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A Caputo fractional derivative of a function with respect to another function
Ricardo Almeida
TL;DR
The paper addresses how to define and analyze a Caputo-type fractional derivative with respect to another function amid limitations in general kernel-based operators. It develops the operator’s properties and a numerical approximation using integer-order derivatives, then applies it to population growth, finding that kernel choice matters for model efficiency.
Problem
General kernel-based fractional operators can lack fundamental derivative laws, motivating a Caputo-type derivative with respect to another function.
Method
The paper defines a Caputo fractional derivative relative to another function, studies its properties, and approximates it by a sum involving integer-order derivatives.
Results
The operator’s properties are derived, numerical approximations reduce problems to ordinary derivatives, and population-growth modeling shows that fractional-derivative choice affects method efficiency.
Takeaways & Limitations
Different kernels can provide more accurate descriptions of population-growth dynamics within the paper’s modeling example.
Takeaways & Limitations
The paper leaves open which kernel best describes dynamics for given experimental data and how a time-varying order α(t) should be selected.
Abstract
from arXiv · showhide
In this paper we consider a Caputo type fractional derivative with respect to another function. Some properties, like the semigroup law, a relationship between the fractional derivative and the fractional integral, Taylor's Theorem, Fermat's Theorem, etc, are studied. Also, a numerical method to deal with such operators, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented. Relying on examples, we show the efficiency and applicability of the method. Finally, an application of the fractional derivative, by considering a Population Growth Model, and showing that we can model more accurately the process using different kernels for the fractional operator is provided.
1 Introduction
Fractional calculus extends ordinary differentiation and integration to arbitrary positive real orders, with multiple operator definitions and kernels. The paper introduces a Caputo derivative relative to another function to study fundamental properties and numerical and modeling applications.
- Motivation: Fractional calculus extends ordinary derivatives and integrals from natural-number orders to arbitrary real orders α > 0.Ordinary calculus is recovered as a particular case.
- Motivation: Fractional integrals and derivatives have multiple definitions, including Riemann–Liouville, Hadamard, and other kernel-based operators.Different choices of kernel and differential operator generate classical fractional operators.
- Problem: General kernel-based operators can limit access to fundamental derivative laws because kernel arbitrariness prevents many basic properties from being obtained.This limitation motivates a more structured approach using derivatives with respect to another function.
- Approach: For an increasing function ψ with nonzero derivative, fractional integrals and derivatives can be defined with respect to ψ, recovering Riemann–Liouville and Hadamard operators for special ψ.The choices ψ(x) = x and ψ(x) = ln x yield these two classical operators.
- Paper scope: The paper studies a Caputo fractional derivative with respect to another function, including integration–differentiation relations, semigroup laws, and numerical and population-growth applications.Its numerical method approximates the fractional derivative using integer-order derivatives.
2 Caputo-type fractional derivative
The paper defines a ψ-Caputo fractional derivative by combining Caputo's ordering of ordinary differentiation and fractional integration with operators defined relative to ψ. It establishes boundedness, relationships with Riemann–Liouville derivatives, and conditions supporting further properties.
- Definition: Caputo’s formulation switches the order of ordinary differentiation and fractional integration, yielding integer-order initial conditions in the Laplace transform.This contrasts with the fractional-order conditions associated with the Riemann–Liouville derivative.
- Definition: The left ψ-Caputo fractional derivative is defined for α > 0 using functions f and ψ in C^n, with ψ increasing and ψ′ nonzero.The order parameter uses n = [α] + 1 for noninteger α and n = α for integer α.
- Scope: The paper restricts its detailed study to noninteger α and proves only the left-sided case because the right-sided arguments are analogous after adjustment.Special choices of ψ recover Caputo, Caputo–Hadamard, and Caputo–Erdélyi–Kober derivatives.
- Properties: The ψ-Caputo fractional derivatives are bounded operators for all α > 0.This result is stated as Theorem 2.
- Properties: Theorem 1 provides a relation between the ψ-Caputo and Riemann–Liouville fractional derivatives with respect to ψ.The relation is obtained using integration by parts.
3 Examples
Examples compute the ψ-Caputo derivative for power and Mittag–Leffler functions and visualize how its values vary with fractional order and kernel choice. The examples include ψ(x) = x, ln(x + 1), and a square-root kernel.
- Power functions: A change of variables and the Beta function are used to compute ψ-Caputo derivatives of example functions.The derivations explicitly invoke the substitution u = (ψ(t) − ψ(a))/(ψ(x) − ψ(a)).
- Power functions: For a quadratic example, the derivative is 2/Γ(3 − α)(ψ(x) − ψ(0))^(2 − α), reducing at α = 1 to 2(ψ(x) − ψ(0)).This formula is reported for the left derivative starting at 0.
- Graphs: Figure 1 plots the example derivative for different fractional orders α and different kernels ψ.The plotted kernels include ψ(x) = x, ψ(x) = ln(x + 1), and a square-root form.
- Mittag–Leffler functions: For the Mittag–Leffler example, the derivative equals the function at α = 0 and becomes exp(ψ(x) − ψ(0)) at α = 1.Figure 2 presents graphs of this derivative for selected orders and kernels.
4 Relation between integration and derivative
The ψ-Caputo fractional derivative is presented as an inverse operation to the fractional integral defined with respect to the same function. This relationship yields a representation involving initial terms and supports a Taylor formula.
