Combined Caputo Derivative Overview

• Combined Caputo derivative is defined as a convex combination of left and right Caputo fractional derivatives, integrating past and future memory effects.
• It enables modeling of nonlocal, memory-laden dynamical systems in fields such as anomalous diffusion, optimal control, and fractional variational calculus.
• Its analytical properties, including linearity and integration by parts, support extensions to variable-order, multidimensional, and probabilistic frameworks.

The combined Caputo derivative is a convex combination of the classical left and right Caputo fractional derivatives, parameterized by fractional orders and a weighting parameter. It has emerged as a fundamental tool in the study of nonlocal, memory-laden dynamical systems, particularly in the context of fractional variational calculus, optimal control, and anomalous diffusion with two-sided memory effects. The formalism generalizes both the one-sided Caputo derivatives and restores the classical derivative in the appropriate integer-order and weight limits.

1. Definition and Mathematical Formalism

Let $f:[a,b]\to\mathbb{R}$ be absolutely continuous. For orders $\alpha,\beta\in(0,1)$ and convex parameter $\gamma\in[0,1]$, the left and right Caputo derivatives are defined as: $${}^C_aD_x^{\alpha} f(x) = \frac{1}{\Gamma(1-\alpha)}\int_a^x (x-t)^{-\alpha} f'(t)\,dt,$$

$${}^C_xD_b^{\beta} f(x) = -\frac{1}{\Gamma(1-\beta)}\int_x^b (t-x)^{-\beta} f'(t)\,dt,$$

where $\Gamma$ denotes the Gamma function. The combined Caputo derivative is given by: $${}^C\!D^{\alpha,\beta}_\gamma f(x) = \gamma\,{}^C_aD_x^{\alpha} f(x) + (1-\gamma)\,{}^C_xD_b^{\beta} f(x),\qquad x\in[a,b],$$ mapping $AC([a,b])$ into $C([a,b])$ (Malinowska et al., 2010, Odzijewicz et al., 2011, Malinowska et al., 2011).

The operator interpolates between purely left-sided ($\gamma=1$) and right-sided ($\alpha,\beta\in(0,1)$0) memory. For $\alpha,\beta\in(0,1)$1 and $\alpha,\beta\in(0,1)$2, one recovers the classical derivative $\alpha,\beta\in(0,1)$3.

2. Analytical Properties

Key properties of the combined Caputo derivative include:

• Linearity: For $\alpha,\beta\in(0,1)$4 and scalars $\alpha,\beta\in(0,1)$5, $\alpha,\beta\in(0,1)$6.
• Nonlocality (memory): Values of $\alpha,\beta\in(0,1)$7 depend on the history ($\alpha,\beta\in(0,1)$8 for $\alpha,\beta\in(0,1)$9) and, via the right Caputo term, on the "future" ($\gamma\in[0,1]$0).
• Existence and continuity: For $\gamma\in[0,1]$1, both left and right Caputo derivatives exist and are continuous, yielding $\gamma\in[0,1]$2.
• Integration by parts: If $\gamma\in[0,1]$3 are sufficiently regular, and denoting the adjoint Riemann–Liouville operator as $\gamma\in[0,1]$4,

$\gamma\in[0,1]$5

where $\gamma\in[0,1]$6 and $\gamma\in[0,1]$7 are the Riemann–Liouville fractional integrals (Malinowska et al., 2010, Malinowska et al., 2011, Malinowska et al., 2011).

The operator does not, in general, satisfy a semigroup law in its fractional orders unless trivial circumstances hold.

3. Extensions: Variable Order, Generalizations, Probabilistic and Multidimensional Settings

Variable Order: The generalized combined Caputo derivative with variable orders $\gamma\in[0,1]$8, $\gamma\in[0,1]$9 and weights $${}^C_aD_x^{\alpha} f(x) = \frac{1}{\Gamma(1-\alpha)}\int_a^x (x-t)^{-\alpha} f'(t)\,dt,$$0, $${}^C_aD_x^{\alpha} f(x) = \frac{1}{\Gamma(1-\alpha)}\int_a^x (x-t)^{-\alpha} f'(t)\,dt,$$1 is defined as

$${}^C_aD_x^{\alpha} f(x) = \frac{1}{\Gamma(1-\alpha)}\int_a^x (x-t)^{-\alpha} f'(t)\,dt,$$2

enabling adaptation to heterogeneous memory (Tavares et al., 2015).

Multidimensional and Fully Mixed Forms: The combined Caputo framework extends to multidimensional and fully mixed settings, with operators such as

$${}^C_aD_x^{\alpha} f(x) = \frac{1}{\Gamma(1-\alpha)}\int_a^x (x-t)^{-\alpha} f'(t)\,dt,$$3

where the weights and orders can be chosen to construct arbitrary linear combinations, with probabilistic interpretations as generators of interrupted/stopped jump processes associated to nonlocal stochastic dynamics (Kolokoltsov, 2015).

4. Role in Fractional Variational Calculus

Combined Caputo derivatives provide the basis for a class of fractional variational problems with Lagrangians depending on nonlocal, two-sided derivatives: $${}^C_aD_x^{\alpha} f(x) = \frac{1}{\Gamma(1-\alpha)}\int_a^x (x-t)^{-\alpha} f'(t)\,dt,$$4 For such functionals, first variation and integration by parts yield Euler–Lagrange systems (Odzijewicz et al., 2011, Malinowska et al., 2010, Malinowska et al., 2011): $${}^C_aD_x^{\alpha} f(x) = \frac{1}{\Gamma(1-\alpha)}\int_a^x (x-t)^{-\alpha} f'(t)\,dt,$$5

When isoperimetric or holonomic constraints are present, Lagrange multipliers are introduced and similar fractional optimality systems are obtained. Transversality conditions (natural boundary conditions) involve fractional integrals, arising from the vanishing of boundary terms in the integration by parts process: $${}^C_aD_x^{\alpha} f(x) = \frac{1}{\Gamma(1-\alpha)}\int_a^x (x-t)^{-\alpha} f'(t)\,dt,$$6 (Malinowska et al., 2010, Malinowska et al., 2011, Malinowska et al., 2011).

