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

Discrete Fractional Calculus (2015)

2. Discrete Delta Fractional Calculus and Laplace Transforms

2.11. Exercises

2.1. Show that $$f: \mathbb{N}_{a} \rightarrow \mathbb{R}$$ is of exponential order r = 1 iff f is bounded on $$\mathbb{N}_{a}$$ .

2.2. Prove that if $$f: \mathbb{N}_{a} \rightarrow \mathbb{R}$$ is of exponential order r > 0, then $$\Delta ^{n}f: \mathbb{N}_{a} \rightarrow \mathbb{R}$$ is also of exponential order r for $$n \in \mathbb{N}_{0}.$$

2.3. Show that if $$f: \mathbb{N}_{a} \rightarrow \mathbb{R}$$ is of exponential order r > 1, then $$h(t):=\int _{ a}^{t}$$ $$f(\tau )\Delta \tau$$ , $$t \in \mathbb{N}_{a}$$ is also of exponential order r.

2.4. Show that h 0(t, a) is of exponential order 1 and for each n ≥ 0, h n (t, a) is of exponential order 1 +ε for all ε > 0.

2.5. Prove formula (i) in Theorem 2.8, that is

$$\displaystyle{\mathcal{L}_{a}\{\cosh _{p}(t,a)\}(s) = \frac{s} {s^{2} - p^{2}}}$$

for $$\vert s + 1\vert >\max \{ \vert 1 + p\vert,\vert 1 - p\vert \}.$$

2.6. Prove formula (ii) in Theorem 2.9, that is

$$\displaystyle{\mathcal{L}_{a}\{\sin _{p}(t,a)\}(s) = \frac{p} {s^{2} + p^{2}}}$$

for $$\vert s + 1\vert >\max \{ \vert 1 + ip\vert,\vert 1 - ip\vert \}.$$

2.7. Prove formula (ii) in Theorem 2.10, that is

$$\displaystyle{\mathcal{L}_{a}\{e_{\alpha }(t,a)\sinh _{ \frac{\beta }{ 1+\alpha } }(t,a)\}(s) = \frac{\beta } {(s-\alpha )^{2} -\beta ^{2}},}$$

for $$\vert s + 1\vert >\max \{ \vert 1 +\alpha +\beta \vert,\vert 1 +\alpha -\beta \vert \}.$$

2.8. Prove Theorem 2.11.

2.9. For each of the following find y(t) given that

(i)

$$Y _{a}(s) = \frac{14-s} {s^{2}+2s-8};$$

(ii)

$$Y _{0}(s) = \frac{2s^{2}} {s^{2}-\sqrt{2}s+1}.$$

2.10. Use Laplace transforms to solve the following IVPs

(i)

$$\displaystyle\begin{array}{rcl} & & y(t + 2) - 7y(t + 1) + 12y(t) = 0,\quad t \in \mathbb{N}_{0}; {}\\ & & y(0) = 2,\quad y(1) = 4. {}\\ \end{array}$$

(ii)

$$\displaystyle\begin{array}{rcl} & & y(t + 1) - 2y(t) = 3^{t},\quad t \in \mathbb{N}_{ 0}; {}\\ & & y(0) = 5. {}\\ \end{array}$$

(iii)

$$\displaystyle\begin{array}{rcl} & & y(t + 2) - 6y(t + 1) + 8y(t) = 20(4)^{t},\quad t \in \mathbb{N}_{ 0} {}\\ & & y(0) = 0,\quad y(1) = 4. {}\\ \end{array}$$

2.11. Use Laplace transforms to solve the IVP

$$\displaystyle\begin{array}{rcl} & & \quad u(t + 1) + v(t) = 0 {}\\ & & -u(t) + v(t + 1) = 0 {}\\ & & \quad u(0) = 1,\quad v(0) = 0. {}\\ \end{array}$$

2.12. Solve each of the following IVPs:

(i)

$$\displaystyle\begin{array}{rcl} & & \Delta y(t) - 2y(t) =\delta _{4}(t),\quad t \in \mathbb{N}_{0}; {}\\ & & y(0) = 2, {}\\ \end{array}$$

(ii)

$$\displaystyle\begin{array}{rcl} & & \Delta y(t) - 5y(t) = 3u_{60}(t),\quad t \in \mathbb{N}_{0} {}\\ & & y(0) = 4,\quad t \in \mathbb{N}_{0}. {}\\ \end{array}$$

2.13. Solve the following summation equations using Laplace transforms:

(i)

$$y(t) = 2 + 4\sum _{r=0}^{t-1}3^{t-r-1}y(r),\quad t \in \mathbb{N}_{0};$$

(ii)

$$y(t) = 3 \cdot 5^{t} - 4\sum _{r=0}^{t-1}5^{t-r-1}y(r),\quad t \in \mathbb{N}_{0};$$

