Department of Mathematics

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On the Construction of Higher Order Conservative Methods
(Library Information Services COMSATS University Islamabad Lahore Campus, 2012) Zareen Akhtar; FA19-RMT-026; LHR TP 7417; Dr. Yousaf Habib
For the numerical solution of conservative differential equations such as Hamiltonian system or differential equations that arise in different physical phenomena with some structure, we use structure preserving numerical methods. In particular, symplectic Runge-Kutta methods, G-symplectic General Linear methods and Variational Integrators have been explored. In this thesis Variational Integrators for degenerate Lagrangian and non- degenerate Lagrangian have been discussed. For degenerate Lagrangian, we obtain multi-step numerical methods which suffer from parasitic corruption. In order to control the parasitism we write Variational Integrators as General Linear methods and then employ the projection technique to project the numerical solution onto the desired manifold
Dispersion Free Finite Difference Scheme for Time Harmonic Wave Equation
(COMSATS University Islamabad, Lahore Campus Library Information Services, CUI Lahore, 2012) CIIT/FA10-MSMATH-011/LHR; Muhammad Sarmad Arshad 05/LHR; Dr. Hafiz Abdul Wajid; FA10
Dispersion Free Finite Difference Scheme for Time Harmonic Wave Equation In this work, we construct novel central finite difference schemes for solving the Helmholtz equation by following the idea of Nehrbass and Yau Shu in one and two dimensions. We present an alternative construction by which not only central f inite difference schemes but all types of schemes (forward, backward) can be made exact for Helmholtz equation. We give explicit expressions for the coefficient of the central node in the standard central finite difference scheme of any order which makes the novel scheme dispersion free in one dimension and optimal in two dimensions. Such schemes add no cost for the implementation and one needs not to write brand new code instead just need to change the central node coefficient with the new one presented in this work to have highly accurate results. Comparison of numerical solutions obtained using novel schemes with standard schemes for Helmholtz equation in single and higher dimensions are given.
Optimal Finite Difference Schemes for Wave Equation
(COMSATS University Islamabad, Lahore Campus Library Information Services, CUI Lahore, 2012) Hifza Iqbal; CIIT/FA10-MSMATH-011/LHR; Master of Science in Mathematics; Dr. Hafiz Abdul Wajid
In this research work the novel optimal central finite difference schemes for solving the transient wave equation, are developed and analyzed. The novel schemes (a) give exact value at the nodes of the grid, meaning that an exact solution is possible for the transient wave equation; (b) the key advantage is that they have the simple structure as that of the standard central finite difference schemes, so we just need to make few amendments in the standard codes, instead of writing the new codes altogether; (c) above all, the beauty lies in the fact that they do not require fine mesh size and works for any wave number, specially for high wave number. Finally, in order to display the actual power of the novel schemes, numerical computations and graphical representations for two model problems are added, by using both novel and standard central finite difference schemes, which support our theoretical results, by showing the superiority of the novel schemes compared to the standard schemes
H-Magic Behaviors of Some Graphs
(Library Information Services, COMSATS University, Lahore Campus, 2015) Imran Zulfiqar Cheema; , FA11-PMATH-001; Dr. Muhammad Hussain,; LHR TP 5297
A graph ( , ) G V E has an H-covering if every edge in E belongs to a subgraph of G isomorphic to H . SupposeG admits an H-covering. AnH-magic labeling is a mapping  from ( ) ( ) E G VG  onto the integers {1,2,...,| ( ) E G VG  ( )|} with the property that, for every subgraph Aof G isomorphic toH , there is a positive integer c such that  A v ( )        v V A ( ) ( ) . e c   e E A ( ) A graph which possess such type of labeling is known as H-magic graph. Further if in a graph vertices are labeled first with smallest positive numbers, then the graph is called H-supermagic. Moreover a graph is said to be H-( , ) ad-anti magic if the magic constant for an arithmetic progression with initial value a and a common difference . d Numerous results on labeling of many families of graphs have been published. In this thesis, research work focuses on to formulate cycle 3 C- ( , ) ad anti-supermagic labeling for the Multi Wheels graph, supermagic labeling for isomorphic copies with its disjoint union of Multi-Wheels graph and cycle ( , ) ad-anti-supermagic labeling for Web graph. Also cycle anti-supermagic labeling for isomorphic copies with its disjoint union of ladder and triangular ladder graphs have been formulated. In addition, investigation of fan, friendship, ladder and wheel line graphs and study of the supermagic and anti-supermagic vertex-edge-face labeling of such graphs and their isomorphic copies have been carried in this thesis. An anti-supermagic labeling of the extension of cycle graphs is also formulated. Lastly the face supermagic labeling of (1,1,1) type of subdivided triangular ladder graph, subdivided mC 4-snake graph and subdivided kmC 4-triangular snake graph with its (1,1,...,1) and (2,2,...,2) string are also the part of this thesis.
