Department of Mathematics
Permanent URI for this communityhttps://repository.cuilahore.edu.pk/handle/123456789/21
Browse
16 results
Search Results
Item Mathematical Modeling and Analysis of Pertussis Cough(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Syed Shan-E- Ahmad; CIIT/SP24-RMT-017/LHR; Dr. Syed Tahir Raza Rizvi; LHR TP 10078Mathematical modeling of dynamical systems often uses ordinary differential equations to represent how key variables evolve over time. Such models support prediction, inter-pretation, and decision-making in biology, engineering, economics, and environmental science by linking mechanisms to observable outcomes. Stability analysis is central in this setting: it determines whether equilibria persist under perturbations and clarifies long-term behavior, resilience, and the potential impact of interventions. We apply these tools to a pertussis (whooping cough) model with five compartments: susceptible, latent-immunity, exposed, infectious, and recovered. The formulation in-corporates waning vaccine-induced immunity, reinfection, and disease-induced mortal-ity, and it is proven to be positive, bounded, and forward invariant. The basic repro-duction number Ro is derived via the next-generation matrix method; sensitivity analy-sis shows transmission, progression to infectivity, recovery, and vaccination rates most strongly influence persistence. Local and global stability of the disease-free and endem-ic equilibria are established. A nonstandard finite difference scheme preserves quali-tative dynamics and supports simulations. Numerical experiments confirm the analy-sis and illustrate how parameters shape outbreak magnitude and duration. Finally, an integral sliding mode control acting on vaccination robustly suppresses exposure and infection and can drive elimination under parameter uncertainty and disturbances. Keywords: Mathematical modeling; pertussis; compartmental model; Vaccination; wan-ing immunity; basic reproduction number Ro; sensitivity analysisItem Study of Stochastic Nonlinear Schr¨odinger Equation for Optical Soliton Solutions(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Iqra Anjum; CIIT/FA23-RMT-014/LHR; Dr. Syed Tahir Raza Rizvi; LHR TP 10063It finds extensive application in the analysis of nonlinear systems for the detection of ap- proximate solutions and stability analysis. In order to understand nonlinear partial differ- ential equations (NLPDEs), physicists and mathematicians need to study exact solutions. Many NLPDEs such as Korteweg de Vries equation (KdV), Sin-Gordon equation, non- linear Schr¨odinger equation (NLSE), and other equations possess solitons solutions. A soliton, which arises as a result of dispersion and nonlinearity, is a wave that retain their original features while propagating from one medium to another. This thesis discusses the stochastic variants of the well-known non linear Kairat-II and Kairat-X (NLKIIKX) equa- tions in (3+1) space dimensions, those being soliton theory’s major models. Adding the Wiener process to simulate random fluctuations, we study the dynamics of these equations within a probabilistic system. With enhanced direct algebraic technique (EDAT) and novel projective Ricatti equation approach (NPREA), we derive exact soliton solutions for these stochastic equations. The behavior of a stochastic differential equation (SDE) is fundamen- tally different from a deterministic equation due to the presence of randomness and gain an understanding about random fluctuation-influenced complex system behavior. At last, by using these two methods, we shall develop three-dimensional and two-dimensional plots in order to show soliton solutions. It is used to simulate randomness-influenced systems, e.g., stock markets or noisy physical systems. We will also study the energy balance approch (EBA) which examines a system by comparing the rate of input and output energy to find the steady-state or dynamic state.Item Numerical And Exact Solutions For Various Nonlinear(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Syed Oan Abbas Bukhari; CIIT/SP20-RMT-039/LHR; Dr. Syed Tahir Raza Rizvi; LHR TP 7641A soliton is a particular type of solitary wave, that is retrieved as a result of integrable NLPDE. There are different type of solutions like analytic and approximate solutions exist. An exact solution is a symbolic representation of a value which solves a given equation exactly, while a numerical solution is numerical representation of a value which solves a given equation exactly or approximately. Numerical solutions are very usefull when we are unable to retrieve analytic solution. It is also used for comparison with analytic solutions. A lot of analytic and numer- ical methods exists in