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On fractional calculus with general analytic kernels
Arran Fernandez, Mehmet Ali Ozarslan, Dumitru Baleanu
TL;DR
The paper addresses the lack of a single framework covering the many fractional-calculus definitions built from modified kernels. It introduces general analytic-kernel operators connected to Riemann–Liouville integrals by an infinite series, then develops their properties, rules, and differential-equation applications. The framework incorporates many existing definitions while remaining subject to stated scope boundaries.
Problem
Fractional calculus contains many proposed derivative and integral definitions, motivating the search for a general class that includes existing operators as particular cases.
Method
The paper defines fractional integral and derivative operators with a general analytic kernel and relates them to Riemann–Liouville fractional integrals through an infinite series.
Results
The framework incorporates many existing fractional integrals and derivatives as special cases and extends the Leibniz and chain rules to the new operators.
Takeaways & Limitations
General results proved in this framework may advance understanding across multiple fractional-calculus models simultaneously.
Takeaways & Limitations
The formulation is not general enough to cover all models of fractional calculus, and further generalisations remain possible.
Abstract
from arXiv · showhide
Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel functions. We demonstrate, under some assumptions, how all of these modifications can be considered as special cases of a single, unifying, model of fractional calculus. We provide a fundamental connection with classical fractional calculus by writing these general fractional operators in terms of the original Riemann-Liouville fractional integral operator. We also consider inversion properties of the new operators, prove analogues of the Leibniz and chain rules in this model of fractional calculus, and solve some fractional differential equations using the new operators.
1 Background
Fractional calculus has many competing definitions, including models that replace the classical power-function kernel. The paper asks whether these operators can be unified in a general framework that supports shared theory and applications.
- 1 Background: Fractional calculus generalises differentiation and integration beyond integer orders, but no single definition is universally accepted.Several definitions have been proposed from different viewpoints, each with advantages and disadvantages.
- 1 Background: The Riemann–Liouville model is a foundational fractional-calculus framework for defining fractional integrals and derivatives.The Caputo model modifies the Riemann–Liouville derivative by interchanging differentiation and fractional integration, while Liouville’s model can include both as special cases.
- 1 Background: Researchers have proposed singular and non-singular kernel replacements to represent real data associated with different complex systems.Examples include the Atangana–Baleanu, generalised proportional fractional, and Prabhakar models.
- 1 Background: The paper poses whether a general class of fractional-calculus operators can contain existing operators as particular cases.A unified formalism could allow analogous results to be proved once for the general model rather than repeatedly for individual models.
- 1 Background: Applications motivate a simple, efficient fractional-calculus structure that can model many real-life processes, while mathematical generalisations often introduce additional parameters.The authors frame the challenge as balancing complex generalised operators with the simplicity of laws of nature.
- 1 Background: The paper develops a general framework whose operators include many proposed models and can be expressed through classical Riemann–Liouville operators using a series formula.The framework is intended to support analogous theorems for fractional calculus, including results related to basic calculus operations.
2 Definition and basic properties
The paper defines a general fractional integral framework using analytic functions of fractional powers, encompassing several existing models while retaining a connection to classical Riemann–Liouville integrals. Under suitable conditions, the framework yields bounded operators, series representations, composition and semigroup results, and inversion procedures, although its scope does not include every fractional-calculus model.
- Existing models as special cases: The framework contains the ABR, ABC, GPF, Prabhakar, Riemann–Liouville, and classical iterated-integral models as special cases.The correspondences are established for appropriate functions and parameters.
- Definition and motivation: The new operator class uses an analytic function of a fractional power as its kernel, balancing generality with an explicit fractional order.Analyticity supports identities connecting the operators to classical fractional calculus.
- Basic properties: For f ∈ L1[a, b], the general fractional integral is a well-defined bounded operator with norm at most (b −a)^αM.Here M is the supremum of |A(x)| over the relevant disc.
- Basic properties: The general operators admit locally uniformly convergent series representations in terms of classical Riemann–Liouville fractional integrals.This gives the framework a direct connection to the standard Riemann–Liouville model.
- Composition and semigroup properties: Composition results include a commutative family and semigroup properties under specified parameter conditions, while two-parameter semigroup validity cannot hold uniformly.The composition formula also supports solving equations involving both Riemann–Liouville and generalized operators.
- Inversion: The methodology supports inversion of generalized integrals and recovers the Prabhakar and AB fractional derivatives in corresponding special cases.The AB derivation includes inversion properties up to multiplicative constants.
3 Transforms and differential equations
The paper derives transform formulas for the generalised fractional operators and applies them to fractional integral equations. Under the stated assumptions, the example equation has a unique solution.
- Transforms: The transform formulas are obtained from the series representation of the operators and classical transforms of Riemann–Liouville integrals.The series formula avoids requiring a direct transform of the kernel function A.
- Transforms: Theorem 3.1 gives a Laplace-transform formula for the generalised fractional integral AIα,β₀+ f(t).The formula is stated for f ∈ L2[a,b], with a = 0 and b > 0, and uses AΓ from Definition 2.4.
