Department of Mathematics

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    Mathematical Analysis of Delayed Tumor Models
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Maria Sajjad; CIIT/SP17-RMT-030/LHR; Dr. Sana Javed; LHR TP 5463
    This thesis focuses on the current problem of cancer, interlinked with innate immune system. We have taken into account the innate immune system with delay and the delay differential equations are solved using analytical and nu- merical techniques. From the mathematical analysis we have explored the chal- lenging problem of cancer invasion.
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    Edge Irregularity Strength of Rooted Product of Certain Families of Graphs
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Muhammad Usman; CIIT/FA16-RMT-011/LHR; Dr. Faisal Nadeem; LHR TP 5183
    Graph theory is one of the most powerful tool in the field of mathematics. Different mathe- maticians work in this field and consider it as a powerful tool in different directions because in recent times graph theory has confirmed as a paramount mathematical tool in various subjects. Now it has emerged as an important mathematical discipline in its own right. There are several types of labelings in graph theory. In this thesis we intend to find the irregular labeling of double graph, rooted product as well as the some family of the Teoplitz graphs. we have divided this thesis into seven chapters. First chapter is about the introduction as well as basic definitions of different types of graphs. In this chapter we have discussed some graph operations along with their examples. In chapter second we have clarified different types of labeling which was defined by well known scholars. At the end of this chapter we have written some known results of edge irregularity strength for certain families of graph. In chapter three we have discussed results related to edge irregularity strength of double graph of path and cycle. we have also proved these results. we believe that these results are accurate. In chapter four we have developed some results for edge irregularity strength of rooted product of cycle and path graphs. In chapter five we find edge irregular labeling of different families of Teoplitz graphs and calculated the edge irregularity strength of Teoplitz graphs.
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    The Group of Runge-Kutta Methods
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Asad Ullah; CIIT/FA13-BSM-018/LHR; Dr. Yousaf Habib; LHR TP 5251
    In this report we gave a review of Runge Kutta methods and partitioned Runge Kutta meth- ods.We also studied the group properties of these methods.The order conditions of Runge Kutta methods are related to rooted tree ,where as the order condition of partitioned Runge Kutta methods are related to bi-color rooted trees.The compositions of Runge Kutta meth- ods and partitioned Runge Kutta methods are studied as well.
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    Inflationary Dynamics and Thermodynamics of Chaplygin Gas Models
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Muhammad Adeel Sultan; CIIT/SP17-RMT-006/LHR; Dr. Abdul Jawad; LHR TP 5441
    In this thesis, we study the reconstruction of warm inflation model with the help of tensor to scalar ratio (r) and scalar spectral index (ns) in terms of number of e-folds (N). In this regards, we take flat FRW metric and discuss the general forms of dissipative coefficient Γ(φ, T ) as well as effec- tive potential V (φ, T ) for two dissipative regimes i.e., the weak and strong. We use inflationary parameters like slow roll parameters, power spectrum of the curvature perturbation, tensor spectrum, spectral index, scalar to tensor ratio and Hubble parameter to find the generalized form of dissi- pative coefficient and effective potential. Further, we discuss generalized cosmic Chaplygin gas in the scenario of particle creation. We examine energy density (ρ), Hubble parameter (H), declaration parameter (q), tem- perature (T ) and particle number density (n) under three different models of particle creation rate. We also analyze the production of entropy and Bekenstein entropy in the scenario of first law of thermodynamics. We dis- cuss the validity of generalized second law of thermodynamics and thermal equilibrium under three models of Γ and discuss the graphical behavior of above mentioned terms.
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    Two-Layer Flow of Immiscible Non-Newtonian Fluids between Parallel Plates
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Sheikh Zain Majid; CIIT/SP17-RMT-020/LHR; Dr. Qurat-ul-Ain Azim; LHR TP 5439
    In this Thesis, we consider fluid flow through two parallel plates which is steady, laminar, incompressible, unidirectional and fully developed. The incompressible Oldroyd Six con- stant fluid is considered.Three flow problems including the Couette flow, Poiseuille flow and Couette-Poiseuille flow are modeled. The differntial equation for each problem is simplified with modified homotopy perturbation method, and thus result is obtained. The solutions are obtained up to second order of approximation for all three types of flows. The results are then presented graphically and the effects of various parameters on the flows are studied. We have observed that the non-Newtonian fluid parameters greatly affect the flow velocities in both the fluid layers.
