Fractional Sums and Differences - Discrete Delta Fractional Calculus and Laplace Transforms - Discrete Fractional Calculus

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 $$f: \mathbb{N}_{a} \rightarrow \mathbb{R}$$ be given, then

$$\displaystyle{ \int _{a}^{t}\int _{ a}^{\tau _{1} }\cdots \int _{a}^{\tau _{n-1} }f(\tau _{n})\Delta \tau _{n}\cdots \Delta \tau _{2}\Delta \tau _{1} =\int _{ a}^{t}h_{ n-1}(t,\sigma (s))f(s)\Delta s. }$$

(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

$$\displaystyle\begin{array}{rcl} y(t)& &:=\int _{ a}^{t}\int _{ a}^{\tau _{1} }\cdots \int _{a}^{\tau _{n-1} }\int _{a}^{\tau _{n} }f(\tau _{n+1})\Delta \tau _{n+1}\Delta \tau _{n}\cdots \Delta \tau _{2}\Delta \tau _{1}. {}\\ \end{array}$$

Let $$g(\tau _{n}) =\int _{ a}^{\tau _{n}}f(\tau _{n+1})\Delta \tau _{n+1}$$ , then it follows from the induction assumption that

$$\displaystyle\begin{array}{rcl} y(t)& =& \int _{a}^{t}h_{ n-1}(t,\sigma (s))g(s)\Delta s {}\\ & =& \int _{a}^{t}u(s)\Delta v(s)\Delta s, {}\\ \end{array}$$

where

$$\displaystyle{u(s):= g(s),\quad \Delta v(s) = h_{n-1}(t,\sigma (s)).}$$

It follows (using Theorem 1.61, (v)) that

$$\displaystyle{\Delta u(s) = f(s)\quad v(s) = -h_{n}(t,s),\quad v(\sigma (s)) = -h_{n}(t,\sigma (s)).}$$

Hence, integrating by parts, it follows that

$$\displaystyle\begin{array}{rcl} y(t)& & = -h_{n}(t,s)\int _{a}^{s}f(\tau _{ n+1})\Delta \tau _{n+1}\Big\vert _{a}^{t} {}\\ & & \quad +\int _{ a}^{t}h_{ n}(t,\sigma (s))f(s)\Delta s {}\\ & & =\int _{ a}^{t}h_{ n}(t,\sigma (s))f(s)\Delta s. {}\\ \end{array}$$

This completes the proof. □ 

Motivated by (2.7), we define the n-th integer sum $$\Delta _{a}^{-n}f(t)$$ for positive integers n, by

$$\displaystyle\begin{array}{rcl} \Delta _{a}^{-n}f(t)& =& \int _{ a}^{t}h_{ n-1}(t,\sigma (s))f(s)\Delta s. {}\\ & & {}\\ \end{array}$$

But, since

$$\displaystyle{h_{n-1}(t,\sigma (s)) = 0,\quad s = t - 1,t - 2,\cdots \,,t - n + 1,}$$

we obtain

$$\displaystyle\begin{array}{rcl} \Delta _{a}^{-n}f(t)& =& \int _{ a}^{t-n+1}h_{ n-1}(t,\sigma (s))f(s)\Delta s,{}\end{array}$$

(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

$$\displaystyle{h_{\nu }(t,s) = \frac{(t - s)^{\underline{\nu }}} {\Gamma (\nu +1)},}$$

whenever the right-hand side is well defined.

We can now define the ν-th fractional sum.

Definition 2.25.

Assume $$f: \mathbb{N}_{a} \rightarrow \mathbb{R}$$ and ν > 0. Then the ν-th fractional sum of f (based at a) is defined by

$$\displaystyle\begin{array}{rcl} \Delta _{a}^{-\nu }f(t)& &:=\int _{ a}^{t-\nu +1}h_{\nu -1}(t,\sigma (\tau ))f(\tau )\Delta \tau {}\\ & & =\sum _{ \tau =a}^{t-\nu }h_{\nu -1}(t,\sigma (\tau ))f(\tau ), {}\\ \end{array}$$

for $$t \in \mathbb{N}_{a+\nu }.$$ Note that by our convention on delta integrals (sums) we can extend the domain of $$\Delta _{a}^{-\nu }f$$ to $$\mathbb{N}_{a+\nu -N}$$ , where N is the unique positive integer satisfying N − 1 < ν ≤ N, by noting that

$$\displaystyle{\Delta _{a}^{-\nu }f(t) = 0,\quad t \in \mathbb{N}_{ a+\nu -N}^{a+\nu -1}.}$$