- Transforms: Theorem 3.2 gives a Fourier-transform formula for AIα,β+ f(t) on the whole-line setting.It assumes a = −∞, b ∈ R, f ∈ L2[a,b], and uses AΓ from Definition 2.4.
- Differential equations: The paper applies these results to solve a fractional integral equation by Laplace transformation.The transformed unknown has an explicit expression, whose unique inverse Laplace transform yields the solution function f under the required initial condition.
4 Leibniz rule and chain rule
The generalised fractional model retains enough structure from classical fractional calculus to support analogues of the Leibniz and chain rules. These rules yield explicit series expressions for broad classes of functions.
- Generalised Leibniz rule: The proposed model admits a generalised Leibniz rule for f ∈ C[a,b] and g ∈ C∞[a,b].The result applies for α and β with non-negative real parts and is developed from the classical Riemann–Liouville rule.
- Generalised Leibniz rule: The proof combines the Riemann–Liouville Leibniz rule with the series formula for the generalised operators.Uniform convergence permits the relevant summations to be interchanged, while the remainder term tends to zero as N → ∞.
- Examples: Applying the chain-rule identity to f(t)=e^{kt} and g(t)=t produces an explicit series expression for the fractional differintegral of te^{kt}.The expression applies for any k ∈ C and for every fractional-calculus model covered by the framework.
- Generalised chain rule: The paper establishes a generalised chain rule for smooth functions f and g when α and β have non-negative real parts.The derivation uses the Faà di Bruno formula to expand the ordinary derivatives appearing in the operator identity.
- Applications: The two rules apply to functions generated from elementary functions by multiplication and composition.The paper specifically mentions power and exponential functions as examples of the resulting computable class.
5 The solution of a Cauchy problem using Volterra integral equations
The paper converts the generalised Cauchy problem into an equivalent Volterra integral equation and proves existence and uniqueness under a Lipschitz condition. A contraction-mapping argument is extended interval by interval to cover the full domain.
- Problem reduction: The Cauchy-type problem is precisely equivalent to a Volterra integral equation under the paper’s integrability assumptions.This equivalence is the starting point for proving well-posedness of the original problem.
- Volterra equation: If f satisfies the stated Lipschitz condition in its second variable, the Volterra equation has a unique solution y ∈ L1[a,b].The proof first applies the contraction mapping theorem on a sufficiently short interval.
- Global construction: The piecewise solutions agree across the interval sequence and yield a unique L1[a,b] solution on the full domain.The extension ends when the endpoint b is reached.
- Cauchy problem: Consequently, the original Cauchy problem has a unique solution u ∈ L1[a,b].The conclusion follows by combining the Cauchy–Volterra equivalence with the Volterra existence-and-uniqueness theorem.
6 Operators with respect to functions
The paper combines differintegration with respect to functions and general analytic kernels in one framework, recovering several classical fractional models as special cases. It also extends this framework toward ψ-based operators and identifies future generalizations.
- Operators with respect to functions: Differintegration with respect to a function replaces differentiation or integration with respect to t by operations using another function, commonly denoted ψ(t).The paper places this idea within the Riemann–Liouville model before combining it with generalized kernels.
- Operators with respect to functions: Definition 6.1 combines differintegration with respect to functions and generalized kernel functions in a single formalism.The framework assumes f is integrable while ψ is monotonic and continuously differentiable.
- Operators with respect to functions: Choosing ψ(t) = log t recovers a generalized Hadamard fractional model and, with β = 0, the standard Hadamard fractional integral.The specialization connects the new operator to a classical logarithmic-kernel model.
- Operators with respect to functions: Choosing ψ(t) = tρ+1 recovers a generalized Katugampola model and, with β = 0, the standard Katugampola fractional integral.This identifies the Katugampola operator as another specialization of Definition 6.1.
- Operators with respect to functions: Choosing ψ(t) = tσ and replacing f(t) by tσηf(t) recovers one possible generalization of the Erdelyi–Kober model.The paper also states that a corresponding parameter choice recovers the Erdelyi–Kober fractional integral.
- Operators with respect to functions: The framework extends beyond these examples to Hadamard, Katugampola, Erdelyi–Kober, and related classical models with generalized kernels.The authors further propose examining how much of fractional calculus can be extended in this combined setting.
7 Conclusions
The paper introduces a general analytic-kernel framework that incorporates many fractional integrals and derivatives as special cases and connects them to Riemann–Liouville integrals. It extends core fractional-calculus rules, solves selected differential equations, and opens a broader unified route for future results.
- Conclusions: The new framework incorporates many existing definitions of fractional integrals and derivatives as special cases.Its general analytic-kernel integral operator is represented as an infinite series of Riemann–Liouville integrals.
- Conclusions: The paper extends the Leibniz rule and chain rule to the new fractional operators.These extensions are established after proving fundamental properties of the generalized operators.
- Conclusions: Fourier and Laplace transforms are used to analyze and solve some simple ordinary differential equations in the generalized framework.The paper also proves existence and uniqueness for a broader class of Cauchy problems using a contraction mapping theorem and a Volterra integral equation.
- Conclusions: The framework opens the possibility of proving results across many fractional models simultaneously rather than within only one model.The authors identify this broader generalization as a future route for product rules, chain rules, Taylor’s theorem, and related results.