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    Some Studies of Two-Population Neural Field Model
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Zeeshan Afzal; CIIT/FA13-PMATH-009/LHR; Dr. Muhammad Yousaf Bhatti; LHR TP 5949
    We investigate the appearance and disappearance of localized activity states, so called bumps in Wilson-Cowan type two population neural field model under the influence of transient spatio-temporal external input with smooth α-type and piecewise-type temporal functions. This two-population model is composed of two coupled nonlinear differential equations derived for the dynamics of spatially localized populations of both excitatory and inhibitory model neurons. The model with no external input corresponds to atmost two bump pair solutions. Such a system can be interpreted as a minimal cortical model for short term working memory, that is the ability of the brain to actively hold stimulus-related information for some seconds in short term memory and discards once it becomes irrele- vant Initially, if there is no activity in the system, persistent activity state can be evoked by switching on a suitable transient excitatory external input. This activity remains stable even though external input is switched off. The effect of external input on appearance and disappearance of bumps for different spatial, smooth α-type and piecewise-type temporal functions of external input is investigated and found that certain parameters play a key role in the generation of persistent activity states in the network e.g. relative inhibition time constant, total duration and the amplitude of external input. It is found that the minimum values of the amplitude and active time to evoke the activity in the network is smaller than those observed in showing that the present choice of temporal functions in the external in- put is more effective and more close to natural behavior.
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    On Some Combinatorial Properties Of Graphs, Posets
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Muhammad Arfan Ali; FA12-PMATH-002; Dr. Sarfarz Ahmad; LHR TP 5945
    This work investigates fundamental combinatorial properties of graphs and partially ordered sets (posets), focusing on structural relationships, ordering principles, and enumeration techniques. Concepts from Graph Theory and Order Theory are used to analyze connectivity, comparability, chains, antichains, and lattice structures. The study highlights how these discrete frameworks can be applied to solve problems in optimization, data organization, and theoretical computer science. Emphasis is placed on key invariants and combinatorial methods that reveal deeper insights into the organization and behavior of discrete systems.
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    Slip Effects on a Creeping Viscous Flow through a Uniform Porous Tube with Exponential Re-Absorption
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) MUHAMMAD ARSHAD; CIIT/SP16-RMT-003/LHR; Dr. Muhammad Younas; LHR TP 5442
    The purpose of this research is to study slip outcomes on creeping viscous flow across tube which is permeable. The fluid is flowing and exponential . reabsorption The governing equations are solved with the help of stream function technique which is an efficient method to solve non linear partial differential equations. Calculate velocity profile which contains radial and axial velocity. Flow rate is also calculated. The graphical analysis of the problem is also given.
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    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
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    Analytical and Soliton Solutions for Nonlinear Schr dinger Equation
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Badar Nawaz; CIIT/FA13-PMATH-001/LHR; Dr. Kashif Ali; LHR TP 5941
    A soliton which is also known as the traveling wave solution has a unique property,” the collision of two solitons produce the waves whose permanent structure remain the same”. These wave solutions are very stable and having a key role in the mathematical physics, especially in fiber technology. ”A soliton does not change its amplitude, shape and speed for a long distance in a non-local and nonlinear optical media”. The applications of soliton solutions are also in various branches of physics. The soliton solutions can be calculated by using the different types of nonlinearities. The optical solitons are very important in the study of nonlinear optical fibers. In this thesis firstly, we determine different types of soliton solutions of dimensionless form of Quintic Complex Ginzburg-Landau (CGLQ) equation by using modified extended tanh− function method (METFM) and the extended trial equation method (ETEM). Sec- ondly, we obtain bright, singular and Jacobi elliptic soliton solutions for the time fractional perturbed NLSE (TFPNLSE) by using ETEM with Kerr, power and log law nonlineari- ties. Thirdly, we find the combo and dipole soliton solutions for CGLQ model with differ- ent ansatz methods. Next, we construct different soliton solutions for the paraxial NLSE (PNLSE) in Kerr media by ETEM. Then, we get the bright and dark solitons for non- Kerr law NLSE with third order (3OD) and fourth order (4OD) dispersions by Sine-cosine method (SCM)and Bernoullis equation method (BEM) with nonlinearities. In the last, by using Hirota bilinear method (HBM), we obtain the multiple solitons for the nonlinear Telegraph equation (NLTE) and the nonlinear PHI-four equation (NLPFE).