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

Francesco A. Costabile; Maria I. Gualtieri; Anna Napoli; Francesco A. Costabile; Maria I. Gualtieri; Anna Napoli

doi:10.3934/math.20251100

AIMS Mathematics

2025, Volume 10, Issue 10: 24836-24856. doi: 10.3934/math.20251100

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

Department of Mathematics and Computer Science, University of Calabria, 87036 Rende (CS), Italy

Received: 10 July 2025 Revised: 21 September 2025 Accepted: 26 September 2025 Published: 30 October 2025
MSC : 11B83, 11C99, 65L10

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.
- differential problem,
- umbral interpolation,
- linear functional,
- delta-operator,
- Euler polynomial
Citation: Francesco A. Costabile, Maria I. Gualtieri, Anna Napoli. Higher-order Umbral Differential Problems for ODEs: Theoretical foundations and computational methods[J]. AIMS Mathematics, 2025, 10(10): 24836-24856. doi: 10.3934/math.20251100

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Abstract

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.

References

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