Discrete Fractional Calculus (2015)
2. Discrete Delta Fractional Calculus and Laplace Transforms
2.11. Exercises
2.1. Show that is of exponential order r = 1 iff f is bounded on .
2.2. Prove that if is of exponential order r > 0, then is also of exponential order r for
2.3. Show that if is of exponential order r > 1, then , 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
for
2.6. Prove formula (ii) in Theorem 2.9, that is
for
2.7. Prove formula (ii) in Theorem 2.10, that is
for
2.8. Prove Theorem 2.11.
2.9. For each of the following find y(t) given that
(i)
(ii)
2.10. Use Laplace transforms to solve the following IVPs
(i)
(ii)
(iii)
2.11. Use Laplace transforms to solve the IVP
2.12. Solve each of the following IVPs:
(i)
(ii)
2.13. Solve the following summation equations using Laplace transforms:
(i)
(ii)
(iii)
(iv)
2.14. Use Laplace transforms to solve each of the following:
(i)
(ii)
2.15. Show that
(i)
(ii)
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
(ii)
Use the definition of the fractional difference (Definition 2.29) and part (2.32) to find
2.18. Show that the following hold:
(i)
(ii)
2.19. Verify that (2.12) holds.
2.20. Show that for ,
2.21. Evaluate each of the following using Theorem 2.38 and Theorem 2.40
(i)
(ii)
(iii)
(iv)
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)
(ii)
(iii)
2.25. Use Theorems 2.54 and 2.58 to show that . Evaluate the convolution product 1 ∗ 1 and show directly (do not use the convolution theorem) that
2.26. Assume and p ≠ 0. Using the definition of the convolution product (Definition 2.59), find
2.27. Assume and p ≠ q. Using the definition of the convolution product (Definition 2.59), find
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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