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 $$\Delta _{a}^{\sqrt{3}}y(t)$$ and $$\Delta _{a}^{\pi }y(t)$$ 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 $$f(a),f(a + 1),\cdots \,,f(t-\nu ),$$ where the coefficient of f(t −ν) is one. In particular one can check that $$\Delta _{a}^{-\nu }f(t)$$ has the form

$$\displaystyle\begin{array}{rcl} \Delta _{a}^{-\nu }f(t) = h_{\nu -1}(t,\sigma (a))f(a) + \cdots +\nu f(t -\nu -1) + f(t-\nu ).& &{}\end{array}$$

(2.9)

The following formulas concerning the fractional Taylor monomials generalize the integer version of this theorem (Theorem 1.61).

Theorem 2.27.

Let $$t,s \in \mathbb{N}_{a}$$ . Then

(i)

h ν (t,t) = 0

(ii)

$$\Delta h_{\nu }(t,a) = h_{\nu -1}(t,a);$$

(iii)

$$\Delta _{s}h_{\nu }(t,s) = -h_{\nu -1}(t,\sigma (s));$$

(iv)

$$\int h_{\nu }(t,a)\Delta t = h_{\nu +1}(t,a) + C;$$

(v)

$$\int h_{\nu }(t,\sigma (s))\Delta s = -h_{\nu +1}(t,s) + C,$$

whenever these expressions make sense.

Proof.

To see that (iii) holds, note that

$$\displaystyle\begin{array}{rcl} \Delta _{s}h_{\nu }(t,s)& =& h_{\nu }(t,s + 1) - h_{\nu }(t,s) {}\\ & =& \frac{(t - s - 1)^{\underline{\nu }}} {\Gamma (\nu +1)} -\frac{(t - s)^{\underline{\nu }}} {\Gamma (\nu +1)} {}\\ & =& \frac{\Gamma (t - s)} {\Gamma (t - s-\nu )\Gamma (\nu +1)} - \frac{\Gamma (t - s + 1)} {\Gamma (t - s + 1-\nu )\Gamma (\nu +1)} {}\\ & =& \bigg[(t - s-\nu ) - (t - s)\bigg] \frac{\Gamma (t - s)} {\Gamma (\nu +1)\Gamma (t - s -\nu +1)} {}\\ & =& - \frac{(\nu +1)\Gamma (t - s)} {\Gamma (\nu )\Gamma (t - s -\nu +1)} {}\\ & =& - \frac{\Gamma (t - s)} {\Gamma (\nu )\Gamma (t - s -\nu +1)} {}\\ & =& -\frac{(t -\sigma (s))^{\underline{\nu -1}}} {\Gamma (\nu )} {}\\ & =& -h_{\nu -1}(t,\sigma (s)). {}\\ \end{array}$$

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 $$\Delta _{0}^{-\frac{1} {2} }1.$$

Using Theorem 2.27, part (v), we get

$$\displaystyle\begin{array}{rcl} \Delta _{0}^{-\frac{1} {2} }1& =& \int _{0}^{t+\frac{1} {2} }h_{-\frac{1} {2} }(t,\sigma (s)) \cdot 1\;\Delta s {}\\ & =& -h_{\frac{1} {2} }(t,s)\big\vert _{s=0}^{s=t+\frac{1} {2} } {}\\ & =& -h_{\frac{1} {2} }(t,t + \frac{1} {2}) + h_{\frac{1} {2} }(t,0) {}\\ & =& -\frac{(-\frac{1} {2})^{\underline{\frac{1} {2} }}} {\Gamma (\frac{3} {2})} + \frac{t^{\underline{\frac{1} {2} }}} {\Gamma (\frac{3} {2})} {}\\ & =& \frac{2} {\sqrt{\pi }}\;t^{\underline{\frac{1} {2} }}. {}\\ \end{array}$$

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 $$f: \mathbb{N}_{a} \rightarrow \mathbb{R}$$ and ν > 0. Choose a positive integer N such that N − 1 < ν ≤ N. Then we define the ν-th fractional difference by

$$\displaystyle{\Delta _{a}^{\nu }f(t):= \Delta ^{N}\Delta _{ a}^{-(N-\nu )}f(t),\quad t \in \mathbb{N}_{ a+N-\nu }.}$$

