Evolution Equations

Evolution equations can be interpreted as differential laws describing the development of a system.  Beginning with the study of differential equations investigators now look at these in a more abstract setting. Dr. Imran Naeem studies exact solutions of evolution equations using Lie Symmetry methods.  Lie symmetry analysis of differential equations was initiated by the Norwegian mathematician Sophus Lie (1842 – 1899). Today, this area of research is being actively pursued. The Lie approach is a systematic way of unraveling exact solutions of ordinary and partial differential equations. It works for linear as well as nonlinear differential equations. Ordinary and partial differential equations and fractional differential equations are natural tool to model physical phenomena which undergo continuous change with respect to time and space. Especially fractional differential equations have been proved to be a prominent tool to model those phenomena where it is necessary to keep the memory and hereditary properties which are neglected in classical integer order models. Due to precision and accuracy, it is preferred over integer order differential equations in those circumstances where inhomogeneity and uncertainty is the core issue. The exact solutions of these differential equation serve as a criterion for design and analysis of the under lying physical or logical model and as a yardstick for critical evaluation of numerical algorithms. Therefore finding exact solution is one of most fundamental problems in studying differential equations. Dr. Muhammad Ahsan also works in this exciting area from another perspective. His research has been related to a Cauchy problem in a real Hilbert space (Semilinear Evolution equations involving maximal monotone linear operators, and nonlinear Lipschitz monotone operators), and some related perturbed problems. He has derived several results related to existence, uniqueness and higher regularity of the unperturbed and perturbed problems, as well as asymptotic expansions of the perturbed problems. Currently he is working on extending some of the results, and to apply the methods to some physical or geometric problems.