For variable-order settings, the Euler–Lagrange equations and transversality conditions are accordingly modified with nonconstant orders and weights (Tavares et al., 2015).

5. Applications and Physical Relevance

Combined Caputo derivatives model systems with both retentive (past) and anticipative (future) memory, offering flexibility beyond purely one-sided or symmetrized Riemann–Liouville operators.

Key Application Areas:

• Nonconservative Mechanics: Capture bidirectional hereditary effects, i.e., influence from both earlier and later states (Malinowska et al., 2010).
• Anomalous Diffusion: Model transport phenomena where both boundary and initial data influence the process (Kolokoltsov, 2015).
• Control and Systems with Two-Sided Memory: Enable state equations in which both initial and terminal conditions affect dynamics, relevant in generalizations of optimal control and model predictive control (Malinowska et al., 2010, Malinowska et al., 2011).
• Fractional PDEs and Stochastic Analysis: Via their interpretation as generators of killed/interrupted jump processes, combined derivatives unify fractional diffusion operators with boundary effects in multidimensional and probabilistic contexts (Kolokoltsov, 2015).

Table: Special Cases of the Combined Caputo Derivative

$${}^C_aD_x^{\alpha} f(x) = \frac{1}{\Gamma(1-\alpha)}\int_a^x (x-t)^{-\alpha} f'(t)\,dt,$$7	$${}^C_aD_x^{\alpha} f(x) = \frac{1}{\Gamma(1-\alpha)}\int_a^x (x-t)^{-\alpha} f'(t)\,dt,$$8	Operator	Limiting Case
$${}^C_aD_x^{\alpha} f(x) = \frac{1}{\Gamma(1-\alpha)}\int_a^x (x-t)^{-\alpha} f'(t)\,dt,$$9	any	$${}^C_xD_b^{\beta} f(x) = -\frac{1}{\Gamma(1-\beta)}\int_x^b (t-x)^{-\beta} f'(t)\,dt,$$0	Left Caputo derivative of order $${}^C_xD_b^{\beta} f(x) = -\frac{1}{\Gamma(1-\beta)}\int_x^b (t-x)^{-\beta} f'(t)\,dt,$$1
$${}^C_xD_b^{\beta} f(x) = -\frac{1}{\Gamma(1-\beta)}\int_x^b (t-x)^{-\beta} f'(t)\,dt,$$2	any	$${}^C_xD_b^{\beta} f(x) = -\frac{1}{\Gamma(1-\beta)}\int_x^b (t-x)^{-\beta} f'(t)\,dt,$$3	Right Caputo derivative of order $${}^C_xD_b^{\beta} f(x) = -\frac{1}{\Gamma(1-\beta)}\int_x^b (t-x)^{-\beta} f'(t)\,dt,$$4
$${}^C_xD_b^{\beta} f(x) = -\frac{1}{\Gamma(1-\beta)}\int_x^b (t-x)^{-\beta} f'(t)\,dt,$$5	$${}^C_xD_b^{\beta} f(x) = -\frac{1}{\Gamma(1-\beta)}\int_x^b (t-x)^{-\beta} f'(t)\,dt,$$6	$${}^C_xD_b^{\beta} f(x) = -\frac{1}{\Gamma(1-\beta)}\int_x^b (t-x)^{-\beta} f'(t)\,dt,$$7	Symmetrized Caputo derivative
any	$${}^C_xD_b^{\beta} f(x) = -\frac{1}{\Gamma(1-\beta)}\int_x^b (t-x)^{-\beta} f'(t)\,dt,$$8	$${}^C_xD_b^{\beta} f(x) = -\frac{1}{\Gamma(1-\beta)}\int_x^b (t-x)^{-\beta} f'(t)\,dt,$$9	Recovers classical first derivative

6. Product and Chain Rule Structures

Analogous to standard Caputo derivatives, the combined Caputo derivative admits generalized product and chain rules. These involve infinite series expansions over integer-order derivatives of the constituent functions. For example, the Caputo fractional derivative of a product $\Gamma$0 is: $\Gamma$1 Chain rules leverage generalized Faà di Bruno-type formulas with hypergeometric function expressions, critical for concrete calculations in physical models (Shchedrin et al., 2018).

7. Probabilistic and Operator-Theoretic Interpretation

The mixed Caputo derivative, particularly in the fully mixed or multidimensional versions, arises naturally as the generator of interrupted or "stopped" Feller-type Markov jump processes. For jump diffusion or stable processes interrupted at domain boundaries, the combined Caputo operator governs the resulting process, endowing the fractional boundary value problem with a precise probabilistic interpretation and robust well-posedness properties (Kolokoltsov, 2015).

The resolvent problem corresponding to the combined Caputo operator yields unique classical and generalized solutions, with explicit Green's function representations in certain cases.

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References:

(Malinowska et al., 2010, Odzijewicz et al., 2011, Malinowska et al., 2011, Tavares et al., 2015, Kolokoltsov, 2015, Malinowska et al., 2011, Shchedrin et al., 2018)