Department of Mathematics
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Item 4-total edge product cordial labeling of some standard classes of graphs(University of Management and Technology, 2018) Iqbal JaveedOur task in this thesis is focuses on 4-total edge product cordial (4-TEPC) labeling of some standard classes of graphs and the main aim of the study is to discuss path graph, cycle graph, wheel graph, helm graph, banana tree graph, recracker graph, gear graph and tadpole graph in the context of 4-TEPC labeling.Item Modified forms of some convex polytopes with constant metric dimension(University of Management and Technology, 2017) Mehboob, SadiaIn this thesis, we expand the study of metric dimension by adding r times pendant edges and adding r times prisms to outer cycle of some convex polytopes determined in [21, 25]. These modi ed forms of convex polytopes have constant metric dimension and only three vertices appropriately chosen be su ce to resolve all the vertices of these graphs of convex polytopes. 3Item Third-Order Numerical Method for the Solution of Heat Equation with Nonlocal Boundary Conditions(University of management and Technology, 2017) Tariq, KanwalA third order numerical technique is developed for solving one dimensional non-homogenous heat equation with integral boundary conditions. In this method second order spatial derivative is approximated by third order nite di erence approximation. The parallel splitting technique combined with Simpson's 1/3 rule is used to tackle this problem. This method is very precise due to highly accurate results.Item On Topological Properties Of The Line Graphs Of Certain Chemical Structures(University of Management and Technology, 2018) Sohail, RaheelaLet G = (V (G); E(G)) be a simple connected graph, where V (G) is the set of vertices and E(G) is the set of edges. A topological index of a graph is a num-ber related to a graph which is invariant under graph automorphisms. In this thesis, we give theoretical results for some topological indices such as geometric-arithmetic index GA(G), fth geometric-arithmetic index GA5(G), atom-bond con-nectivity index ABC(G), fourth atom-bond connectivity index ABC4(G), Zagreb indices M1(G), M2(G), M3(G), M1(G), M2(G), Zagreb coindices M1(G), M2(G), M2(G), Randic index R(G), hyper-Zagreb index HM(G), general sum-connectivity index (G) and augumented Zagreb index AZI(G) by considering graph G as a line graph of V phenylenic nanotube, V phenylenic nanotorus, linear pentacene and multiple pentacene.Item THE DYNAMICAL STUDY OF COMPACT OBJECTS IN GENERAL RELATIVITY(University of Management and Technology, 2018) Syed Ali Mardan AzmiIn this thesis, we discuss the dynamical stability of charged compact objects with the help of some mathemat¬ical models. For this purpose, we have selected three different models of charged compact objects to discuss the possible occurrence of cracking under different conditions. In first selected model, we discuss charged anisotropic compact objects with a linear equation of state. In second model, we study anisotropic charged compact object PSR J1614-2230 in quadratic regime, while in third model, we study charged compact stars corresponding to embedded class one metric with perfect inner fluid distribution. We investigate the impact of electromagnetic field on the stability regions of charged self-gravitating compact objects by using the con¬cept of cracking. For this, we have applied local density perturbation scheme to the hydrostatic equilibrium equation as well as on physical parameters involved in the model. In particular, we have examined the crack¬ing of charged compact objects (a) PSR J1614-2230, PSR J1903+327, Vela X-1, SMC X-1 and Cen X-3 with linear equation of state (b) PSR J1614-2230 with quadratic equation of state (c) Her X-1, PSR 1937+21, PSR J1614-2230, PSR J0348+0432 and RX J1856-37 corresponding to embedded class one metric. We conclude that these objects exhibit cracking and stability regions decreases with the increase of charge. We also extend two conventional polytropic equations of state to generalized polytropic equations of state for spherical and cylindrical symmetries in the context of general relativity. For this purpose, we formulate the general framework to discuss the physical properties of spherical and cylindrical polytropes with charged anisotropic inner fluid distribution under conformally flat condition. We investigate the stability of general¬ized polytropic models through Tolman-mass and Whittaker formula for spherical and cylindrical symmetries respectively. We also discuss the possible occurrence of cracking in charged anisotropic polytropes devel¬oped under the assumption of generalized polytropic equation of state in two different ways (i) by carrying out local density perturbation under conformally flat condition (ii) by parametric perturbations. We conclud that one of the generalized polytropic equations of state results into a physically viable model and cracking appears for a specific range of density and model parameters.Item Fourth order numerical method for the solution of heat equation with nonlocal boundary condition(University of Management and Technolog, 2017) Zahra, AnamIn