Note that our fractional difference agrees with our prior understanding of whole-order differences—that is, for any $$\nu = N \in \mathbb{N}_{0}$$

$$\displaystyle{ \Delta _{a}^{\nu }f(t):= \Delta ^{N}\Delta _{ a}^{-(N-\nu )}f(t) = \Delta ^{N}\Delta _{ a}^{-0}f(t) = \Delta ^{N}f(t), }$$

(2.10)

for $$t \in \mathbb{N}_{a}$$ . 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 $$\Delta _{a}^{\nu }f(t)$$ depends on the values of f on $$\mathbb{N}_{a+\nu -N}^{t+\nu }$$ . 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 $$\Delta ^{n}f(t)$$ only depends on the values of f at the n + 1 points $$t,t + 1,t + 2,\cdots \,,t + n$$ .

Example 2.31.

Use Definition 2.29 to find $$\Delta _{0}^{\frac{1} {2} }1.$$ Using Example 2.28, we have that

$$\displaystyle\begin{array}{rcl} \Delta _{0}^{\frac{1} {2} }1& =& \Delta \Delta _{0}^{-\frac{1} {2} }1 {}\\ & =& \Delta \frac{2} {\sqrt{\pi }}t^{\underline{\frac{1} {2} }} {}\\ & =& \frac{1} {\sqrt{\pi }}t^{\underline{-\frac{1} {2} }}. {}\\ \end{array}$$

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 $$f: \mathbb{N}_{a+\mu } \times \mathbb{N}_{a} \rightarrow \mathbb{R}$$ . Then

$$\displaystyle\begin{array}{rcl} \Delta \bigg[\int _{a}^{t-\mu +1}f(t,\tau )\Delta \tau \bigg] =\int _{ a}^{t-\mu +1}\Delta _{ t}f(t,\tau )\Delta \tau + f(t + 1,t -\mu +1)& &{}\end{array}$$

(2.11)

and

$$\displaystyle\begin{array}{rcl} \Delta \bigg[\int _{a}^{t-\mu +1}f(t,\tau )\Delta \tau \bigg] =\int _{ a}^{t-\mu +2}\Delta _{ t}f(t,\tau )\Delta \tau + f(t,t -\mu +1)& &{}\end{array}$$

(2.12)

for $$t \in \mathbb{N}_{a+\mu },$$ where the $$\Delta _{t}f(t,s)$$ inside the integral means the difference of f(t,τ) with respect to t.

Proof.

To see that (2.11) holds, note that, for $$t \in \mathbb{N}_{a+\mu }$$ ,

$$\displaystyle\begin{array}{rcl} \Delta \bigg[\int _{a}^{t-\mu +1}f(t,\tau )\Delta \tau \bigg]& =& \int _{ a}^{t-\mu +2}f(t + 1,\tau )\Delta \tau -\int _{ a}^{t-\mu +1}f(t,\tau )\Delta \tau {}\\ & =& \int _{a}^{t-\mu +1}\Delta _{ t}f(t,\tau )\Delta \tau + f(t + 1,t + 1-\mu ). {}\\ \end{array}$$

The proof of (2.12) is Exercise 2.19. □ 

In the next theorem we give a very useful formula for $$\Delta _{a}^{\nu }f(t)$$ . We call this formula the alternate definition of $$\Delta _{a}^{\nu }f(t)$$ (see Holm [123, 124]).

Theorem 2.33.

Let $$f: \mathbb{N}_{a}\rightarrow \mathbb{R}$$ and ν > 0 be given, with N − 1 < ν ≤ N. Then

$$\displaystyle\begin{array}{rcl} \Delta _{a}^{\nu }f(t):= \left \{\begin{array}{l} \int _{a}^{t+\nu +1}h_{ -\nu -1}(t,\sigma (\tau ))f(\tau )\Delta \tau,\quad N - 1 <\nu < N \\ \Delta ^{N}f(t),\quad \quad \nu = N \end{array} \right.& &{}\end{array}$$

(2.13)

for $$t \in \mathbb{N}_{a+N-\nu }.$$

Proof.

