pressure does not vanish; instead, it exhibits a dependence on a dynamically evolving field term, highlighting the persistent influence of exotic components overtime. Additionally, the study explores the interaction between dark matter and gravitational waves in this extended gravity regime. The results indicate that the presence of exotic terms leads to a suppression of gravitational wave transmission, suggesting that such matter components can act as a barrier to wave propagation, thereby altering the observational signatures of gravitational phenomena in regions dominated by exotic or dark matter.