this thesis, a numerical technique is develop for solving one dimensional parabolic partial differential equation (PDE) with integral boundary conditions. Spatial derivative is approximated by finite difference (FD) scheme and by ap- plying method of lines, resulting a system of ordinary differential equations (ODEs). Simpson’s 1/3 rule is used to tackle integral boundary conditions and it also help in removing additional variables to produce a system of N equations with N variables. This numerical method can be coded on sequential as well as parallel computing environmentItem Convective flow of rotating MHD second grade fluid over an oscillating plate in a porous medium(University of Management and Technology Lahore, 2017) Saba IramIn this research, the analytical solutions for unsteady free convection flow of rotating second grade fluid over an isothermal oscillating vertical plate are gained. The influence of Magneto hydrodynamics (MHD) flow is also deliberated in a porous medium. The governing equation for momentum is sculpted in a rotating system such that both fluid and plate revolve in unison with uniform angular velocity. The phenomenon is sculpted in the froth of partial differential equation unruffled in the preliminary and boundary condition. Some appropriate non-dimensional variable are familiarized. The analogous non-dimensional momentum and energy equations with conditions are deciphered via Laplace transform technique. Expressions for velocity and temperature fields are found and demonstrated graphically for dissimilar values of second grade fluid , rotation , magnetic and porosity parameters. End result acquired gratified all the initial and boundary conditions.Item Some standard classes of graphs labeled by 3-total edge product cordial labeling(University of Management and Technology Lahore, 2017) Muhammad BilalOur task in this thesis is to study 3-total edge product cordial (3-TEPC) labeling of some standard classes of graphs. We discuss tadpole graph, helm graph, gear graph, banana tree graph and firecracker graph in the context of 3-TEPC labeling. We also discussed 3-TEPC labeling of some particular examples of graph operations like corona graphs.Item Edge version of harmonic index and polynomial(University of Management and Technology Lahore, 2017) Rabia NazirIn this thesis, we introduced the edge version of harmonic index and harmonic poly- nomial on the basis of the edges of a graph. Then we computed the edge version of harmonic index and polynomial for some standard classes of graphs. We also dis- cussed some examples of rooted product graphs and bridge graphs in the context of harmonic index and polynomial.Item A study of Generalized Variational-like Inequalities in Normed Spaces for Pseudo-Monotone Type iii Operators(University of Management and Technology, 2015) Talha Nazir, MuhammadIn this thesis, we shall review some existence results for solution of generlized variational like inequalities for Pseudo monotone type III operators. These results will be extensively and thourghly studied in normed spaces on non-compact sets.Item A Study of TOPSIS in Classical, Fuzzy, Intuitionistic Fuzzy and Neutrosophic Environments(University of Management and Technology, 2014) Muzammil AliMulti Criteria Decision Making uses different techniques to find a best alternative from multi-alternative and multi-criteria conditions. TOPSIS is an important practical technique for ranking and selection of different alternatives by using distance measures. Classical TOPSIS uses crisp techniques for the linguistic assessments, but due to imprecise and fuzziness nature of the linguistic assessments, there must be some tools to deal with the vague information. Therefore, it is necessary to involve fuzzy techniques (FS, IFS and NS). In this thesis, the algorithms of crisp TOSIS, Fuzzy TOPSIS, Intuitionistic Fuzzy TOPSIS and Neutrosophic TOPSIS are discussed. Examples related to each type of TOPSIS are solved. Finally, it is tried to compare all the discussed techniques.Item General solutions of convective flows of MHD casson fluid with slip and radiative Heat transfer at the boundary(University of Management and Technology Lahore, 2017) Sana EjazThe purpose of this work is to nd General solutions for Magnetohydrodynamic(MHD) natural convection ow of an incompressible viscous uid over an in nite verti- cal plate by considering Radiative heat transfer, Porous e ects and slip conditions. These solutions, are obtained by Laplace transform technique. They satisfy all imposed initial and boundary conditions and generate large class of exact solu- tions. For illustration, three special cases are considered and some interesting re- sults from literature are revived as limiting cases. The in uence of di erent param- eters on the uid motion are graphically illustrated. This work has been published International Journal of Computational Thermal Sciences, Vol: 9, pp. 1-11, (2017). viItem N-framed soft set with basic operations(University of Management and Technology Lahore, 2016) Azhar HussainThe main purpose of this thesis is to introduce the basic operations of union, intersection, difference and relative complement of double framed soft sets, discussed