Symmetry analysis is a powerful tool that enables the user to construct exact solutions of a given differential equation in a fairly systematic way. For this reason, the Lie point symmetry groups of most well-known differential equations have been catalogued. It is widely believed that the set of symmetries of an initial-value problem (or boundary-value problem) is a subset of the set of symmet...

Diagonal Padd approximations to the time evolution operator for initial value problems are applied in a novel way to the numerical solution of these problems by explicitly factoring the polynomials of the approximation. A remarkable gain over conventional methods in efficiency and accuracy of solution is obtained.