Dynamical Properties of Specific Black Holes
(Library Information Services, COMSATS University, Lahore Campus, 2015) Muhammad Umair Shahzad,; FA13-PMATH-005; Dr. Abdul Jawad, Assistant Profesor; LHR TP 5606
Dynamical Properties of Specific Black Holes The present thesis comprises the study of three dynamical phenomenon such as thermal fluctuations, accretion and tidal forces of black holes/regular black holes. We consider the logarithmic corrected entropy in order to analyze the thermal fluctuations. We examine the effects of thermal fluctuations on a regular black hole of the non-minimal Einstein-Yang-Mill theory with gauge field of magnetic Wu-Yang type and a cosmological constant. We investigate the first law of thermodynamics in the presence of logarithmic corrected entropy and non-minimal regular black hole. Furthermore, we discuss the thermal fluctuation problem by utilizing the higher order corrected entropy. We examine the thermodynamical behavior of two well-known black holes such as Reissner-Nordström Anti de Sitter black hole with global monopole and f(R) black hole in the presence of higher order corrected entropy. We also discuss the accretion problem in two phases. In first phase, we analyze the accretion onto static spherically symmetric regular black holes for specific choices of the equation of state parameter. The underlying regular black holes are charged regular black holes using the Fermi-Dirac distribution, logistic distribution, non-linear electrodynamics, respectively, and Kehagias-Sftesos asymptotically flat regular black holes. In second phase, we develop the Hamiltonian dynamical system to tackle the accretion problem. We investigate the accretion of test fluids onto regular black holes such as Kehagias-Sftesos black hole and regular black holes with Dagum distribution function. We analyze the accretion process when different test fluids are falling onto these regular black holes. The behavior of fluid flow and the existence of sonic points is being checked for these regular black holes. Finally, we investigate the tidal forces occurring in a Kiselev black hole surrounded by radiation and dust fluids. We also solve the geodesic deviation equation for radially free-falling bodies toward Kiselev black hole. We explain the geodesic deviation vector graphically and point out the location of the event and Cauchy horizons for specific values of the radiation and dust parameters.
Group H-magic labelings of graphs
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2015) Syed Tahir Raza Rizvi; FA11-PMT-002; Dr. Kashif Ali; LHR TP 5283
A simple graph G admits an H-covering if each edge in E G is a member of a sub- graph of G isomorphic to H. An H-magic labeling of a graph G possessing an H-covering is a bijective function from the vertex-set and the edge-set of the graph G onto the set 1 2 V G E G if there occurs a positive integer , called the magic sum, such that for every subgraph H of G isomorphic to H, the sum wt H v V H v e E H e is equal to . The sum wt H is termed as, the H-weight. A graph is called H-magic (H-supermagic) if it possesses an H-magic (H-supermagic) labeling. If H is isomorphic to K2, then resulting labeling is known as edge-magic total labeling. This dissertation studies about the formulation of cycle-supermagic labelings for the disjoint union of isomorphic and non isomorphic copies of different families of graphs. We also evaluate the K2-supermagic labelings of some families of alpha trees. More- over, we ﬁnd the cycle-(super)magic labelings of uniform subdivided graph and cycle- supermagic labelings for non uniform subdivisions of fans and triangular ladders graphs. We also articulate about the introduction of H-groupmagic total labeling of a graph G over a ﬁnite abelian group A . In this study, we determine the H-groupmagic total labelings of fan graphs and disjoint union of isomorphic as well as non-isomorphic copies of fan graphs over ﬁnite abelian group A 3 n, where n 3
Chromatic Equivalence and Uniqueness of Subdivided K4 Graphs
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2016) Areeb; FA14-MSMATH-009; LHR TP 6935; Dr. Sana Javed
In this thesis, various chromatically unique and correspondent pairs of several families of graphs with same girths have been attained. Subdivided complete graphs are related to these families, indeed. Our major goal in this paper is to carry on the study of chromaticity of the subdivided complete graphs. We are dealing particularly with four vertices which is called 𝑘4-homeomorph. Distinctiveness of two particular classes of 𝑘4-homeomorph graphs with girth 11 has been explained in detail. The cycles of different lengths from graph has been removed during the discussion. In order to make them chromatically unique, vital and satisfactory conditions have been established. All well known definitions, results and terms are given in Chapter 1. Chapter 2 is covering a brief literature review about chromatic uniqueness and correspondence. While the chapter 3 is focused on the chromatic uniqueness along with chromatic equivalence of two 𝐾4-homeomorph graphs with same girth. At the end of the thesis, few open problems have been recommended as well.