(iii)

$$y(t) = t +\sum _{ r=0}^{t-1}y(r),\quad t \in \mathbb{N}_{0};$$

(iv)

$$y(t) = 2^{t-a} +\sum _{ r=a}^{t-1}4^{t-r-1}y(r),\quad t \in \mathbb{N}_{a}.$$

2.14. Use Laplace transforms to solve each of the following:

(i)

$$y(t) = 3^{t} +\sum _{ m=0}^{t-1}3^{k-m-1}y_{m},\quad t \in \mathbb{N}_{0};$$

(ii)

$$y(t) = 3^{t} +\sum _{ m=0}^{t-1}4^{k-m-1}y_{m},\quad t \in \mathbb{N}_{0}.$$

2.15. Show that

(i)

$$\Delta _{a}^{-\nu }f(a+\nu ) = f(a);$$

(ii)

$$\Delta _{a}^{-\nu }f(a +\nu +1) =\nu f(a) + f(a + 1).$$

2.16. Complete the proof of Theorem 2.27.

2.17. Work each of the following:

(i)

Use the definition of the ν-th fractional sum (Definition 2.25) to find $$\Delta _{a}^{-\frac{1} {3} }1;$$

(ii)

Use the definition of the fractional difference (Definition 2.29) and part (2.32) to find $$\Delta _{a}^{\frac{2} {3} }1.$$

2.18. Show that the following hold:

(i)

$$\Delta _{a+\mu }^{-\nu }(t - a)^{\underline{\mu }} =\mu ^{\underline{-\nu }}(t - a)^{\underline{\mu +\nu }},\quad t \in \mathbb{N}_{a+\mu +\nu };$$

(ii)

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

2.19. Verify that (2.12) holds.

2.20. Show that $$h_{\mu }(t,t -\mu +k) = 0$$ for $$k \in \mathbb{N}_{1}$$ , $$\mu -k + 1\notin \{0,-1,-2,\cdots \,\}.$$

2.21. Evaluate each of the following using Theorem 2.38 and Theorem 2.40

(i)

$$\Delta _{\frac{3} {2} }^{-1}(t - 1)^{\underline{\frac{1} {2} }},\quad t \in \mathbb{N}_{\frac{5} {2} };$$

(ii)

$$\Delta _{4}^{-.7}(t - 1.7)^{\underline{2.3}},\quad t \in \mathbb{N}_{4.7};$$

(iii)

$$\Delta _{5.5}^{.5}(t - 3)^{\underline{2.5}},\quad t \in \mathbb{N}_{5};$$

(iv)

$$\Delta _{3}^{\frac{1} {2} }t(t - 1)(t - 2),\quad t \in \mathbb{N}_{\frac{5} {2} }.$$

2.22. Prove that part (ii) of Theorem 2.42, follows from Theorem 2.40.

2.23. Prove (2.26).

2.24. Solve each of the following IVPs:

(i)

$$\Delta _{-0.3}^{2.7}x(t) = t^{\underline{2}},\quad t \in \mathbb{N}_{0}x(-0.3) = x(0.7) = x(1.7) = 0;$$

(ii)

$$\Delta _{-0.4}^{1.6}x(t) = t^{\underline{4}},\quad t \in \mathbb{N}_{0}x(-0.4) = x(0.6) = 0;$$

(iii)

$$\Delta _{-0.1}^{0.9}x(t) = t^{\underline{5}},\quad t \in \mathbb{N}_{0}x(-0.1) = 0.$$

2.25. Use Theorems 2.54 and 2.58 to show that $$\mathcal{L}_{a}\{h_{1}(t,a)\} = \frac{1} {s^{2}}$$ . Evaluate the convolution product 1 ∗ 1 and show directly (do not use the convolution theorem) that $$\mathcal{L}_{a}\{1 {\ast} 1\}(s) = \mathcal{L}_{a}\{1\}(s)$$   $$\mathcal{L}_{a}\{1\}(s).$$

2.26. Assume $$p \in \mathcal{R}$$ and p ≠ 0. Using the definition of the convolution product (Definition 2.59), find

$$\displaystyle{[h_{1}(t,a) {\ast} e_{p}(t,a)](t).}$$

2.27. Assume $$p,q \in \mathcal{R}$$ and p ≠ q. Using the definition of the convolution product (Definition 2.59), find

$$\displaystyle{[e_{p}(t,a) {\ast} e_{q}(t,a)](t).}$$

2.28. For N a positive integer, use the definition of the Laplace transform to prove that (2.4) holds (that is, (2.52) holds when ν = N).

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