May 21 2021 12:00 pm
Approximate Noether-type Symmetries, Conservation Laws and Exact Solutions of Some Perturbed Differential Equations
Dr. Imran Naeem
Zoom Meetings (Online)
MS Thesis defense
The approximate conservation laws for (1+1)-Timoshenko Prescott equation are derived via approximate Noether-type Symmetry approach associated with the Standard Lagrangian with small parameters. The partial Noether approach is employed to compute the partial Noether operators and the associated conservation laws for perturbed partial differential equations. These approximate Noether-type symmetry operators do not form a Lie algebra in general. We find approximate conserved vectors for mixed derivatives, and it is observed that some extras terms appear in the approximate conservation law. Taking differentials and adding them to the conserved flows, satisfy the divergence relation and some new conservation laws arise. The classical Lie point symmetries are computed for the Timoshenko Prescott equation and the symmetry conservation laws relation is employed which shows that which symmetry is associated with the approximate conservation law. The double reduction theory is used to reduce the number of independent variables and the order of the Timoshenko Prescott equation via similarity variables. The inverse similarity transformations give rise to exact solutions of the Timoshenko Prescott equation. In addition to the above, different combinations of Lie symmetries are used to derive several independent solutions. The analysis will be extended to a class of perturbed partial differential equations involving arbitrary functions. This study points out new ways of finding exact solutions as well as the reduction in the number of independent variables for perturbed partial differential equations that arise in different areas of sciences, business, economics, etc.