Department of Mathematics

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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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    Thermodynamics of FRW Universe in Various Theories of Gravity Inspired by Entropy Corrections
    (Library Information Services, COMSATS University, Lahore Campus, 2018) Salman Rafique; , SP16-RMT-004; Dr. Shamaila Rani, Assistant Profesor; lhr tp 5349
    In this thesis, we discuss the thermodynamical analysis for gravitationally induced particle creation scenario in the framework of DGP braneworld and Chern-Simons gravity. For this purpose, we consider apparent horizon as the boundary of the universe. We take three types of entropy such as Bakenstein, logarithmic corrected and power law corrected entropies with ordinary creation rate Γ. We analyze the first law and generalized second law of thermodynamics for these entropies which hold under some con straints. The behavior of total entropy in each case is also discussed which implies the validity of generalized second law of thermodynamics. Also, we check the thermodynamical equilibrium condition for two phases of creation rate Γ, that is constant and variable and found its vitality in all cases of entropy
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    Mathematical Model of Targeted Bone Remodeling
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Abdul Sattar; CIIT/SP16-RMT-009/LHR; Dr.Sana Javed; LHR TP 5199
    In this thesis stability analysis of the mathematical models for osteoporotic and osteomyelitic bone turnover is checked. The study of diseases in silico has been becoming very popular because of its rapid outcomes. This study is also beneficial from economical point of view. Several diseases have been investigated through mathematical modeling like cancer, chronic kidney diseases, liver disorders etc. Osteoporosis and osteomyelitis are very common bone disorders having different incident causes. Osteoporosis occurs due to several factors like age, estrogen deficiency or an imbalance in bone remodeling process, while osteomyelitis is caused by a bacterium called Staphylococcus aureus, a type of staph bacteria. We examined the expository signaling among the bone cells named as osteoclast and osteoblast. Main functioning of osteoblasts is bone formation while osteoclasts are bone removal cells. Mathematical frame works for both pathologies comprising of the communication between osteoclasts and osteoblasts have been presented to exhibit the conditions for stability in normal and dysregulated bone turnover. The percentage ratios of the population of osteoblast/osteoclast have been determined via numerical simulations. The remedial upshots of targeting both osteomyelitic and osteoporotic cells participating in such processes are examined. This work not only studies the dynamics of osteoporotic and osteomyelitic bone turnover but also exemplifies how therapeutic approaches might be inspected via such computational methodologies. This thesis is divided into five chapters. First chapter is about the introduction of bone biology, second chapter is comprised of the existing literature of different mathematical models of bone remodeling. Chapter 3 is based on our main results. Discussions, open problems and references are provided in chapter 4 and 5 respectively.
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    Different Cases of Singular Domains in Two Population Synaptic Drive Model
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Wasim Shahid; SP16-RMT-008; Dr. Muhammad Yousaf; LHR TP 5198
    In this work we have investigated the dynamics of synaptic drive two-population neural firing rate model for the case of steep firing rate function by applying the theory of switching dynamical systems. The switching dynamic theory simplifies the dynamical system and receive the approximation of the solution for the system with certain modes of parameters. In such type of outlook, the phase space is divided into regular as well as singular domains. These singular domains are characterized into black, white and transparent on the basis of the behavior of trajectories on the sides of singular domain. Moreover, there has been investigation of the existence and uniqueness solutions with corresponding singular and regular stationary points. It has been observed that regular stationary point, which are located within regular domain, are always constant and this characteristic is secured for smooth and sufficiently steep firing rate function. Yousaf.et al investigated the dynamics for the same model. They investigated the dynamics only for some particular cases. We have explored some more cases of singular domains and investigated the dynamics in regular and singular domains. Moreover, general formulae for the stationary points of the system have been calculated and the existence of singular stationary points which belong to the singular domains have been investigated. Furthermore, these singular domains in new dynamical system have also been transformed. Besides, the stability of singular stationary points has been investigated. In the wake of outcomes, the stationary points are attended by implementing the singular perturbation analysis and Thikhonov theorem.
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    Topological Indices On Chemical Graphs
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Tayyaba Razzaq; SP16-RMT-002; Dr. Muhammad Hussain; LHR TP 5345
    Topological index is a numeric parameter which describe some silent features (con nectivity, boiling point, stability and melting point) of a Molecular graph representing some chemical compounds. These indices characterize topology and are usually hold mathematical properties of Graph invariant.The process of transforming chemical information of chemical molecules into useful numerical results is known as molec ular descriptor. These molecular descriptors are extensively used in the study of Quantitative structure-activity relationships (QSARs) which give insight to the bio logical properties based upon chemical structures. We will investigate R˜andic indices Rλ(G), where λ ∈ R. First, second and third Z˜agreb indices, First and second Z˜agreb Polynomials, Multi Z˜agreb indices, Atom bond connectivity ABC index, Geometric arithmetic GA index, fourth version of ABC index, Harmonic index and fifth version of GA index to estimate biological properties of chemical graphs