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On the fractional relaxation equation with Scarpi derivative

Matija Adam Horvat, Nikola Sarajlija

Abstract

In this article we solve the Cauchy problem for the relaxation equation posed in a framework of variable order fractional calculus. Thus, we solve the relaxation equation in, what seems to be, the most general case. After introducing some general mathematical theory we establish concepts of Scarpi derivative and transition functions which make essentials of our problem. Next, we completely solve our initial value problem for an arbitrary transition function, and we calculate the solution in the case of an exponential-type transition function, as well as in the case of a Mittag-Leffler transition.

On the fractional relaxation equation with Scarpi derivative

Abstract

In this article we solve the Cauchy problem for the relaxation equation posed in a framework of variable order fractional calculus. Thus, we solve the relaxation equation in, what seems to be, the most general case. After introducing some general mathematical theory we establish concepts of Scarpi derivative and transition functions which make essentials of our problem. Next, we completely solve our initial value problem for an arbitrary transition function, and we calculate the solution in the case of an exponential-type transition function, as well as in the case of a Mittag-Leffler transition.

Paper Structure

This paper contains 8 sections, 4 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mathematical prerequisites
  3. Transition function $\alpha(t)$
  4. Definition of Scarpi's operators
  5. Existence and uniqueness of a solution
  6. Special cases for transition function $\alpha(t)$
  7. Exponential transition
  8. Order transition of Mittag-Leffler type

Key Result

Theorem 2.1

For function $f$ as described above the following is true: provided that $f'$ has an integrable singularity at $t=0$. Suppose that every pole of $F(s)$ is either in the open left half plane or at the origin, and that $F(s)$ has at most a single pole at the origin. Then

Figures (1)

  • Figure 1: Contour of integration $\Gamma$

Theorems & Definitions (5)

  • Theorem 2.1
  • Definition 2.2
  • Theorem 3.1
  • Theorem 4.1
  • Theorem 4.2