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Difference between Riesz derivative and fractional Laplacian on the proper subset of ℝ

  • Caiyu Jiao , Abdul Khaliq , Changpin Li EMAIL logo and Hexiang Wang
Published/Copyright: November 22, 2021
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 24 Issue 6

Abstract

In general, the Riesz derivative and the fractional Laplacian are equivalent on ℝ. But they generally are not equivalent with each other on any proper subset of ℝ. In this paper, we focus on the difference between them on the proper subset of ℝ.

Key Words and Phrases: Riemann-Liouville derivative; Riesz derivative; fractional Laplacian
MSC 2010: Primary 26A33; Secondary 47B06

Acknowledgements

The work was partially supported by the National Natural Science Foundation of China under Grant nos. 11926319 and 11926336.

References

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6 Appendix

Theorem 6.1

If f(x) ∈ ACm(Ωa−) ∩ L1(Ωa−) then RLD−∞,xαf(x), (m − 1 ≤ α < m, m ∈ Z+) exists almost everywhere on Ωa−.

Proof

For simplicity, we first prove the case of α ∈ (0, 1) for the left-sided Riemann-Liouville derivative. For arbitrary c1 > 0, we have

RLD−∞,xαf(x)=1Γ(1−α)ddx∫x−c1x+∫−∞x−c1f(y)(x−y)αdy. (6.1)

Since f(y) ∈ BV(Ωa−), there exists two bounded monotone increasing functions g1(y) and g2(y) on Ωa− such that

f(y)=g1(y)−g2(y). (6.2)

Then we have

∫x−c1xf(y)(x−y)αdy=∫0c1f(x−y)yαdy=∫0c1g1(x−y)yαdy−∫0c1g2(x−y)yαdy:=G1(x)−G2(x). (6.3)

It is obvious that G1(x) and G2(x) are bounded functions on Ωa−. For arbitrary x1, x2 ∈ Ωa− satisfying x1 < x2, it holds that

G1(x2)−G1(x1)=∫0c1g1(x2−y)−g1(x1−y)yαdy≥0. (6.4)

With similar way for G2(x), we can derive both G1(x) and G2(x) are bounded monotone increasing functions. Thus, we have

F(x)=∫x−c1xf(y)(x−y)αdy∈BV(Ωa−), (6.5)

so that F(x) is differentiable almost everywhere on Ωa−. For the second part, we have

ddx∫−∞x−c1f(y)(x−y)αdy=limΔx→01Δx∫−∞x+Δx−c1f(y)(x+Δx−y)αdy−∫−∞x−c1f(y)(x−y)αdy=limΔx→01Δx∫x−c1x+Δx−c1f(y)(x+Δx−y)αdy−α∫−∞x−c1f(y)(x−y)α+1dy=f(x−c1)limΔx→01Δx∫x−c1x+Δx−c1dy(x+Δx−y)α−α∫−∞x−c1f(y)(x−y)α+1dy=f(x−c1)limΔx→011−α(c1+Δx)1−α−(c1)1−αΔx−α∫−∞x−c1f(y)(x−y)α+1dy=f(x−c1)(c1)α−α∫−∞x−c1f(y)(x−y)α+1dy, (6.6)

where integral mean value theorem and dominated convergence theorem are utilized. Combining (6.5) and (6.6), we obtain RLD−∞,xαf(x) exists almost everywhere on Ωa− for α ∈ (0, 1).

When α ∈ (m − 1, m), since f(x) ∈ ACm(Ωa−), f(x) has continuous derivative up to m − 1 on closed real line ℝ and f(m−1)(x) ∈ AC(Ωa−). Then, we obtain

dm−1dxm−1∫x−c1xf(y)(x−y)α−m+1dy=∫x−c1xfm−1(y)(x−y)α−m+1dy∈BV(Ωa−), (6.7)

where c1 is a positive constant. Thus, 1Γ(m−α)dmdxm∫x−c1xf(y)(x−y)α−m+1dy exists almost everywhere on Ωa−. Also, we have

dmdxm∫−∞x−c1f(y)(x−y)α−m+1dy=dm−1dxm−1f(x−c1)(c1)α−m+1−(α−m+1)∫−∞x−c1f(y)(x−y)α−m+2dy=dm−2dxm−2[f′(x−c1)(c1)α−m+1−(α−m+1)f(x−c1)(c1)α−m+2+(α−m+1)(α−m+2)∫−∞x−c1f(y)(x−y)α−m+3dy]=∑k=0m−1Γ(m−α)Γ(k−α+1)f(k)(x−c1)(c1)α−k+Γ(m−α)Γ(−α)∫−∞x−c1f(y)(x−y)α+1dy. (6.8)

Combining equations (6.7) and (6.8), we get

RLD−∞,xαf(x)=dmdxmRLD−∞,x−(m−α)f(x)=1Γ(m−α){dmdxm∫x−c1xf(y)(x−y)α+1−mdy+∑k=0m−1Γ(m−α)Γ(k−α+1)f(k)(x−c1)(c1)α−k+Γ(m−α)Γ(−α)∫−∞x−c1f(y)(x−y)α+1dy}. (6.9)

By means of the condition of f(x),RLD−∞,xαf(x) exists almost everywhere on Ωa−. For the other cases, we can derive the result similarly. The proof is completed. □
Received: 2021-05-21
Revised: 2021-10-14
Published Online: 2021-11-22
Published in Print: 2021-12-20

© 2021 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 24–6–2021)
  4. Research Paper
  5. Weighted fractional Hardy operators and their commutators on generalized Morrey spaces over quasi-metric measure spaces
  6. B-spline collocation discretizations of caputo and Riemann-Liouville derivatives: A matrix comparison
  7. A strong maximum principle for the fractional laplace equation with mixed boundary condition
  8. Difference between Riesz derivative and fractional Laplacian on the proper subset of ℝ
  9. Some properties of the fractal convolution of functions
  10. Continuous dependence of fuzzy mild solutions on parameters for IVP of fractional fuzzy evolution equations
  11. Discrete fractional boundary value problems and inequalities
  12. On the generalized fractional Laplacian
  13. Recent developments on the realization of fractance device
  14. Explicit representation of discrete fractional resolvent families in Banach spaces
  15. Convergence rate estimates for the kernelized predictor corrector method for fractional order initial value problems
  16. Inverse problems for diffusion equation with fractional Dzherbashian-Nersesian operator
  17. An inverse problem approach to determine possible memory length of fractional differential equations
https://doi.org/10.1515/fca-2021-0074
Keywords for this article
Riemann-Liouville derivative; Riesz derivative; fractional Laplacian

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 24–6–2021)
  4. Research Paper
  5. Weighted fractional Hardy operators and their commutators on generalized Morrey spaces over quasi-metric measure spaces
  6. B-spline collocation discretizations of caputo and Riemann-Liouville derivatives: A matrix comparison
  7. A strong maximum principle for the fractional laplace equation with mixed boundary condition
  8. Difference between Riesz derivative and fractional Laplacian on the proper subset of ℝ
  9. Some properties of the fractal convolution of functions
  10. Continuous dependence of fuzzy mild solutions on parameters for IVP of fractional fuzzy evolution equations
  11. Discrete fractional boundary value problems and inequalities
  12. On the generalized fractional Laplacian
  13. Recent developments on the realization of fractance device
  14. Explicit representation of discrete fractional resolvent families in Banach spaces
  15. Convergence rate estimates for the kernelized predictor corrector method for fractional order initial value problems
  16. Inverse problems for diffusion equation with fractional Dzherbashian-Nersesian operator
  17. An inverse problem approach to determine possible memory length of fractional differential equations
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