literature to solve the NLPDEs such as Bernouli’s equation approach, extended tanh method, modified simple equation method and RK methods etc. In this work, our purpose is to attain bell-type, kink and anti kink type, singular solitons and power series solutions for various forms of NLEEs. We’ll also obtain Weierstrass elliptic func- tions, Jacobi elliptic functions in terms of hyperbolic, trigonometric and rational functions. In this thesis, we’ll study various forms of NLEEs to attained these types of solutions with the help of various method like sine-cosine method (SCM), csch method, tan-cot method, sub-ode technique and residual power series method (RPSM). We will also compare our exact solutions with numerical solutions and then represent these solutions graphically.Item Solitary Wave Solutions and Stability nalysis for Nonlinear Evolution Equation(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Asmavia Shahid; SP23-RMT-005; Dr. Syed Tahir Raza Rizvi; LHR TP 9587In order to understand nonlinear partial differential equations (NLPDEs), physicists and mathematicians need to study exact solutions. Many NLPDEs such as Korteweg de Vries equation (KdV), Sin-Gordon equation, nonlinear Schr¨odinger equation (NLSE), and other equations possess solitons solutions. A soliton, which arises as a result of dispersion and nonlinearity, is a wave that retain their original features while propagating from one medium to another. The trial equation method proposed by Liu it is an advanced analyt- ical method for finding the exact solution of NLPDE in the study of solitons. Another method complete discrimination system for Polynomial Methods (CDSPM) is a collection of explicit expressions developed with the coefficients of a polynomial that contains real or symbolic variables. Using this method CDSPM, we find both the quasi periodic behaviour and sensitivity of nonlinear equation. Additionally, in a mathematical framework, sensitiv- ity analysis examines to see how modifications to a system’s parameters affect its behavior. This thesis examines nonlinear Kakutani-Matsuuchi (NKM) model of internal gravity waves. The evolution of lengthy internal gravity waves in a stratified fluid medium is meticulously captured by the (1+1)-dimensional NKM model. By using trial approach we explore several solutions, such as SW, rational and Jacobi elliptic function (JEF). The CDSPM technique is utilized to examine quasi-periodic behavior, critical solution conditions, bifurcation behav- ior and sensitivity analysis. We also explores the quasi periodic behaviour and sensitivity analysis of our governing models at various initial values. Key words: Kakutani-Matsuuchi equation; solitons, complete discriminant system; quali- tative behaviour; quasi periodic behaviour; sensitivity analysis.Item Jacobi Elliptic and Solitons for Nonlinear Schrödinger Equations(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Marwa Ahmad; CIIT/SP17-RMT-016/LHR; Dr. Syed Tahir Raza Rizvi; LHR TP 5440The Nonlinear Schrödinger Equation plays a central role in describing wave propagation in nonlinear media such as optical fibers, plasmas, and Bose–Einstein condensates. This study explores exact solutions of the equation using Jacobi Elliptic Functions, which provide periodic wave structures, and their limiting forms that lead to Solitons. The analysis highlights how different parameter regimes yield distinct wave profiles, ranging from periodic patterns to localized solitary waves. These solutions are important for understanding stability, modulation, and nonlinear wave dynamics in various physical systems.Item Topological Indices Related to Graph Operations(COMSATS University Islamabad, Lahore Campus Library Information Services, CUI Lahore, 2020) Urooj Akram,; FA16-RMT-005; Dr. Syed Tahir Raza Rizvi, Assistant Profesor; lhr tp 5179Topological Indices Related to Graph Operations In this thesis we find out the atomic bond connectivity index, Randic index, augmented Zegreb index, geometric arithmetic index and product connectivity index for sum and corona of any two connected graphs. By using neighborhood degree we formulate fourth version of atomic bond connectivity index ABC4 and fifth version of geometric arithmetic index GA5 of sum of two graphs. We also formulate atomic bond connec tivity index for product of any two regular graphs . Furthermore, we formulate new atomic bond connectivity index of F-sum of cycle (Cm) and path (Pn) graphs. At the end we calculate new atomic bond connectivity index ABC5 depending on graphs kC8 where k are the stages ofC8 depends on straight chain and pairwise zigzag chainItem New Solitary Wave Solution for Nonlinear Schrodinger Equation(COMSATS University Islamabad, Lahore Campus Library