Higher-order Umbral Differential Problems for ODEs: Theoretical foundations and computational methods

In this paper, we consider the higher-order Umbral Differential Problem (UDP). It consists of a nonlinear ordinary differential equation of order $ r+1 $, $ r > 0 $, associated with an Umbral Interpolation Problem (UIP) with $ r+1 $ conditions. We prove the existence and uniqueness of the solution to the UIP and study the error using Peano's Kernel. This class includes, for example, the classical Initial Value Problem (or Cauchy Problem) and a more recent case, the so-called Bernoulli boundary problem, also referred to as a non-classical boundary problem. We then use a Birkoff-Lagrange collocation method for obtaining numerical solutions. Two new examples of UDPs are presented and analyzed. The first is the Euler Umbral Differential Problem, so named because the solution to the associated UIP is expressed in terms of Euler polynomials. The second example is a higher-order problem whose interpolation conditions are expressed using iterated forward finite differences. These conditions are equivalent to multipoint conditions; therefore, the related UDP is called Multipoint Value Problem. Finally, we present some numerical tests. The results confirm the effectiveness of the proposed method. Conclusions, along with directions for future research, are provided.