Department of Mathematics
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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 Application of the Bilinear Neural Network Approach to Kadomtsev-Petviashvili Model for Nonlinear Waves(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Numan Ahmed; CIIT/SP24-RMT-003/LHR; Dr. Syed Tahir Raza Rizvi; LHR TP 10066Nonlinear evolution equations (NLEEs) serve as fundamental models for wave phe-nomena in many physical systems, including fluid dynamics, plasma physics, optical pulse propagation, and atmospheric processes. They describe the spatiotemporal evolu-tion of waves in media where nonlinear interactions and dispersive effects act together. A prominent feature of such equations is the emergence of solitons, which are localized travelling waves capable of preserving their shape and speed over long distances. This remarkable property is generally attributed to a balance between nonlinear steepening and dispersive spreading. Solitons appear across a broad spectrum of physical contexts, ranging from hydrodynamics, nonlinear optics, and plasma physics to solid-state and lat-tice systems, where they often display particle-like behavior while retaining their wave nature. They have also been observed in more complex settings such as astrophysical, condensed-matter, nuclear, and other nonlinear media, highlighting their wide applica-bility and importance in both theoretical studies and practical applications of nonlinear wave dynamics. Different types of solitons exist based on their shape and behavior. Kink solitons rep-resent a transition from one state to another, similar to steps in a staircase. Breather solitons are localized waves that expand and contract over time. Rogue waves are ex-tremely large and sudden waves that can be much higher than normal waves. Lump solitons are localized in all directions and gradually disappear with distance. These d-ifferent wave structures help researchers better understand nonlinear wave behavior in real-world systems. This thesis presents a bilinear neural network modeling (BNNM) framework that cou-ples the Hirota bilinear approach with trainable neural parameters to obtain closed-form solutions of the (3+1)-dimensional generalized Kadomtsev-Petviashvili equationItem 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 Application of CDSPM and Qualitative Analysis for Dynamical Models(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Sana Shabbir; FA22-RMT-033; Dr. Syed Tahir Raza Rizvi; LHR TP 9353In order to understand nonlinear partial differential equations (NLPDEs), physicists and mathematicians need to study exact solutions. Many analytical methods have been pro posed to acquire solutions for NLPDEs such as Korteweg de Vries equation (KdV), Sin Gordon equation, nonlinear Schrodinger equation (NLSE), and all these 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 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. Liu was the first who suggested CDSPM technique. Using this method CDSPM we find both the exact, solitary wave (SW) solutions and sensitivity of nonlinear equation. Additionally, in a mathematical framework, sensitivity analysis examines to see how modifications to a system’s parameters affect its behavior. This thesis examines the complex wave patterns of the Gerjikov-Ivanov equation (GIE), commonly known as the derivative nonlinear Schrodinger equation (DNLSE) and analyt- ¨ ical solitons solutions for the cubic-quintic time-fractional nonlinear non-paraxial pulse transmission model. These modifiable model are significant because they are applied in fiber optics communication, nonlinear optics, and optical processing of signals. By using CDSPM approach we explore several solutions, such as rational, SW, and Jacobi elliptic function (JEF). The CDSPM technique is utilized to examine quasi-periodic behavior, bi furcation behavior, critical solution conditions and sensitivity analysis. We also explores the sensitivity analysis and quasi periodic behaviour of our governing models at various initial values. In addition, a number of methods for detecting quasi periodic behaviour will be addressed, including 3D and 2D graphs, time series, and Poincare mapsItem Study of Breathers and M-Shaped Solutions to Nonlinear Models(Library Information Services, CUI Lahore, 2022) Samia Ahmed; SP21-RMT-041; Dr. Syed Tahir Raza RizviA soliton is a localised solitary wave with a constant structure that recovers its shape after col- lision. We obtain a soliton solution as a result of the balance dispersion and nonlinear effects. Solitons are generated in a variety of material and have several applications. Solitons have a broad range of applications, which suggests that they frequently occur in a variety of physics domains.In this thesis, exponential, hyperbolic and trigonometric function will be combined to analyse various NLEEs to get some solutions. Our goal in this work is to find rogue soliton wave solu- tion (RSW), lump soliton wave solution (LSW), lump with one (LSW1K) and two kink soliton wave solution (LSW2K), periodic soliton waves solution (PSW), correspondence between lump kink and periodic soliton wave (LPKSW) for different types of NLEEs. Additionally, M type interaction with periodic and kink soliton waves solution (MSPKSW), M type correspondence with rogue and kink soliton waves solution (MSRKSW), M type rational soliton wave solution (MSRSW), M type rational interaction with one kink wave (MSR1K) and two kink (MSR2K), periodic cross rational soliton wave solution (PCRSW), kink cross rational soliton wave solu- tion (KCRSW), periodic cross kink soliton wave solution (PCKSW), periodic cross lump soli- ton wave solution (PCLSW), breather soliton wave (BSW), homoclinic breather soliton wave solution (HBSW) and multiwave soliton wave solutions (MWSW) will be obtained. We also observed the degeneration of waves through bilinear form and also reviewed the graphical be- haviour of our solutions for quadratic nonlinearity (QN) model and cold bosonic atoms in a zig zag optical lattice (CBAZZ) model.Item Applications of Variational Integrators and Solitons Solutions for Evaluation Equations(Library Information Services COMSATS University Lahore Campus, 2024-03-14) Sana Ghafoor; FA22-RMT-006; LHR TP 9343; Dr. Syed Tahir Raza Rizvin extensive variety of fields, including biology, chemistry, physics, fiber optics, mechan ics, atmospheric science, and electronics science, use nonlinear partial differential equa tions (NLPDEs), which makes them important. NLPDEs of a particular type are the non linear Schrodinger equations (NLSE). Every NLPDE that is integrable and nonlinear has ¨ a