in chapter 3. Extension of double framed soft sets into triple framed soft sets and to propose the basic operations of union, intersection, difference and relative complement of triple framed soft sets, discussed in chapter no 4. Finally the introduction of n-framed soft sets, as a generalization of double framed soft sets, the basic operations of union, intersection, difference and relative complement of n-framed soft sets, discussed in chapterItem Existence results for a system of fractional differential equations subject to coupled integral boundary conditions(University of Management and Technology Lahore, 2016) Mughal, Abdul AleemRecently, Johnny Henderson and Rodica Luca [1], have presented some new existence and uniqueness results, and for this, they have used a variety of theorems. They have worked on fractional differential equations, and have investigated the uniqueness and existence for non-negative solutions of a system of nonlinear Riemann-Liouville fractional differential equations α v1(t) + λ1f (t, v1(t), v2(t)) = 0, where t ∈ (0, 1) and n − 1 < α ™ n β v2(t) + λ2g(t, v1(t), v2(t)) = 0, where t ∈ (0, 1), and m − 1 < β ™ m with the coupled integral boundary conditions j n−2 ∫ 1 j m−2 ∫ 1 where m, n ∈ N ; m, n “ 3; Dα and Dβ are the derivatives from Riemann-Liouville with orders α, β respectively. Further, the integrals in the boundary conditions are the integrals from Riemann-Stieltjes. Some adequate conditions will be given on the parameters λ1, λ2, and nonlinear functions f and g, so that non-negative solutions of the above problem exist. This thesis is detailed review of the results presented in [1].Item Dynamical analysis of compact objects IN F(R) and F(R,T ) theories(University of Management and Technology Lahore, 2016) Noureen, IfraThe astrophysics and astronomical theories are invigorated largely by the gravitational evolution and instability range explorations of gravitating sources. Gravitational collapse is the fundamental phenomenon to account evolution within galaxies and assemble supergiant structures. This dis- sertation is based on the explorations regarding dynamical instability of gravitating sources in f (R) and f (R, T ) theories of gravity. The considered modified gravitational theories provide dark energy substitutes constitut- ing large negative pressure and thought to be responsible for the cosmic speed-up. The dynamical systems are studied by considering spherically and axi- ally symmetric backgrounds with anisotropic matter distribution. The mod- ified field equations and conservation equations for spherically symmetric dynamical system are constructed in f (R, T ) gravity. The variations in gravitating system are estimated by implementation of first order pertur- bations on dynamical equations. Insertion of perturbed physical quanti- ties derived from perturbed field equations in perturbed Bianchi identities leads to the evolution equation. The expression for adiabatic index is con- structed from evolution equation to investigate the variation in pressure stresses with the given energy density. Moreover, terms lying in Newtonian and post Newtonian eras are identified to establish the corrections to weak field limit. We have also studied the dynamics of spherically symmetric anisotropic stars under the influence of shear-free condition. The modified field equa- tions accompanying vanishing shear scalar are obtained. On establishment of evolution equation, it is observed that the flow variables are less con- strained in shear-free case and so leads to a wider range of stability. The corrections to Newtonian and post Newtonian approximations are estimated as well. The dynamics of spherical stars evolving under expansion-free condition in f (R, T ) gravity is explored by taking anisotropic matter configuration. The collapse equation is acquired from linearly perturbed dynamical equa- tions. It is concluded that in zero-expansion case, the unequal stresses and density profile defines instability rage rather than the adiabatic index. However, the physical quantities are constrained to maintain positivity of energy density and stable stellar configuration. Motivating from the incidental deviations from spherical symmetry of gravitating objects, we study the dynamical instability of axially symmet- ric sources (avoiding reflection and rotation terms about symmetry axis). Furthermore, the evolution equation is settled for both the considered mod- ified theories leading to the instability range of axially symmetric dynamical system in Newtonian and post Newtonian regimes.Item Comparative study of fuzzy soft matrix (FSM) and interval valued fuzzy soft matrix (IVFSM) in Decision making(University of Management and Technology Lahore, 2016) Zulqarnain, MuhammadIn this work, we study soft matrices (SMs), FSM and IVFSM. First of all, we use FSM and the IVFSM in decision making problem with examples. Secondly, we compare results which are found by FSM and IVFSM and see that the values by FSM method do not lie in the intervals which are found by IVFSM. Finally, for perfect comparison of FSM and IVFSM in decision making, we redefined the product of IVFSM. Now we see that FSM is more appropriate for decision making.