Department of Mathematics
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Item Cosmological and Dynamical Instability Analysis in Some Modified Theories of Gravity(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Tanzeela Nawaz; CIIT/FA13-PMATH-007/LHR; Dr. Shamaila Rani; LHR TP 7732The present thesis comprises the study of three dynamical phenomena such as evolution of the universe through cosmological parameters, thermodynamics via the generalized second law of thermodynamics and dynamical instability of collapsing star taking into account adiabatic index. Firstly, we consider a new holographic dark energy model in the framework of dynamical Chern-Simons modified theory of gravity to explore various cosmological parameters (like equation of state, squared speed of sound, Om-diagnostic) and cosmological planes ( d d ' where d ' is the evolutionary equation of state parameter, statefinders). We examine their graphical results and also comprise with the different observational data like Planck, 0 H , SNLS and WMAP. Secondly, the thermodynamics through generalized second law of thermodynamics in the framework of dynamical Chern-Simons theory of gravity by taking interaction between dark energy and dark matter is discussed. In the first part, we discuss the scalar field phenomenon of dark energy model having a non-canonical kinetic term in the flat universe and investigate the stability of the model through squared speed of sound. The generalized second law of thermodynamics by taking usual entropy and its correction forms like logarithmic and power law at apparent horizon is studied. In second part, this law is investigated by taking the interaction of dark matter and dark energy with holographic dark energy model and different cutoffs like Hubble horizon, event horizon, their combination, Ricci scalar and its generalized form. The generalized form of second law of thermodynamics in terms of coincidence parameter is being evolved. Next, we choose the framework of modified Brans-Dicke gravity to study generalized second law of thermodynamics by developing the expressions tot S ' for usual entropy as well as its corrected forms, logarithmic and power law corrected on the apparent and event horizons. Also, the power law forms of scalar field and scale factor are utilized in order to analyze the clear view of thermodynamic law. Finally, we find the instability ranges of spherical and cylindrical symmetric structures of a collapsing object in FT, theory where, T denotes the torsion scalar and is the trace of energy-momentum tensor. The perturbation method is used upto first order in the modified field equations to construct dynamical collapse equations. These equations with adiabatic index are utilized to check the instability ranges in Newtonian and post-Newtonian limits. It is found that the instability ranges depend on the dynamical variables of collapsing object.Item 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 5251In 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.Item 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 5949We 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.Item 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 5941A 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).Item Dynamical Properties of Specific Black Holes(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2017) Muhammad Umair Shahzad; CIIT/FA13-PMATH-005/LHR; Dr. Abdul Jawad; LHR TP 5606The 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.Item Topological In Variants Of Graphs(Library Information Services, COMSATS University, Lahore Campus, 2017) Imran Nadeem,; FA13-PMATH-003; Dr. Hani Shaker, Assistant Profesor [Supervisor]Chemical graph theory provides useful tools such as molecular descriptors to develop strong intrinsic relationship between the physicochemical features of chemical com pounds and their molecular graphs. The study of molecular descriptors provides a the oretical basis for the fabrication of chemical materials and is helpful in making up for the lack of chemical experiments. There are two prominent types of molecular descrip tors; the topological indices and counting polynomials. Further, topological indices can be categorized in two major classes: one class is based on degree and the other class is based on distance. In this thesis, we present the study of certain topological indices belonging to degree- and distance-based classes and counting polynomials for some well-known nanostructures. We also present a comparative study between dif ferent topological indices belonging to degree- and distance-based classes for general graphs. In addition, we study the para-line transformation of graphs and obtain the gen eral expressions of certain topological indices for this transformation. We achieve the lower and upper bounds of certain distance-based topological indices for the para-line transformation of graphs