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United lattice fractional integro-differentiation

  • Vasily E. Tarasov EMAIL logo
Published/Copyright: June 28, 2016
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 19 Issue 3

Abstract

New fractional operators of non-integer and integer orders for N-dimensional lattice ℤN are suggested. The proposed lattice fractional integro-differentiation of positive order can be considered as a lattice analog of partial derivatives of non-integer orders. For negative values of order, it can be considered as a lattice fractional integration. For integer orders, the continuum limit of the united lattice derivatives gives the usual partial derivatives of integer orders. In contrast to the usual lattice derivatives, which are proposed in our recent paper (“Lattice fractional calculus”, Appl. Math. Comp. 257 (2015), 12–33), the suggested lattice operators give the usual partial derivatives for all integer orders. The united lattice integro-differentiation is represented by finite differences with infinite series that describe long-range lattice interactions. The suggested lattice operators of positive orders can be considered as an exact discretization of derivatives of corresponding orders. The basic mathematical feature of these united lattice operators is an implementation of the same algebraic properties that have the differentiation operator.

MSC 2010: Primary 26A33; Secondary 35R11; 39A12; 39A70
Key Words and Phrases: fractional calculus; Riesz derivative; fractional integral; lattice operator; finite difference; discretization

Dedicated to Professor Stefan G. Samko on the occasion of his 75th anniversary

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Received: 2015-7-25
Published Online: 2016-6-28
Published in Print: 2016-6-1

© 2016 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA-volume 19-3-2016)
  4. Survey Paper
  5. Fractional integrals and derivatives: mapping properties
  6. Research Paper
  7. Riesz fractional integrals in grand lebesgue spaces on ℝn
  8. Survey Paper
  9. United lattice fractional integro-differentiation
  10. Research Paper
  11. Integral equations of fractional order in Lebesgue spaces
  12. Research Paper
  13. General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems
  14. Research Paper
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  16. Research Paper
  17. Fractional integration operator on some radial rays and intertwining for the Dunkl operator
  18. Research Paper
  19. Pseudo almost automorphy of semilinear fractional differential equations in Banach Spaces
  20. Research Paper
  21. Existence and uniqueness of global solutions of caputo-type fractional differential equations
  22. Research Paper
  23. Perfect nonlinear observers of fractional descriptor continuous-time nonlinear systems
https://doi.org/10.1515/fca-2016-0034
Keywords for this article
fractional calculus; Riesz derivative; fractional integral; lattice operator; finite difference; discretization

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA-volume 19-3-2016)
  4. Survey Paper
  5. Fractional integrals and derivatives: mapping properties
  6. Research Paper
  7. Riesz fractional integrals in grand lebesgue spaces on ℝn
  8. Survey Paper
  9. United lattice fractional integro-differentiation
  10. Research Paper
  11. Integral equations of fractional order in Lebesgue spaces
  12. Research Paper
  13. General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems
  14. Research Paper
  15. Multilinear integral operators in weighted grand Lebesgue spaces
  16. Research Paper
  17. Fractional integration operator on some radial rays and intertwining for the Dunkl operator
  18. Research Paper
  19. Pseudo almost automorphy of semilinear fractional differential equations in Banach Spaces
  20. Research Paper
  21. Existence and uniqueness of global solutions of caputo-type fractional differential equations
  22. Research Paper
  23. Perfect nonlinear observers of fractional descriptor continuous-time nonlinear systems
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