Researches Differential Equations

Differential Equations

Differential Equations are mathematical equations that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

The group is further subdivided in subgroups given below:

  1. Symmetry Methods and Differential Equations Group

Several real-world models are formulated in terms of ordinary differential equations (ODEs) and partial differential equations (PDEs). Depending on the nature of the problem being investigated, one might be interested in obtaining numerical or analytical solutions and comparing them with the experimental results. Solving differential equations (DEs) can be quite difficult, though some techniques for ODEs have been developed to determine the solutions analytically. But no general method is available which is applicable to every DE. To address problems of finding the exact solutions, Lie symmetry method could be usefully applied to circumvent difficulties in solving DEs (both linear and nonlinear types). Lie symmetry method is helpful in integrating as well as reducing the order of underlying DE. One Lie symmetry reduces the order of equation by one, two symmetries reduce the order by two, and so on. An other important aspect of a Lie symmetry emerges when the underlying equation possesses a variational form. That is, a Lagrangian function exists that when insert in the Euler-Lagrange (LE) equation yields the given equation. When a Lie symmetry leaves the action integral of that EL-equation invariant than it becomes a Noether symmetry of that DE. So, a Noether symmetry not only provides a double reduction of order but also a conservation law for that DE. This research group is interested in finding Lie and Noether symmetries to study physical aspects as well as to get analytical solutions of underlying dynamical models in various fields, e.g. electrical engineering, mechanical engineering, general relativity, cosmology, etc.

Research Group Members

Name Research Interest Personal Page Profile Picture
Dr M. Umar Farooq Symmetry Methods for Differential Equations,
Dr Asim Aziz Differential equations
Dr Hina Munir Dutt (Associate Member from SEECS- NUST)​ Symmetry analysis, Differential equations
Dr Safia Akram (Associate Member from MCS- NUST)​ Numerical solution of PDE’s
Dr Adnan Aslam​ (Associate Member from SEECS- NUST)​ Symmetry analysis
Dr. Sajid Iqbal (Associate Member from MCS- NUST)​ Non-linear PDE’s
Dr. M. Safdar (Associate Member from SMME- NUST) Symmetry methods for differential equations
  1. Differential Equations and Stochastic Dynamics Group

The stochastic processes are key tools for modelling and reasoning of numerous physical and engineering systems such as neuroscience, bioinformatics, image processing, financial markets, information theory, cryptography, and telecommunications.  Although theory and its applications of stochastic differential equations have made huge progress in recent years, there are still a lot of new and challenging problems existing in the areas of stochastic control, Markov chains, renewal process, and actuarial science etc.

Research Group Members

Name Research Interest Personal Page Profile Picture
Dr.Faizullah Stochastic analysis, Nonlinear Stochastic differential equations
Dr. Safia Akram (Associate Member from MCS- NUST)​ Numerical solution of PDE’s
Dr. Sajid Iqbal Assoc. Professor (Associate Member from MCS- NUST) Non-linear PDE’s
Umer Saeed (Associate Member from SCEE- NUST)​ Numerical and Wavelets methods, Fractional Differential equations