Introduction - Lesson 2 - Discrete Delta Fractional Calculus and Laplace Transforms - Discrete Fractional Calculus

Discrete Fractional Calculus (2015)

2. Discrete Delta Fractional Calculus and Laplace Transforms

Christopher Goodrich1 and Allan C. Peterson2

(1)

Department of Mathematics, Creighton Preparatory School, Omaha, NE, USA

(2)

Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE, USA

2.1. Introduction

At the outset of this chapter we will be concerned with the (delta) Laplace transform, which is a special case of the Laplace transform studied in the book by Bohner and Peterson [62]. We will not assume the reader has any knowledge of the material in that book. The delta Laplace transform is equivalent under a transformation to the Z-transform, but we prefer the definition of the Laplace transform given here, which has the property that many of the Laplace transform formulas will be analogous to the Laplace transform formulas in the continuous setting. We will show how we can use the (delta) Laplace transform to solve initial value problems for difference equations and to solve summation equations. We then develop the discrete delta fractional calculus. Finally, we apply the Laplace transform method to solve fractional initial value problems and fractional summation equations.

The continuous fractional calculus has been well developed (see the books by Miller and Ross [147], Oldham and Spanier [152], and Podlubny [153]). But only recently has there been a great deal of interest in the discrete fractional calculus (see the papers by Atici and Eloe [3236], Goodrich [8896], Miller and Ross [146], and M. Holm [123125]). More specifically, the discrete delta fractional calculus has been recently studied by a variety of authors such as Atici and Eloe [31, 32, 34, 35], Goodrich [88, 89, 91, 92, 94, 95], Miller and Ross [147], and M. Holm [123125]. As we shall see in this chapter, one of the peculiarities of the delta fractional difference is its domain shifting properties. This property makes, in certain ways, the study of the delta fractional difference more complicated than its nabla counterpart, as a comparison of the present chapter to Chap. 3 will demonstrate.