Fractional Order Model of Tumor-Induced Angiogenesis
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2016) Sufian Asgher; SP15-RMT-001; LHR TP 6936; Dr. Ayesha Sohail
Methods of blood vessel formation is termed angiogenesis. Two distinguishing features of cancer, uncontrollable cell growth and metastasis, cannot be sustained in the absence of neovascularisation. Angiogenesis enhances the arrangement of solid tumors and their metastasis. During this research, present progress in understanding the procedure of angiogenesis in dynamic tumor development is talked about. The development of strong tumors, including neck and head squamous cell carcinomas, relies on upon the foundation of a blood supply inside the tumor (neovascularization or angiogenesis). For this procedure to occur, tumors create development components which stimulate the expansion, movement, and separation of endothelial cells. Nonlinear pde model of tumour induced angiogenesis is discussed during this research. The semi-analytic solution of this model is presented.
DEGREE BASE TOPOLOGICAL INDICES OF SOME FAMILIES OF GRAPHS
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2016) Mahrukh; SP15-RMT-009; LHR TP 6828; Dr. Kashif Ali
Atopological index is a numerical value associated with chemical constitution for cor relation of chemical structure with various physical properties, chemical reactivity or biological activity. In this thesis, we will discuss the different degree based topological indices i.e. Randic index, Zagreb index, Atom-bond connectivity index, Geometric arithmetic index, Augmented Zagreb index, Harmonic index, Sum-connectivity index for benzene BCZ, Benzene BTA diamond like graph, line graph of ladder and line graph of nanostar dendrimers. We will also compute ABC4 and GA5 for benzene BCZ, Benzene BTA diamond like graph
Anti-Magic Labeling on Harary Graph and Subdivided Caterpiller
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2016) Muhammad Mubashar; FA14-RMT-029; LHR TP 6839; Dr. Muhammad Hussain
Graphs play an important and vital role in Mathematics. Graph labeling creates new ideas graph theory. Can labeling be helped us to solve problems in Mathematics and other fields? The answer of this question increases the deep value of Graph Labeling. A magic labeling of a graph is described by Kotzig and Rosa [22] in 1970 as an important iniative. A graph labeling (or valuation) is a unique way that gives the labels to graph elements (commonly, these are non negative or positive integers). if we consider the domain set of all vertices and edges; such labelings are called total labelings. Sometime we use labelings for vertex set or the edge set alone and these labelings would be called as vertex-labeling and edge-labeling respectively. Many other domains are also possible. There are some famous labelings like as harmonious, cordial, graceful and antimagic. In 2000, Baˇca et al. [2] discussed a (a,d)-vertex-antimagic total labeling. A (a,d) vertex-antimagic total labeling is called super (a,d)-vertex-antimagic total labeling, when we give smallest labels to the vertices. An (a,d)-edge-antimagic total labeling was defined by Simanjuntak et al. [41]. An (a,d)-edge-antimagic total labeling is called a super (a,d)-edge-antimagic total labeling, when we give the least labels to the vertices. A graph G, which is made up of cycles with chords is known as Harary Graph. A brief discussion exposed important results on super (a,d)-edge-antimagic total labeling Harary graph by M. Baˇca and M. Murugan [1, 6]. This work is designed for the construction of (a,d)-vertex-antimagic total labeling of Harary graph as well as for super (a,d)-vertex antimagic total labeling of Harary graph for the different value of d.
Effect of Slip Velocity on Membrane Surface in Channel Flow System Using Power Series Method
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2016) Saba Qummer; SP15-RMT-013; LHR TP 6990
This thesis presents the general solution for two dimensional, laminar and fully developed flow in the parallel flat membrane system. The fourth order, non linear differential equation contain a free parameter Rw. This parametric Equation is solved by using a very general method i.e power series method. From this general solution expression are developed for velocity components, Normalized velocity and pressure distribution .We also discussed the effect of slip velocity at a porous surface. The effect of slip coefficient on velocity profiles is also examined graphically.