- Inverse relation: The ψ-Caputo fractional derivative is an inverse operation for the fractional integral with respect to the same function ψ.This inverse relationship is the central result of the section.
- Inverse relation: Theorem 4 establishes the derivative–integral relationship for f ∈ C^n[a, b] and α > 0.The result is developed using semigroup and integration-by-parts properties.
- Consequences: The inverse relation leads to a representation containing coefficients c_k determined by differences between functions and their derivatives.The coefficients are introduced when proving equality from a vanishing ψ-Caputo derivative.
- Consequences: The functions (ψ(x) − ψ(a))^k, for k = 0, 1, . . . , n − 1, have zero left ψ-Caputo derivative in the stated result.These terms appear as the relevant initial-function components.
5 Semigroup laws
The section examines composition and semigroup properties of the ψ-Caputo fractional derivative, showing that semigroup behavior is generally limited and holds under specific conditions. It also reduces higher-order fractional derivatives to a fractional order in (0,1).
- The semigroup law generally fails for the ψ-Caputo fractional derivative, although it is valid in some specific cases.
- Theorems 7–9 establish composition relations between fractional integrals and ψ-Caputo fractional derivatives.
- Any fractional derivative of order α > 0 can be computed from the derivative of order β = α − (n − 1) ∈ (0, 1).
- Theorem 10 gives an additional composition result when α, β > 0 satisfy a shared integer-interval condition.
- The integer-existence assumption in Theorem 10 is essential and fails for ψ(x) = x.
6 Miscelious results
This section develops analytical properties of the ψ-Caputo fractional derivative, including integration by parts, low-level fractionality, fractional Fermat and mean value theorems, and Taylor-type formulas. The results connect fractional derivatives with monotonicity, extrema, and integral representations.
- The section establishes integration by parts, a near-one-order relation, fractional Fermat and mean value theorems, and a Taylor formula.
- Theorem 12 provides an integration by parts formula for the fractional operator.
- When α ≈ 1−, the relation between the fractional and ordinary first derivatives is known as low-level fractionality.
- For α ∈ (0, 1), a maximum yields the fractional inequalities stated by the fractional version of Fermat’s theorem.
- The ψ-Caputo derivative has the sign of the ordinary derivative when the function is respectively non-decreasing or non-increasing on the relevant interval.
- Theorem 16 supplies a fractional mean value theorem, while Theorems 17–18 provide the basis for a fractional Taylor formula.
7 A numerical tool
The paper develops a numerical method that replaces ψ-Caputo fractional derivatives with sums involving only integer-order derivatives, converting fractional problems into ordinary differential systems. The approximation is illustrated for a Caputo–Hadamard kernel and a nonlinear fractional differential equation, with accuracy improving as the truncation parameter increases.
- Approximation method: Theorem 19 approximates ψ-Caputo fractional derivatives by a sum of functions involving only the first-order derivative of f.This provides the paper's central numerical tool for fractional-type problems.
- Approximation method: The approximation error decreases as N →∞, allowing the fractional problem to be rewritten as an ordinary one.After this transformation, analytical or numerical methods for ordinary differential equations can be applied.
- Caputo–Hadamard example: For ψ(x) = ln(x + 1) and α ∈(0, 1), the method approximates the fractional derivative using integer-order derivatives for functions of class C2.The example is considered on x ∈[0, 5].
- Caputo–Hadamard example: Figure 3 compares the exact fractional derivative of f(x) = ln2(x + 1) with numerical approximations from Eq. (4) for different N values and α = 0.5.The comparison is made over x ∈[0, 5].
- Cauchy problem example: For the nonlinear fractional differential equation with α = 0.5 and ψ(x) = ln(x + 1), increasing N makes the ordinary-system solution approach the fractional differential-equation solution.Figure 4 reports this comparison for N = 2, 4, 6.
8 An application
The paper applies fractional population-growth models to world-population data, comparing classical and fractional formulations across kernels and time periods. The fractional models fit the data more accurately than the ordinary or classical model, with kernel choice affecting the reported error.
- Model formulation: The Malthusian population-growth law is reformulated by replacing the first-order derivative with a ψ-Caputo fractional derivative.The resulting fractional differential equation is solved with an initial population condition.
- Kernel comparison: The fractional model fits the population data better than the ordinary model, motivating tests of alternative kernels.
- Kernel comparison: For ψ(x) = (x + 1)b, the best reported parameters are λ ≈0.26821, α ≈2.05784, and b ≈0.66734, with error E ≈1.26039 × 105.
- Kernel comparison: For ψ(x) = ln(x+1), reported fits include errors E ≈8.2257 × 104 and E ≈5.3735 × 104 with different λ and α values.
Conclusion and future work
The paper develops a Caputo-type fractional derivative with respect to another function, establishes important properties and numerical approximations, and applies it to world population growth. Future work includes identifying kernels and time-varying orders that best fit experimental dynamics.
- The paper derives important properties of the new fractional derivative and establishes conditions under which semigroup laws are valid.
- It provides numerical approximations that reduce problems involving the new operator to ones depending only on ordinary derivatives.
- The derivative is applied to world population growth, where the choice of fractional derivative affects method efficiency.
- Future work will investigate which kernel best describes a problem's dynamics given experimental data.
- A further proposed generalization is to make the fractional order time-dependent as α(t) and determine the order that best fits the model.