Information Services, CUI Lahore, 2020) Urooj Akram,; FA16-RMT-005; Dr. Syed Tahir Raza Rizvi, Assistant Profesor; lhr tp 5179A nonsingular and localized wave which propagates without change of velocity and shape is called a solitary wave. A soliton is a solution of NLPDE which has a per manent form and is localized. A solitary traveling wave does not obey superposition principle; it does not disperse. Solitary waves retain its permanent structure even when it collides with other soliton [25]. Soliton has many applications in other fields like engineering and biology. Soliton solutions can be calculated with the help of differ ent nonlinearities. These nonlinearities are Kerr law, power law, parabolic law, dual power law, exponential law etc. In this thesis we will study, the soliton solutions of Lakshmanan-Porsezian-Danial (LPD) model. In order to obtain exact soliton solutions several powerful methods have been proposed. We will use modified trail equation method with Kerr and parabolic law to obtain some hyperbolic and rational function solutions to LPD model. We also find new and more general exact traveling wave solution by using improved G′/G-expansion method with Kerr and parabolic laws of nonlinearitiesItem Stationary Solutions And Optical Solitons For Nonl(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Insibat Afzal; FA17-RMT -012; Dr. Syed Tahir Raza Rizvi; LHR TP 5649Solitons are vastly growing field of research not only in the field of nonlinear optics but they also appear in applied mathematics, telecommunication industry and in various branches of physics and biology. Specially, optical solitons arises in many different f ields including nonlinear optics. They are the light pulses that do not diffract in the mediumthrough which they passes. These days, optical solitons play a significant role in many branches of modern telecommunication mediums including internet industry, twitter, Facebook and many more. Due to the property of not changing shape and path after collision, optical solitons are used to transfer data without using repeaters. A lot of work have been done in retrieving optical and exact solutions of the solitary waves in nonlinear optics. In this dissertation, we will obtain the stationary solutions of Gerdjikov-Ivanov (GI) equation, cubic-quintic NLSE and paraxial equation by using the help of Lie group analysis. And also, we will get chirped sub-pico bright, dark and dark singular combo optical soliton solutions for Triki-Biswas equation by using some constraints. To solve Triki-Biswas equation we will use two different integrating methods namely, sine Gordon method and modified extended direct algebraic method.Item PT-Symmetric and W-Shaped Solitons for Nonlinear Schrödinger Equations(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Ujala Safdar; FA18-RMT-035; LHR TP 6548; Dr. Syed Tahir Raza RizviSolitons are the special types of solitary waves who does not change their speed and shape during propagation. When a light pulse in space or time travels in some host medium, it does not dispersive or diffract is called optical Solitons. Optical solitons are produced due to interaction between group velocity dispersion and self-phase modulation. In this thesis, we studied the different forms of optical solitons in non-Kerr law media, parabolic, power, quadratic-cubic and anti-cubic laws with parity-time (PT) symmetric nonlinear and linear lattices. We obtain various types of soliton solutions with the as sistance of the inverse engineering scheme (IES). Under diverse physical conditions, we produce bright, dark and singular solitons. We will also obtained the dynamical behaviour of W-shaped soliton in cigar-shaped Bose Einstein condensate (BEC) in various nonlinear itiesItem Lump and Interaction Solution for the Nonlinear Schrödinger Equation(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Iram Javed; FA18-RMT-075; LHR TP 6550; Dr. Syed Tahir Raza RizviSolitons are the special types of solitary waves which do not vary their shape and frequency during propagation speed and is also an attractive area of current research in nonlinear physics and mathematics as well as in fiber optics and communication technologies. There are many forms of solitons such as singular soliton, bright soliton, dark soliton and lump soliton. In this thesis, we acquire lumpsolitonsolutionsfor paraxial nonlinear Schr¨odinger equation (NLSE) with the aid of Hirota bilinear method (HBM). In space, lump soliton solutions are rationally positioned in every direction. We also get optical dromions for paraxial NLSE by using modified extended Tanh method. We get domain walls, singular dromions and combined dark-singular dromions. We also list the constraint conditions