Discrete Fractional Calculus (2015)
3. Nabla Fractional Calculus
3.13. Further Properties of the Nabla Laplace Transform
In this section we want to find the Laplace transform of a ν-th order fractional difference of a function, where 0 < ν < 1.
Theorem 3.87.
Assume is of exponential order r > 0 and 0 < ν < 1. Then
for |s − 1| < r.
Proof.
Using Theorems 3.82 and 3.83 we have that
for | s − 1 | < 1. □
Next we state and prove a useful lemma (see Hein et al. [119] for n = 1 and see Ahrendt et al. [3] for general n).
Lemma 3.88 (Shifting Base Lemma).
Given and , we have that
Proof.
Consider
which is what we wanted to prove. □
With this, we are ready to provide the general form of the Laplace transform of a ν-th order fractional difference of a function f, where 0 < ν < 1.
Theorem 3.89.
Given and 0 < ν < 1. Then we have
Proof.
Consider
From this and Lemma 3.88, we have that
Applying Lemma 3.88 again we obtain
which is the desired result. □
The following theorem was proved by Jia Baoguo.
Theorem 3.90.
Let and N − 1 < ν < N be given. Then we have
Proof.
We first calculate
Since
we have
□
Remark 3.91.
When N = 1, Theorem 3.90 becomes the Theorem 3.89. When N = 2, we can get the following Corollary.
Corollary 3.92.
Let and 1 < ν < 2 be given. Then we have