Analysis of Plasma Modified KdV Waves using Discrete Periodic Inverse Scattering Transformation
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2016) Muhammad Noaman Akbar; SP15-RMT-005; LHR TP 6989; Dr. Ayesha Sohail
Discrete Periodic Inverse Scattering Transformation (DPIST) are presented in this paper for solution of the well-known Korteweg-de Vries equations that has been hypothesized hypothetically such that non-linear dispersive waves spreading on interface of length only a joined of products fluid profundity fulfill a development condition with more extreme non-linearity than ordinary Korteweg-de Vries (KdV) equations for shallow water hypothesis. In this paper non-linear Fourier investigation is processed on time and space arrangement of surface relocation create at two areas downstream of a wavering film. Comes about because of a DPIST in view of the KdV equation, that decays a flag into soliton, sinusoidal and cnoidal parts, demonstrate while the amplitude of direct modes are preserved as unsettling influence spreads among two sensors, amplitudes of nonlinear modes increment. This recommends non-linearity of such surface waves is without a doubt more grounded than that anticipated by the KdV equations.
Effect of Slip Velocity at a Porous Surface in Tube Flow System using Series Solution Method
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2016) Zil-e-Huma; SP15-RMT-002; LHR TP 6987; Dr. Muhammad Younas
In this thesis, an attempt has been made on the effect of slip velocity at a mem brane surface in a tube flow system. The Navier-Stokes equations in cylindrical coordinates reduced to fourth order non-linear ordinary differential equation, con taining a parameter Rew, the wall Reynolds number. The semi-analytical solution of the set of governing equations with suitable slip flow boundary conditions is presented by using power series method to obtain velocity profile, pressure gradi ent and friction coefficient. Also the effect of slip coefficient on velocity profiles and pressure gradient is studied and their graphical solutions are also provided
On Some Topological Indices and Algebraic Properties of the Polyomino Chains of 4K-Cycles
(LHR TP 6988, 2016) Farah Iqbal; SP15-RMT-007; LHR TP 6988
In the present age of science and technology, topological indices have sig nificant role in the study of Quantitative Structure-activity (QSPR) and Structure-property (QSPR) relationships, to understand the chemical prop erties of the chemical compounds. In mathematical chemistry distance-based molecular descriptors are used to depict the structure of various molecules. For a simple connected graph G=(V,E), the Harary index is described as the sum of reciprocals of dis tances between all pairs of the vertices of the graph G. In this work, we derived some conjectures about the Harary index of the linear polyomino chain of 4k-cycles provided that k = 2 for both the cases i.e. linear and zig-zag chains, also there are expressions to get the sum of distances of the vertices while keeping one vertex as fix by dividing the chain into four dif ferent levels. This technique provide a basis to extend the idea for some other values of k
Face Labeling of Graph Embedded on the Surface of Klein Bottle
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2016) Sharafat Ali; SP15-RMT-004; LHR TP 7004; Dr. Saad Ihsan Butt
We deal with the problem of labeling the Klein Grid’s Km n vertices, edges and faces with mn squares by the consecutive positive integers from 1 up to jV(Km n )j + jE(Km n )j + jF(Km n )j in such a way that if we add up all the labels of a 4-sided face that becomes the weight of that face. With common difference d these weight of the faces form an arithmetic progression. We will ﬁnd out the existence of such type of labelings for several differences of d.
FACE ANTI-MAGIC LABELINGS AND DEFICIENCY OF GRAPHS
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2016) Ali Tabraiz; FA12-PMATH-001; LHR TP 4824; Dr. Muhammad Hussain
Labeling of graph is a beautiful and rich technique of Mathematics. It has extensive usage in applications such as data transmission, x-ray, circuit design, astronomy, communication links and coding theory. A labeling of a graph is a mapping that carries some set of graph elements into (usually positive integers). An ) , ( d a face anti-magic total labeling of a graph, with p vertices, q edges and f faces, is a one-to-one mapping that takes the vertices, edges and faces into }, , , 2 , 1 { f q p such that the sums of the label on the edges, vertices and faces form an arithmetic progression starting at a and having difference . d An edge magic total labeling of a graph G is a one-to-one mapping from ) ( ) ( G E G V onto the integers }, , , 2 , 1 { q p with the property that there is an integer constant k such that k y xy x ) ( ) ( ) ( for any ). (G E y x Furthermore, the edge-magic deficiency of a graph , G ) (G is defined as the minimum non-negative integer n such that 1 K n G is edge-magic.
