Source-linked AI summary
New Approach To A Generalized Fractional Integral
Udita N. Katugampola
TL;DR
The paper addresses how to place the Riemann-Liouville and Hadamard fractional integrals within one generalized framework. It derives the generalized operator, establishes boundedness and a semigroup property, and defines generalized fractional derivatives, while leaving several transform and derivative investigations for future work.
Problem
The paper seeks a single fractional integration form that generalizes the Riemann-Liouville and Hadamard fractional integrals.
Method
It derives a generalized fractional integration operator parameterized by ρ and studies its action in an extended Lebesgue measurable space.
Results
The paper establishes boundedness conditions for the generalized operator, proves a semigroup property, and gives a general definition of fractional derivatives.
Takeaways & Limitations
The generalized framework contains the Riemann-Liouville and Hadamard integrals as special cases and extends the paper's treatment to boundedness, semigroup properties, and fractional derivatives.
Takeaways & Limitations
Laplace, Fourier, and Mellin transform formulas and further generalized-derivative properties are left for future work.
Abstract
from arXiv · showhide
The paper presents a new formula for the fractional integration, which generalizes the Riemann-Liouville and Hadamard fractional integrals into a single form, which when a parameter fixed at different values, produces the above integrals as special cases. Conditions are given for such a generalized fractional integration operator to be bounded in an extended Lebesgue measurable space. Semigroup property for the above operator is also proved. Finally, we give a general definition of the Fractional derivatives.
1. Introduction
Fractional calculus has developed multiple fractional derivatives and integrals, including the extensively studied Riemann-Liouville and Hadamard forms. The paper introduces a generalized integration approach and states goals of establishing boundedness, semigroup properties, and generalized fractional derivatives.
- Fractional calculus includes several derivative forms, including Riemann-Liouville, Hadamard, Grunwald-Letnikov, Riesz, and Caputo derivatives.
- Fractional derivatives are defined through fractional integrals, with Riemann-Liouville and Hadamard integrals identified as extensively studied forms.
- The proposed fractional integration generalizes the Riemann-Liouville and Hadamard integrals into a single form.
- The paper seeks conditions under which the generalized integration operator is bounded in an extended Lebesgue measurable space.
- It also establishes a semigroup property for the generalized integration operator and gives a general definition of fractional derivatives.
2. Generalization of the fractional integration
The paper derives a generalized fractional integral from an n-fold integral and studies its relation to established fractional integrals. Setting ρ to specific values recovers the Riemann-Liouville and Hadamard forms.
- The operator acts on complex-valued Lebesgue measurable functions in the extended space Xp_c(a,b), which coincides with classical Lp(a,b) in a stated case.
- The generalized integral is developed from an n-fold integral for natural n, real ρ, and a ≥ 0.
- When ρ = 0, the generalized expression becomes the standard Riemann-Liouville fractional integral.
- Taking ρ → −1+ yields the Hadamard fractional integral.
- The resulting framework connects to studied topics including composition and semigroup properties, Mellin transforms, integration by parts, and G-transform representations.
3. Boundedness in the space Xp
The generalized fractional integration operator is shown to be well-defined and bounded in the extended space Xp under conditions on α, p, ρ, and c, with a corresponding Lp(a,b) result.
- Boundedness in Xp: Theorem 3.1 establishes boundedness of the generalized fractional integration operator in Xp when α > 0, 1 ≤ p ≤ ∞, and ρ ≥ c.The proof treats finite p using the generalized Minkowski inequality and p = ∞ separately.
- Relation to prior work: The boundedness proof follows the approach of Theorem 2.1 in reference, while the ρ → −1+ case had previously been established there.The paper distinguishes its general parameter regime from the earlier special case.
- Boundedness in Xp: The operator is also bounded in Xp for the case ρ = c, with an explicit bound involving the incomplete gamma-function when ρ > c.The bound is stated separately for ρ = c and ρ > c.
- Boundedness in Lp(a,b): Taking c = 1/p yields boundedness of the generalized operator in Lp(a,b) whenever ρ ≥ 1/p.This is presented as Corollary 3.4 and identified with an earlier special-case corollary.
4. Semigroup property
The generalized fractional integration operator satisfies a semigroup property under the stated positivity, integrability, interval, and parameter conditions. The paper also extends the construction to complex orders and right-sided integrals.
- Semigroup property on Xp: For α > 0, β > 0, 1 ≤ p ≤ ∞, 0 < a < b < ∞, and ρ ≥ c, the semigroup property holds on Xp_c(a,b).The proof uses Fubini's theorem, a change of variables, and a beta-function identity before extending the result by boundedness.
- Proof strategy: The semigroup result applies to sufficiently good functions first and then to the full space through boundedness of the operators.The operator boundedness condition ρ ≥ c is used to complete the theorem.
- Semigroup property on Lp: When ρ ≥ 1/p, the semigroup property holds for f ∈ Lp(a,b).This is stated as Corollary 4.2 for the Lp setting.
- Extensions: The generalized integral is extended to arbitrary complex order α ∈ C and to right-sided generalized fractional integrals.Left-sided integrals are defined for x > a and right-sided integrals for x < b when Re(α) > 0.
- Extensions: The framework is used to define generalized fractional derivatives of Riemann-Liouville type and, similarly, Caputo type.The derivative definitions are intended for arbitrary complex order with Re(α) > 0.
5. Generalized fractional derivatives
The paper defines generalized Riemann-Liouville and Caputo-type fractional derivatives from the generalized integrals, including complex orders and left- and right-sided forms. Power-function examples show that derivative characteristics vary with ρ, while transform formulas and further derivative properties remain future work.
- Derivative definitions: The generalized integrals define corresponding Riemann-Liouville-type derivatives, with Hadamard-type derivatives recovered as ρ → −1+ and Caputo-type derivatives defined similarly.The derivative framework covers left- and right-sided forms and arbitrary complex orders with Re(α) > 0.
- Caputo-type derivatives: The generalized Caputo-type derivative is defined for α ∈ C with Re(α) > 0 and n = ⌈α⌉, provided the right-hand sides exist.The definition includes separate left- and right-sided cases.
- Power-function example: For f(x) = x^ν, the generalized derivative is evaluated using a substitution and the beta function; at ρ = 0 it becomes the Riemann-Liouville derivative.The power-function calculation is presented for 0 < α < 1 and a = 0, with ρ > −1 for the resulting formula.
- Numerical illustrations: Figures 1 and 2 compare the generalized derivative for ρ = −0.4, 0.0, 0.4 and multiple values of ν.Figure 1 uses ν = 1.0, 2.0; Figure 2 uses ν = 0.5, 1.5.
- Open problem: An open problem is to find an exact formula for the left-sided generalized fractional derivative of (x − a)^ω for ω ∈ R.The stated setting includes α ∈ C, n = ⌈Re(α)⌉, ρ ≠ −1, and x > a.
- Conclusion: The conclusion states that the characteristics of the fractional derivative are highly affected by ρ, suggesting a direction for control applications.This conclusion is based on the comparisons in Figures 1 and 2.
- Future work: Formulae for Laplace, Fourier, and Mellin transforms, along with further properties of the generalized derivatives, are deferred to future work.The authors also plan to investigate the effect of the parameter ρ.