Seminars

This talk shows how to derive all possible analytical functions, y = f(x), subject to n constraints on the function and its derivatives defined at any specified values. These expressions, called “constrained expressions” can be then adopted describe trajectories satisfying specific constraints (e.g., path planning). The talk first shows general explicit function passing through a single point in three distinct ways, linear, rational, and additive. Then, functions with constraints on single, two, and multiple points are introduced. In particular, a generalization of the Waring's interpolation form is derived to obtain expressions passing through a set of points.

The capability of deriving constrained expressions allows to obtain least-squares solutions to initial, boundary, and multivalues problems of nonlinear differential equations of any order. The constrained expressions contain a freely to choose function, g(x), which is then expanded in terms of basis functions (e.g., Chebyshev and Legendre orthogonal polynomials). The procedure leads to a set of equations in terms of the unknown coefficients vector that is then computed by least-squares. Numerical comparisons are then provided quantifying speed and accuracy versus the-state-of-art differential equation solvers.

The Theory of Connection is then applied to obtain all possible surfaces connecting functions and all manifolds connecting surfaces. This extension is then used to solve partial differential equations such as Poisson, Wave, and Heat equations.