First note that if $$\nu = N \in \mathbb{N}_{0},$$ then using (2.10), we have that

$$\displaystyle{ \Delta _{a}^{\nu }f(t) = \Delta ^{N}\Delta _{ a}^{-\left (N-\nu \right )}f(t) = \Delta ^{N}\Delta _{ a}^{-0}f(t) = \Delta ^{N}f(t). }$$

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 $$t \in \mathbb{N}_{a+N-\nu },$$

$$\displaystyle\begin{array}{rcl} \Delta _{a}^{\nu }f(t)& =& \Delta ^{N}\Delta _{ a}^{-\left (N-\nu \right )}f(t) {}\\ & =& \Delta ^{N}\left [\int _{ a}^{t-(N-\nu )+1}h_{ N-\nu -1}(t,\sigma (\tau ))f(\tau )\Delta \tau \right ] {}\\ & =& \Delta ^{N-1} \cdot \Delta \bigg[\int _{ a}^{t-(N-\nu )+1}h_{ N-\nu -1}(t,\sigma (\tau ))f(\tau )\Delta \tau \bigg]. {}\\ \end{array}$$

Using the Leibniz rule (2.12), we get

$$\displaystyle\begin{array}{rcl} \Delta _{a}^{\nu }f(t)& & = \Delta ^{N-1}\Bigg[\int _{ a}^{t-(N-\nu -1)+1}h_{ N-\nu -2}(t,\sigma (\tau ))f(\tau )\Delta \tau {}\\ & & \quad \quad \quad \quad + h_{N-\nu -1}(t,t - (N -\nu -2))f(t - (N -\nu -1))\Bigg] {}\\ & & = \Delta ^{N-1}\left [\int _{ a}^{t-(N-\nu -1)+1}h_{ N-\nu -2}(t,\sigma (\tau ))f(\tau )\Delta \tau \right ]. {}\\ & & {}\\ \end{array}$$

Applying the Leibniz formula (2.12) again we get

$$\displaystyle\begin{array}{rcl} \Delta _{a}^{\nu }f(t)& & = \Delta ^{N-2}\Bigg[\int _{ a}^{t-(N-\nu -2)+1}h_{ N-\nu -3}(t,\sigma (\tau ))f(\tau )\Delta \tau {}\\ & & \quad \quad \quad \quad + h_{N-\nu -2}(t,t - (N -\nu -3))f(t - (N -\nu -2))\Bigg] {}\\ & & = \Delta ^{N-2}\left [\int _{ a}^{t-(N-\nu -2)+1}h_{ N-\nu -3}(t,\sigma (\tau ))f(\tau )\Delta \tau \right ]. {}\\ & & {}\\ \end{array}$$

Repeating these steps N − 2 more times, we find that

$$\displaystyle\begin{array}{rcl} \Delta _{a}^{\nu }f(t)& & = \Delta ^{N-N}\Bigg[\int _{ a}^{t-(N-\nu -N)+1}h_{ N-\nu -N-1}(t,\sigma (\tau ))f(\tau )\Delta \tau {}\\ & & \quad \quad + h_{N-\nu -N}(t,t - (N -\nu -(N + 1))f(t - (N -\nu -N))\Bigg] {}\\ & & =\int _{ a}^{t+\nu +1}h_{ -\nu -1}(t,\sigma (\tau ))f(\tau )\Delta \tau + h_{-\nu }(t,t +\nu +1)f(t+\nu ) {}\\ & & =\int _{ a}^{t+\nu +1}h_{ -\nu -1}(t,\sigma (\tau ))f(\tau )\Delta \tau. {}\\ \end{array}$$

This completes the proof. □ 

Remark 2.34.

By Theorem 2.33 we get for all ν > 0, $$\nu \notin \mathbb{N}_{1}$$ that the formula for $$\Delta _{a}^{\nu }f(t)$$ can be obtained from the formula for $$\Delta _{a}^{-\nu }f(t)$$ in Definition 2.25 by replacing ν by −ν and vice-versa, but the domains are different.

Theorem 2.35 (Existence-Uniqueness Theorem).

Assume $$q,f: \mathbb{N}_{0} \rightarrow \mathbb{R}$$ , ν > 0 and N is a positive integer such that N − 1 < ν ≤ N. Then the initial value problem

$$\displaystyle\begin{array}{rcl} \Delta _{\nu -N}^{\nu }y(t) + q(t)y(t +\nu -N) = f(t),& & t \in \mathbb{N}_{ 0}{}\end{array}$$

(2.14)

$$\displaystyle\begin{array}{rcl} y(\nu -N + i) = A_{i},& & 0 \leq i \leq N - 1,{}\end{array}$$

(2.15)

where A i , 0 ≤ i ≤ N − 1, are given constants, has a unique solution on $$\mathbb{N}_{\nu -N}.$$

Proof.

