Abstract:Knowledge of conservation laws of nonlinear systems of differential equations (DEs) provides essential information for analysis and solution of such equations in particular, it is needed for the application of modern numerical methods. Noether's theorem gives a well-known method of construction of conservation laws of self-adjoint (Lagrangian) systems of differential equations (DEs), through the computation of variational symmetries that preserve the action. For non-self-adjoint systems of DEs, Noether's theorem is not applicable. In this talk, I will discuss the details of the Direct Method of construction of conservation laws, developed originally by Bluman and Anco. The Direct Method is a systematic procedure to obtain conservation law multipliers (integrating factors). It yields conserved (divergence) forms of partial differential equations (PDE), and conserved quantities (first integrals) for ordinary differential equations (ODE). The Direct Method is applicable to virtually any physical DE system, linear or nonlinear. Symbolic implementation of the Direct Method in Maple will be discussed, as well as numerous examples of its applicationsto various physical systems that were considered in my recent research.