Computing Degree Based Topological Indices Of HAC5
(COMSATS University Islamabad, Lahore Campus Library Information Services, CUI Lahore, 2016) Ahsan Raza,; FA14-MSMAT-010; Dr. M. Faisal Nadeem, Assistant Profesor [Supervisor]; LHR TP 4624
Computing Degree Based Topological Indices of HAC5C7 and VC5C7 Nanotubes There are certain types of topological indices such as degree based topological indices, distance based topological indices and spectrum based topological indices, which are used by researchers in quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) to predict physico-chemical properties of chemical compounds. A large number of topological indices are based on vertex-degree of molecular graphs. The purpose of current study is the discussion of the line graphs of HAC5C7[p,q] and VC5C7[p,q] nanotubes constructed by joinig left right side and as well as constructed by joining upside down, and computing some vertex-degree based topological indices of the line graphs of these nanotubes. Graph theory is a very powerful tool of applied mathematics. It concerns with different fields of science and has wide applications. In the first chapter we give some basic introduction and some definitions of graph theory. In the second chapter we give brief description about vertex-degree based topological indices. In the third chapter of this dissertation, we discussed the line graphs of HAC5C7[p,q] nanotube constructed by joinig left right side and as well as constructed by joining upside down. We also computed some well known degree based topological indices of the line graphs of HAC5C7[p,q] nanotube. This dissertation also includes discussion of the line graph of VC5C7[p,q] nanotube constructed by joinig left right side and as well as constructed by joining upside down. We also computed some well known degree based topological indices of the line graphs of VC5C7[p,q] nanotube given in last chapter.
On Fuzzy Generalizations Of Some Results In Finite
(COMSATS University Islamabad, Lahore Campus Library Information Services, CUI Lahore, 2016) Ghous Ali,; FA14-MSMAT-025; Dr. Adeel Farooq, Assistant Profesor [Supervisor]; LHR TP 4631
In this thesis, we have investigated some basic results of fuzzy group theory by constructing various examples. In particular, we have studied the notions of fuzzy subgroup, normal fuzzy subgroup, fuzzy coset and fuzzy quotient group and have elaborated these concepts through examples. We have proved that if ξ1 and ξ2 are fuzzy subgroups of a group G, then intersection of ξ1 and ξ2 is again a fuzzy subgroup of G. Also, we have shown that for two fuzzy subgroups ξ1 and ξ2 of a group G, if ξ1 is a normal fuzzy subgroup of ξ1, then ξ1 ∩ ξ2 is a normal fuzzy subgroup of ξ1 ∩ ν. Furthermore, we have defined the concept of a fuzzy subgroup of a fuzzy group. We have formulated the notion of an m-polar fuzzy group as a generalization of a fuzzy group. Moreover, we have introduced m-polar fuzzy analogs of some results in fuzzy group theory
Dynamics Of Two Population Synaptic Drive Model
(COMSATS University Islamabad, Lahore Campus Library Information Services, CUI Lahore, 2016) Iqra Khan,; FA14-MSMAT-018; Dr. Muhammad Yousaf Bhatti, Assistant Profesor [Supervisor]; LHR TP 4627
In this work we have investigated the two population synaptic drive model with a relatively complex temporal kernal for steep firing rate function. We have done this justification by deriving four dimensional autonomous dynamical system from Volterra equations using linear chain trick. Both of the local and global wellposedness of the model has been in vestigated. The existence of stationary point has also been investigated and the general formula for stationary point is derived. We get two different models for delay and non delay case. Theorem on the existence and stability has also been formulated. The stability issue has also been addressed for both cases for the steepness parameter q = 0, and q > 0, which leads us to the conclusion that delays in the system with weak kernel does not effect the stability properties of the model
Super Edge Magic Deficiencies Of Forests Of Alpha
(COMSATS University Islamabad, Lahore Campus Library Information Services, CUI Lahore, 2016) Abdul Rau; f, FA14-MSMAT-008; Assistant Profesor [Supervisor]; LHR TP 4633
Let G be a graph which is admitting an edge-magic total valuation. The edge magic deficiency [38] of this graph G is the positive numbers α required to make it a edge magic total graph. The general notation for edge-magic deficiency is μ(G). If there does not exist such α then edge magic deficiency will be ∞. The super edge magic deficiency of graph G [17], follows the same definition and it is denoted by μs(G). This dissertation studies about the super edge-magic deficiencies of forests of alpha families of trees. Firstly, We show that alpha trees with caterpillars as its components have super edge-magic deficiency is μs =0. We also prove that if the alpha tree has w-trees and alpha trees as its components then its super edge-magic deficiency is μs =0. At the end, we prove that forest of alpha trees with components as banana tree and stars preserve super edge-magic deficiency μs =1