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Some Applications of Fractional Velocities

  • Dimiter Prodanov EMAIL logo
Published/Copyright: March 9, 2016
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 19 Issue 1

Abstract

Fractional velocity is defined as the limit of the difference quotient of the increments of a function and its argument raised to a fractional power. The fractional velocity can be suitable for characterizing singular behavior of derivatives of Hölderian functions and non differentiable functions. Relations to integer-order derivatives and other integral-based definitions are discussed.It is demonstrated that for Hölder functions under certain conditions the product rules deviates from the Leibniz rule. This deviation is expressed by another quantity, fractional co-variation.

MSC: Primary 26A27; Secondary 26A33; 26A12; 26A16; 26A30; 35R11; 47G30
key words and phrases: fractional calculus; non-differentiable functions; singular functions; Hölder classes; pseudo differential operators

Acknowledgments

The work has been supported in part by a grant from Research Fund-Flanders (FWO), contract number 0880.212.840.

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Received: 2015-2-11
Revised: 2015-12-1
Published Online: 2016-3-9
Published in Print: 2016-2-1

© 2016 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA Related News, Events and Books (FCAA–Volume 19–1–2016)
  4. Research Paper
  5. Smallest Eigenvalues for a Right Focal Boundary Value Problem
  6. Research Paper
  7. High-Order Algorithms for Riesz Derivative and their Applications (III)
  8. Research Paper
  9. Existence and Uniqueness for a Class of Stochastic Time Fractional Space Pseudo-Differential Equations
  10. Research Paper
  11. Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data
  12. Research Paper
  13. Bogolyubov-Type Theorem with Constraints Generated by a Fractional Control System
  14. Research Paper
  15. Numerical Solution of Nonstationary Problems for a Space-Fractional Diffusion Equation
  16. Research Paper
  17. Solving 3D Time-Fractional Diffusion Equations by High-Performance Parallel Computing
  18. Discussion Survey
  19. Physical and Geometrical Interpretation of Grünwald-Letnikov Differintegrals: Measurement of Path and Acceleration
  20. Discussion Survey
  21. Some Applications of Fractional Velocities
  22. Research Paper
  23. Maximum Principles for Multi-Term Space-Time Variable-Order Fractional Diffusion Equations and their Applications
  24. Survey Paper
  25. Atypical Case of the Dielectric Relaxation Responses and its Fractional Kinetic Equation
  26. Research Paper
  27. Operator Method for Construction of Solutions of Linear Fractional Differential Equations with Constant Coefficients
  28. Research Article
  29. On the Existence and Multiplicity of Solutions for Dirichlet’s problem for Fractional Differential equations
  30. Research Paper
  31. Approximate controllability for semilinear composite fractional relaxation equations
https://doi.org/10.1515/fca-2016-0010
Keywords for this article
fractional calculus; non-differentiable functions; singular functions; Hölder classes; pseudo differential operators

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA Related News, Events and Books (FCAA–Volume 19–1–2016)
  4. Research Paper
  5. Smallest Eigenvalues for a Right Focal Boundary Value Problem
  6. Research Paper
  7. High-Order Algorithms for Riesz Derivative and their Applications (III)
  8. Research Paper
  9. Existence and Uniqueness for a Class of Stochastic Time Fractional Space Pseudo-Differential Equations
  10. Research Paper
  11. Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data
  12. Research Paper
  13. Bogolyubov-Type Theorem with Constraints Generated by a Fractional Control System
  14. Research Paper
  15. Numerical Solution of Nonstationary Problems for a Space-Fractional Diffusion Equation
  16. Research Paper
  17. Solving 3D Time-Fractional Diffusion Equations by High-Performance Parallel Computing
  18. Discussion Survey
  19. Physical and Geometrical Interpretation of Grünwald-Letnikov Differintegrals: Measurement of Path and Acceleration
  20. Discussion Survey
  21. Some Applications of Fractional Velocities
  22. Research Paper
  23. Maximum Principles for Multi-Term Space-Time Variable-Order Fractional Diffusion Equations and their Applications
  24. Survey Paper
  25. Atypical Case of the Dielectric Relaxation Responses and its Fractional Kinetic Equation
  26. Research Paper
  27. Operator Method for Construction of Solutions of Linear Fractional Differential Equations with Constant Coefficients
  28. Research Article
  29. On the Existence and Multiplicity of Solutions for Dirichlet’s problem for Fractional Differential equations
  30. Research Paper
  31. Approximate controllability for semilinear composite fractional relaxation equations
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