Note that by Remark 2.26, for each fixed t, $$\Delta _{\nu -N}^{-(N-\nu )}y(t)$$ is a linear combination of $$y(\nu -N),y(\nu -N + 1),\cdots \,,y(t - N+\nu )$$ with the coefficient of $$y(t - N+\nu )$$ being one. Since

$$\displaystyle{\Delta _{\nu -N}^{\nu }y(t) = \Delta ^{N}\Delta _{\nu -N}^{-(N-\nu )}y(t),}$$

we have for each fixed t, $$\Delta _{\nu -N}^{\nu }y(t)$$ is a linear combination of $$y(\nu -N),y(\nu -N + 1)$$ , ⋯ , y(t +ν), where the coefficient of y(t +ν) is one. Now define y(t) on $$\mathbb{N}_{\nu -N}^{\nu -1}$$ by the initial conditions (2.15). Then note that y(t) satisfies the fractional difference equation (2.14) at t = 0 iff

$$\displaystyle\begin{array}{rcl} \Delta _{\nu -N}^{\nu }y(0) + q(0)y(\nu -N) = f(0).& & {}\\ \end{array}$$

But this holds iff

$$\displaystyle{(\cdots \,)y(\nu -N) + (\cdots \,)y(\nu -N + 1) + \cdots + y(\nu ) + q(0)y(\nu -N) = f(0),}$$

which is equivalent to the equation

$$\displaystyle{(\cdots \,)A_{0} + (\cdots \,)A_{1} +\ldots +(\cdots \,)A_{n-1} + y(\nu ) + q(0)A_{0} = f(0).}$$

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 $$\nu -N + i$$ , 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 $$\mathbb{N}_{\nu -N}^{\nu }$$ uniquely determine the value of the solution at ν + 1. An induction argument shows that the solution is uniquely determined on $$\mathbb{N}_{\nu -N}.$$  □ 

Remark 2.36.

We could easily extend Theorem 2.35 to the case when $$f,q: \mathbb{N}_{a} \rightarrow \mathbb{R}$$ instead of the special case a = 0 that we considered in Theorem 2.35. Also, the term $$q(t)y(t +\nu -N)$$ in equation (2.14) could be replaced by $$q(t)y(t +\nu -N + i)$$ for any 0 ≤ i ≤ N − 1. Note that we picked the nice set $$\mathbb{N}_{0}$$ so that the fractional difference equation needs to be satisfied for all $$t \in \mathbb{N}_{0}$$ , but then solutions are defined on the shifted set $$\mathbb{N}_{\nu -N}.$$ By shifting the set on which the fractional difference equation is defined, we can evidently obtain solutions that are defined on the nicer set $$\mathbb{N}_{0}$$ . In this book our convention when considering fractional difference equations is to assume the fractional difference equation is satisfied for $$t \in \mathbb{N}_{a}$$ and the solutions are defined on $$\mathbb{N}_{a+\nu -N}.$$

In a standard manner one gets the following result that follows from Theorem 2.35.

Theorem 2.37.

Assume $$q: \mathbb{N}_{0} \rightarrow \mathbb{R}$$ . Then the homogeneous fractional difference equation

$$\displaystyle{\Delta _{\nu -N}^{\nu }u(t) + q(t)u(t +\nu -N) = 0,\quad t \in \mathbb{N}_{\nu -N}}$$

has N linearly independent solutions u i (t), 1 ≤ i ≤ N, on $$\mathbb{N}_{0}$$ and

$$\displaystyle{u(t) = c_{1}u_{1}(t) + c_{2}u_{2}(t) + \cdots + c_{N}u_{N}(t),}$$

where c 1 ,c 2 ,⋯ ,c N are arbitrary constants, is a general solution of this homogeneous fractional difference equation on $$\mathbb{N}_{0}$$ . Furthermore, if in addition, y p (t) is a particular solution of the nonhomogeneous fractional difference equation ( 2.14) on $$\mathbb{N}_{0}$$ , then

$$\displaystyle{y(t) = c_{1}u_{1}(t) + c_{2}u_{2}(t) + \cdots + c_{N}u_{N}(t) + y_{p}(t),}$$

where c 1 ,c 2 ,⋯ ,c N are arbitrary constants, is a general solution of the nonhomogeneous fractional difference equation ( 2.14).