Discrete Fractional Calculus (2015)
2. Discrete Delta Fractional Calculus and Laplace Transforms
2.3. Fractional Sums and Differences
The following theorem will motivate the definition of the n-th integer sum, which will in turn motivate the definition of the ν-th fractional sum. We will then define the ν-th fractional difference in terms of the ν-th fractional sum.
Theorem 2.23 (Repeated Summation Rule).
Let be given, then
(2.7)
Proof.
We will prove this by induction on n for n ≥ 1. The case n = 1 is trivially true. Assume (2.7) holds for some n ≥ 1. It remains to show that (2.7) then holds when n is replaced by n + 1. To this end, let
Let , then it follows from the induction assumption that
where
It follows (using Theorem 1.61, (v)) that
Hence, integrating by parts, it follows that
This completes the proof. □
Motivated by (2.7), we define the n-th integer sum for positive integers n, by
But, since
we obtain
(2.8)
which we consider the correct form of the n-th integer sum of f(t). Before we use the definition (2.8) of the n-th integer sum to motivate the definition of the ν-th fractional sum, we define the ν-th fractional Taylor monomial as follows.
Definition 2.24.
The ν-th fractional Taylor monomial based at s is defined by
whenever the right-hand side is well defined.
We can now define the ν-th fractional sum.
Definition 2.25.
Assume and ν > 0. Then the ν-th fractional sum of f (based at a) is defined by
for Note that by our convention on delta integrals (sums) we can extend the domain of to , where N is the unique positive integer satisfying N − 1 < ν ≤ N, by noting that
The expression “fractional sum” is actually is misnomer as we define the ν-th fractional sum of a function for any ν > 0. Expressions like and are well defined.
Remark 2.26.
Note that the value of the ν-th fractional sum of f based at a is a linear combination of where the coefficient of f(t −ν) is one. In particular one can check that has the form
(2.9)
The following formulas concerning the fractional Taylor monomials generalize the integer version of this theorem (Theorem 1.61).
Theorem 2.27.
Let . Then
(i)
h ν (t,t) = 0
(ii)
(iii)
(iv)
(v)
whenever these expressions make sense.
Proof.
To see that (iii) holds, note that
The rest of the proof of this theorem is Exercise 2.16. □
Example 2.28.
Using the definition of the fractional sum (Definition 2.25), find
Using Theorem 2.27, part (v), we get
Later we will give a formula (2.16) that also gives us this result.
Next we define the fractional difference in terms of the fractional sum.
Definition 2.29.
Assume and ν > 0. Choose a positive integer N such that N − 1 < ν ≤ N. Then we define the ν-th fractional difference by
Note that our fractional difference agrees with our prior understanding of whole-order differences—that is, for any
(2.10)
for . This is called the Riemann–Liouville definition of the ν-th delta fractional difference.
Remark 2.30.
We will see in the proof of Theorem 2.35 below that the value of the fractional difference depends on the values of f on . This full history nature of the value of the ν-th fractional difference of f is one of the important features of this fractional difference. In contrast if one is studying an n-th order difference equation, the term only depends on the values of f at the n + 1 points .
Example 2.31.
Use Definition 2.29 to find Using Example 2.28, we have that
Later we will give a formula (see (2.22)) that also gives us this result.
The following Leibniz formulas will be very useful.
Lemma 2.32 (Leibniz Formulas).
Assume . Then
(2.11)
and
(2.12)
for where the inside the integral means the difference of f(t,τ) with respect to t.
Proof.
To see that (2.11) holds, note that, for ,
The proof of (2.12) is Exercise 2.19. □
In the next theorem we give a very useful formula for . We call this formula the alternate definition of (see Holm [123, 124]).
Theorem 2.33.
Let and ν > 0 be given, with N − 1 < ν ≤ N. Then
(2.13)
for
Proof.
First note that if then using (2.10), we have that
Now assume N − 1 < ν < N. Our proof of (2.13) will follow from N applications of the Leibniz formula (2.12). To see this we have for
Using the Leibniz rule (2.12), we get
Applying the Leibniz formula (2.12) again we get
Repeating these steps N − 2 more times, we find that
This completes the proof. □
Remark 2.34.
By Theorem 2.33 we get for all ν > 0, that the formula for can be obtained from the formula for in Definition 2.25 by replacing ν by −ν and vice-versa, but the domains are different.
Theorem 2.35 (Existence-Uniqueness Theorem).
Assume , ν > 0 and N is a positive integer such that N − 1 < ν ≤ N. Then the initial value problem
(2.14)
(2.15)
where A i , 0 ≤ i ≤ N − 1, are given constants, has a unique solution on
Proof.
Note that by Remark 2.26, for each fixed t, is a linear combination of with the coefficient of being one. Since
we have for each fixed t, is a linear combination of , ⋯ , y(t +ν), where the coefficient of y(t +ν) is one. Now define y(t) on by the initial conditions (2.15). Then note that y(t) satisfies the fractional difference equation (2.14) at t = 0 iff
But this holds iff
which is equivalent to the equation
Hence if we define y(ν) to be the solution of this last equation, then y(t) satisfies the fractional difference equation at t = 0. Summarizing, we have shown that knowing y(t) at the points , 0 ≤ i ≤ N − 1 uniquely determines what the value of the solution is at the next point ν. Next one uses the fact that the values of y(t) on uniquely determine the value of the solution at ν + 1. An induction argument shows that the solution is uniquely determined on □
Remark 2.36.
We could easily extend Theorem 2.35 to the case when instead of the special case a = 0 that we considered in Theorem 2.35. Also, the term in equation (2.14) could be replaced by for any 0 ≤ i ≤ N − 1. Note that we picked the nice set so that the fractional difference equation needs to be satisfied for all , but then solutions are defined on the shifted set By shifting the set on which the fractional difference equation is defined, we can evidently obtain solutions that are defined on the nicer set . In this book our convention when considering fractional difference equations is to assume the fractional difference equation is satisfied for and the solutions are defined on
In a standard manner one gets the following result that follows from Theorem 2.35.
Theorem 2.37.
Assume . Then the homogeneous fractional difference equation
has N linearly independent solutions u i (t), 1 ≤ i ≤ N, on and
where c 1 ,c 2 ,⋯ ,c N are arbitrary constants, is a general solution of this homogeneous fractional difference equation on . Furthermore, if in addition, y p (t) is a particular solution of the nonhomogeneous fractional difference equation ( 2.14) on , then
where c 1 ,c 2 ,⋯ ,c N are arbitrary constants, is a general solution of the nonhomogeneous fractional difference equation ( 2.14).