M.Phil / MS

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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 10063
It 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.
Applications of Extended Physics Informed Neural Network Method for Nonlinear Differential Equations
(Library Information Services COMSATS University Islamabad Lahore Campus, 2025) Muhammad Ismail (FA23-RMT-045); Dr. Syed Tahir Raza Rizvi; LHR TP 9788
Nonlinear partial differential equations featuring nonlinear terms occur in a number of engineering and physical problems, including reaction-diffusion processes, heat conduction, and population dynamics. Classical numerical techniques struggle to efficiently tackle high-dimensional problems and observe intricate nonlinear phenomena. Physics-Informed Neural Networks (PINNs) present a new opportunity in solving PDEs by incorporating physical boundary conditions inside the learning problem. Discontinuities, domain decomposition, and multi-scale phenomena, however, pose a problem to traditional PINNs. Extended PINNs (XPINNs) and conservative PINNs (cPINNs) have been proposed to overcome these drawbacks. In this thesis, we examine the use of PINN, XPINN, and cPINN in solving a reactiondiffusion equation with an exponential nonlinearity. We compare and examine their efficiency, accuracy, and stability in different parameter regimes. We will also examine the dynamics of mean square and absolute error in the given models. We will consider certain specific initial and boundary conditions to train the NN for governing model and will calculate the error of the equations by comparing the result extracted using PINN, cPINN and XPINN with the exact solution. With this, the results depict a sufficiently good agreement in the approximate solution of the governing model, the mean square error of which is of the order of 10−1 to 10−4 in the simulated equations. To the best of our knowledge, we apply this method to this model for the first time and the results are new and novel.

M.Phil / MS

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