Narrow Results

Variational Integrator (VI) is a numerical technique, in which the Lagrangian of the system is used as the action integral. It is a special type of numerical solution that preserves the energy and momentum of the system. In this paper, we retrieve numerical solutions for heat and wave equation with the help of all possible combinations of finite difference scheme like forward–forward, forward–backward, forward–centered, backward–forward, backward–backward, backward–centered, centered–forward, centered–backward, centered–centered. We also use Lagrangian approach along with the projection technique to obtain approximate solutions of these linear models. This approach provides the best approximate solutions as well as preserves the energy of the system while the finite difference scheme gives only the numerical solutions. We also draw a comparison of existing exact solution with all approximate solutions for both models and provide graphical representation of these solutions.

Several applications of Lie symmetries and its generalisation are presented: from turning butterflies into tornados, to its applications in epidemics, population dynamics, and ultimately converting classical problems into the quantum realm. Applications of nonclassical symmetries are also illustrated.