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Symmetrical Sonin kernels in terms of the hypergeometric functions

Yuri Luchko

Abstract

In this paper, we introduce a new class of the kernels of the integral transforms of the Laplace convolution type that we call symmetrical Sonin kernels. For a symmetrical Sonin kernel given in terms of some elementary or special functions, its associated kernel has the same form with possibly different parameter values. Several known and new kernels of this type are derived by means of the Sonin method in the time domain and using the Laplace integral transform in the frequency domain. The new symmetrical Sonin kernels are provided in terms of the Wright function and some extensions of the Horn confluent hypergeometric functions in two variables.

Symmetrical Sonin kernels in terms of the hypergeometric functions

Abstract

In this paper, we introduce a new class of the kernels of the integral transforms of the Laplace convolution type that we call symmetrical Sonin kernels. For a symmetrical Sonin kernel given in terms of some elementary or special functions, its associated kernel has the same form with possibly different parameter values. Several known and new kernels of this type are derived by means of the Sonin method in the time domain and using the Laplace integral transform in the frequency domain. The new symmetrical Sonin kernels are provided in terms of the Wright function and some extensions of the Horn confluent hypergeometric functions in two variables.
Paper Structure (4 sections, 2 theorems, 129 equations)

This paper contains 4 sections, 2 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Definitions and examples
  3. Sonin's kernels in the time domain
  4. Sonin's kernels in the Laplace domain

Key Result

Lemma 3.1

Let $\kappa_1$ and $k_1$ be analytical functions and the conditions be satisfied. The functions and build a pair of the Sonin kernels iff the coefficients $a_n,\ n=0,1,2,\dots$ and $b_n, \ n=0,1,2,\dots$ of the functions $\kappa_1$ and $k_1$ satisfy the following triangular system of equations

Theorems & Definitions (14)

  • Definition 2.1
  • Lemma 3.1
  • Remark 3.1
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Example 4.1
  • ...and 4 more