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Time Scales Analysis

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Time Scales Analysis

Time scale theory was first initiated by Stefan Hilger in 1988 in his PhD thesis to unify both approaches of dynamic modelling: difference and differential equations. Similar ideas have been used before and go back in the introduction of the Riemann-Stieltjes integral which unifies sums and integrals. Many results to differential equations carry over easily to corresponding results for difference equations, while other results seem to be totally different in nature. Because of these reasons, tim

Mathematics
Author
Svetlin G. Georgiev
Publisher
Sorbonne University
Source
LibreTexts
License
CC BY-NC-SA
Language
EN
Category
Basic Sciences
Read online Download PDF (open licence)
Contents· 39 sections
  1. 01: Time Scales Analysis PDF
  2. 02: Definition. Examples
  3. 03: Forward Jump Operators Backward Jump Operators and Graininess Functions
  4. 04: A Classification of Points
  5. 05: The Topology of Time Scales
  6. 06: Functions and Jump Operators
  7. 07: The Induction Principle
  8. 08: Differentiable Functions of One Variable
  9. 09: Higher Order Delta Differentiation
  10. 10: Nabla Derivatives
  11. 11: Regulated rd-Continuous and Pre-Differentiable Functions
  12. 12: The Delta Integral
  13. 13: Improper Integrals of the First Kind
  14. 14: Improper Integrals of the Second Kind
  15. 15: Sequences of Functions
  16. 16: Series of Functions
  17. 17: Exam
  18. 18: Homeworks
  19. 19: A Project
  20. 20: A Supplementary Material- Matrix Exponential Functions on Time Scales
  21. 20.01: Regressive Matrix-Valued Functions
  22. 20.02: Definition for a Matrix Exponential Function. Properties
  23. 20.03: The Putzer Algorithm on Time Scales
  24. 21: A Supplementary Material- Generalized Zeros
  25. 21.01: Definition. Properties
  26. 21.02: The Rolle Theorem
  27. 21.03: The Knesser Theorem
  28. 22: A Supplementary Material- Linear Integral Inequalities
  29. 22.01: Gronwall-Type Integral Inequalities
  30. 22.02: Pachpatte Integral Inequalities
  31. 22.03: Dragomir-Type Integral Inequalities
  32. 23: A Supplementary Material- Diamond Alpha Calculus on Time Scales
  33. 23.01: Diamond Alpha Differential Calculus
  34. 23.02: Diamond Alpha Integral Calculus
  35. 23.03: Some Steffensen Inequalities
  36. 24: A Supplementary Material- Inequalities for Convex Functions
  37. 24.01: Inequalities for Exponentially Convex Functions
  38. 24.02: Inequalities for Strongly r-Convex Functions
  39. 24.03: Inequalities for Exponentially s-Convex Functions