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Student Projects
​Student Name ​Ridha Waheed
Project Title​ ​Periodic Solutions of Non-Linear Ordinary Differential Equations
Desciption ​The existence of periodic orbits in the solution of a non-linear ordinary differential equation has always been of special interest in the theory of non-linear ordinary differential equation. Periodic orbits are an non-linear phenomenon. They cannot occur in linear systems. Such non-linear systems often appear in different branches of science. If such orbits exist then a lot can be predicted about the nature of the solutions of the problem. Also long term predictions can be made on the basis of such solutions. Hence it is in our interest to know if in a given system such orbits exist or not and if they do how many are there? The aim of our work involves the discussion of such non-linear systems which exhibit periodic orbits. However we shall be only concerned with second-order autonomous ordinary differential equations. We start by defining such systems and then obtain a representation on phase plane to observe the solutions graphically. Then results like Bendixson’s negative criterion are used to check the on-existence of periodic orbits of simply connected regions in plane. Poincare Bendixson theorem is used for closed and bounded regions to establish existence of such orbits along with some other results that hold for particular systems. Finally, some approximate methods are presented related to finding the number of periodic solutions of Lienard equation along with their comparison on the accuracy and fast convergence to the results.

 

Student Name ​Ghazala Nazeer
Project Title​
Abstract​ ​In this thesis, by including the variable cosmological term, we extend the work presented in Charged analogue of Vlasenko-Pronin superdense star by Maurya and Gupta. We have obtained a set of three static spherically symmetric solutions of the Einstein Maxwell field equations. Motivated by Tiwari’s work in paper titled Relativistic electromagnetic mass models with cosmological variable in spherically symmetric anisotropic source, particular values of cosmological term are taken. We have obtained well behaved solutions of the Einstein Maxwell field equations. All the results produced by Maurya and Gupta can be obtained as particular cases of our solutions by choosing cosmological term equals to zero.

 

Student Name ​Sumbul Azeem
Project Title​ Relationship between M-natural convexity and ultramodularity​
Desciption The concepts of M♮-convex functions due to Murota and Shioura are considered as a backbone in Discrete Convex Analysis. On the other hand ultramodular functions is a class of functions that generalizes scalar convexity. Ultramodualr functions arise naturally in some Economic and Statisitcal applications. In our research will shall study the relationship between these two intersting classes of functions​

 

Student Name ​Rabia Aziz
Project Title​ Lie Group method in Geometric integration ​
Desciption​ to understand the behavior of available geometric integrator among the class of lie group methods for the solution of D.Eqs that posses certain invariant like energy conservation in the case of Hamiltonian system and existence of the numerical solution to lie on the same manifold on which exact solution lie.
Objective​ ​One of the aim when building a classical numerical integrator is to minimize the error produced by the method.
​Advantages Geometric integrator are not only good for qualitative analysis But also a good competitors of classical integrator where qualitative analysis is do to measure the error.​
Areas of applications​ lie group methods are applied to solve several system of differential equations including planetary motion , weather prediction D.eqs arises from Quantum mechanics .​

 

Student Name ​Bismah Jamil
Project Title​ The classification of a class of plane symmetric static spacetimes according to their noether symmetries​
Desciption In this dissertation, the symmetry methods have been used to classify a class of plane symmetric static spacetimes according to their Noether symmetries and metrics. The method adopted here provides all those plane symmetric static spacetimes which we obtain during the classification of these spacetimes according to their isometries. The Lie algebra of infinitesimal generators corresponding to each metric has also been discussed here.

 

Student Name ​Rafay Mustafa
Project Title​ Experimental Behavior Of Some Symplectic And G-Symplectic Methods For The Numerical Solution Of Conservative Systems
Desciption Hamiltonian equations define a range of physical systems from planetary motion to harmonic oscillator and many more.  Numerical methods provides an approximate flow of these Hamiltonian equations.
For the long term integration of Hamiltonian systems, one should use numerical methods that preserve the qualitative features of such systems like energy and symplecticity of the flow. In this thesis we investigate the symplecticity and G-symplecticity of numerical methods for Hamiltonian systems. In particular we deal with the symplectic Implicit Runge–Kutta, General Linear methods and Composition method to remove the parasites that arise due to the multinvalue nature of the General Linear methods.