soliton solution. A specific type of solitary wave defined as a soliton has the ability to maintain its original structure even after interacting with another soliton. They are special wave packets which have the capacity to travel long distances without suffering any dis tortion. Solitons are often used in communication due to they are able to transmit signals with no errors throughout long distances and contain an abundance of data. In general, A nonlinear partial differential equation can be resolved to generate a soliton in many appli cations. Variational Integrators (VIs) is a numerical technique in which Lagrangian of the system is used in the action integral. VIs discretized the Lagrangian to obtain a discrete Euler Lagrange equation with the help of the Hamiltonian principle of stationary action. VIs are renowned for their capacity to preserve a distinct multi-symplectic structure while demonstrating desirable long-term energy characteristics. Finite-difference scheme (FDS) are a class of numerical techniques utilized by numerical analysis that approximate deriva tives by employing finite differences in order to solve differential equations (DE). During the past two decades, research on nonlinear models has frequently focused on soliton, a vital description in the NLSE analysis. In particular, the establishment of optical solitons has given rise to a theoretical basis for nonlinear optics. In this thesis, Our objective is to study the nonlinear Klein-Gordon model (NLKGM) and the Stochastic Biswas-Milovic equation (SBME) with parabolic law nonlinearity using the VIs by the use of projection technique, forward, backward, and central difference schemes and the Sub-OdE method. Utilizing the projection technique, we study the VIs of our gov erning model, NLKGM; Additionally, we investigate some of the numerical solutions by ix the use of central difference, forward, and backward techniques. The Sub-OdE approach is also utilized to obtain soliton solutions, that include avariety of solutions:three positive solitons, three Jacobian elliptic function solutions(JEFS), bright solitons, dark solitons, pe riodic solitons, rational solitons, and hyperbolic function solutions. Bose-Einstein conden sation, fiber optic sensors, plasma physics, optical communication, and other fields belong to the applications for these solitons.Item Study of Multiple Solitons and Nonlinear Waves(Library Information Services COMSATS University Lahore Campus, 2024-03-14) Saria Khizar; FA22-RMT-055; Dr. Syed Tahir Raza Rizvi; LHR TP 9345Soliton theory is a significant topic in mathematical physics and applied mathematics with its rapid development since the 1960s. Solitons, also known as SWs, are found in the solutions of many kinds of nonlinear evolution equation (NLEE) or nonlinear partial dif ferential equations (NLPDEs). They have several unique properties and can be used to explaina number of significant physical phenomena. A soliton is a distinct type of SW that possesses the unique characteristic of maintaining its shape and velocity unchanged across collisions. The balance between the nonlinear effects and dispersion leads to a soliton so lution. Solitons are used widely in both pure and applied mathematics, particularly in areas such as differential equations, algebraic and differential geometry, Lie groups, and Lie al gebras. The SIR (Susceptible-Infected-Removed) model is a simple mathematical model of epi demic outbreaks. The present thesis surveys new analytical results about the SIR model. The SIR model is one of the most basic models for describing the temporal dynamics of an infectious disease in a population. It compartmentalizes population into one of three categories: those who are Susceptible to the disease, those who are currently Infectious, and those who have Recovered (with immunity). At its most basic level, the SIR model is a set of ODEs that describes the number (or proportion) of people in each compartment over time. This thesis will analyze the SIR model and provide some solutions through the use of exponential, hyperbolic, and trigonometric functions. Our goal in this work is to find lump soliton solution (LS), rogue wave solution (RW), lump with one kink (LSK), peri odic waves solution (PW), interaction solution between lump, periodic and one kink as well as interaction solution between lump, periodic and two kink soliton wave for different types of NLPDEs. Additionally, M type interaction with periodic, M type rational soliton wave solution, periodic cross kink wave solution, breather wave, homoclinic breather wave, Kuznetsov Ma breathers, W shaped soliton, mixed type solutions will be obtained.Item Study of Chirp Solitons and Rogue Wave Solutions(Library Information Services COMSATS University Lahore Campus, 2023-03-13) Muhammad Sohail; LHR TP 8738; Dr. Syed Tahir Raza RizviNonlinear Evolution Equations (NLEEs) are a type of mathematical equation, commonly a partial differential equation (PDE), that describe the evolution of physical phenomena with nonlinear behaviours across time. These equations are critical in many scientific disci plines, including mathematical modelling and soliton theory. The solutions of the NLEEs represent the solitary waves (SW) called solitons. The solitary waves or solitons have a very unique history.Solitons are observed as a dispersion and nonlinearity balance. They are common in many different sectors and have intriguing qualities that make them use ful in a variety of applications. The chirped solitons are a recently popular solitary wave phenomenon. Chirp is used in spread spectrum communications as well as some sonar and radar devices. In this thesis, firstly we will obtain some chirped periodic and solitons wave by using Jaco bian elliptic function (JEF) for for higher order NLSE with anomalous dispersion regime. We also obtain some solitary waves (SW) like dark, bright, kink, hyperbolic, periodic and other solutions for the governing model. The chirp that corresponds to each of these optical solitons is also determined. We will also display the graph of our solutions in different di mensions. Secondly we will study different analytical solutions for Susceptible-Infectious Recovered (SIR) epidemic model with specific nonlinear incidence rate and spatial diffu sion like lump waves (LW), rogue waves (RW), periodic wave (PW) and periodic-cross lump waves (PCLW). This model offers useful information for containment methods by simulating and understanding the geographical spread of infectious illnesses. It advances our knowledge of how the dynamics of an epidemic are influenced by the movement of infected and susceptible people across space