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General Airy-type equations, heat-type equations and pseudo-processes

Fabrizio Cinque, Enzo Orsingher

Abstract

We present a systematic study of higher-order Airy-type differential equations providing the explicit form of the solutions, deriving their power series expansions and a probabilistic interpretation. Under suitable convergence hypotheses, we compute their integral on the real line and, by means of complex integration, we provide alternative explicit forms. We then focus on the differential equations governing their derivatives, their products, their convolutions and higher-order Scorer type equations. Then, we study higher-order heat-type fractional Cauchy, showing that their fundamental solutions can be expressed in terms of Airy-type functions and their convolutions, recovering as special cases several results of appeared in previous papers. Furthermore, pseudo-processes theory permits us to give nice interpretations of the results, extending them in the case of equations involving different fractional operators and study the moments of the solutions.

General Airy-type equations, heat-type equations and pseudo-processes

Abstract

We present a systematic study of higher-order Airy-type differential equations providing the explicit form of the solutions, deriving their power series expansions and a probabilistic interpretation. Under suitable convergence hypotheses, we compute their integral on the real line and, by means of complex integration, we provide alternative explicit forms. We then focus on the differential equations governing their derivatives, their products, their convolutions and higher-order Scorer type equations. Then, we study higher-order heat-type fractional Cauchy, showing that their fundamental solutions can be expressed in terms of Airy-type functions and their convolutions, recovering as special cases several results of appeared in previous papers. Furthermore, pseudo-processes theory permits us to give nice interpretations of the results, extending them in the case of equations involving different fractional operators and study the moments of the solutions.

Paper Structure

This paper contains 3 sections, 6 theorems, 100 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Generalized ordinary Airy-type differential equations
  3. Heat-type equations and pseudo-processes

Key Result

Theorem 2.1

Let $n\ge2$ be an integer number. The ordinary differential equation admits the following $n-1$ linearly independent solutions:

Figures (3)

  • Figure 1: Curves of the complex plane used to build the solutions to equation (\ref{['equazioneDifferenzialeGenerale']}) with $n=5$.
  • Figure 2: Curves of the complex plane used to build the solutions to equation (\ref{['equazioneDifferenzialeGenerale']}) with $n=6$ and $c>0$.
  • Figure 3: Example of the contours of integration.

Theorems & Definitions (22)

  • Theorem 2.1
  • proof
  • Remark 2.1: Complex constant $c$
  • Example 2.1
  • Remark 2.2
  • Example 2.2
  • Remark 2.3
  • Corollary 2.1: Power series of the solutions
  • proof
  • Remark 2.4: Differential equation for the derivatives
  • ...and 12 more