Variable Order Caputo Derivatives

• Variable order Caputo derivatives are fractional operators defined with a spatially or temporally varying order function, generalizing classical derivatives.
• They underpin rigorous integration by parts formulas that facilitate deriving Euler–Lagrange equations in fractional variational frameworks.
• Numerical approaches using MATLAB Chebfun and spectral collocation schemes validate their application in modeling adaptive memory and nonlocal dynamics.

A variable order Caputo derivative is a fractional differential operator where the order of differentiation $\alpha$ is a function, typically of the position and/or integration variable, rather than a fixed constant. These operators generalize classical Caputo derivatives and are central to modeling systems with time- and state-dependent memory, as found in variable-order anomalous diffusion, viscoelasticity, or control systems with adaptive dynamics. The most rigorous development of variable order Caputo derivatives is based on the framework and results in (Tavares et al., 2017).

1. Definitions and Types

Let $a<b$ be real, $n\in\mathbb N$, and let $\alpha_n:[a,b]\times[a,b]\to(n-1,n)$ be continuous. For $x\in C^n([a,b])$, the left higher-order Caputo derivative of variable order $\alpha_n(t,\tau)$ is defined by

$$^C_aD_t^{\,\alpha_n(\cdot,\cdot)}x(t) =\int_{a}^{t} \frac{1}{\Gamma\left(n-\alpha_n(t,\tau)\right)} (t-\tau)^{n-1-\alpha_n(t,\tau)} x^{(n)}(\tau) d\tau.$$

The right Caputo derivative is given by

$$^C_tD_b^{\,\alpha_n(\cdot,\cdot)}x(t) = (-1)^n \int_{t}^{b}\frac{1}{\Gamma\left(n-\alpha_n(\tau,t)\right)} (\tau-t)^{n-1-\alpha_n(\tau,t)} x^{(n)}(\tau) d\tau.$$

When $n=1$ this reduces to

$$^C_aD_t^{\alpha(t,\tau)}x(t) = \int_a^t \frac{1}{\Gamma\left(1-\alpha(t,\tau)\right)} (t-\tau)^{-\alpha(t,\tau)} x'(\tau) d\tau.$$

The order function $a<b$0 allows for dependence on both the evaluation point $a<b$1 and the integration variable $a<b$2, enabling "fully variable" order settings. In all cases, $a<b$3 must be sufficiently smooth (at least $a<b$4) and the kernel $a<b$5 continuous on $a<b$6 (Tavares et al., 2017).

A combined Caputo derivative of variable order, important in variational formulations, is defined by

$a<b$7

which is a convex combination of left and right variable-order Caputo derivatives (Tavares et al., 2017, Tavares et al., 2015).

2. Integration by Parts and Adjointness

A crucial analytic result is the higher-order integration by parts formula for variable-order Caputo derivatives [(Tavares et al., 2017), Theorem 2.6]. For $a<b$8: $a<b$9 where

$n\in\mathbb N$0

and

$n\in\mathbb N$1

If $n\in\mathbb N$2 and its derivatives up to order $n\in\mathbb N$3 vanish at $n\in\mathbb N$4 and $n\in\mathbb N$5, all boundary terms vanish, yielding a formal adjointness relation between the variable-order Caputo and dual Riemann–Liouville operators.

This formula is fundamental for deriving necessary conditions in fractional variational calculus, such as Euler–Lagrange equations and boundary transversality conditions (Tavares et al., 2017, Almeida et al., 2018).

3. Applications in Variational Problems

Consider functionals involving combined Caputo derivatives of variable order: $n\in\mathbb N$6 where the combined derivatives are as above and $n\in\mathbb N$7 is a terminal payoff (Tavares et al., 2017). Upon variation, and using the integration by parts formula, necessary optimality conditions are obtained:

• A fractional Euler–Lagrange system on $n\in\mathbb N$8,

$n\in\mathbb N$9

where $\alpha_n:[a,b]\times[a,b]\to(n-1,n)$0 involves the dual combined Riemann–Liouville operators.

• A transmission condition on $\alpha_n:[a,b]\times[a,b]\to(n-1,n)$1,

$\alpha_n:[a,b]\times[a,b]\to(n-1,n)$2

• Boundary transversality conditions at $\alpha_n:[a,b]\times[a,b]\to(n-1,n)$3 and $\alpha_n:[a,b]\times[a,b]\to(n-1,n)$4 ensuring the proper vanishing of boundary terms as dictated by the structure of the nonlocal operators (see equation (3.12) in (Tavares et al., 2017)).

These structures generalize classical variational calculus and allow for flexible modeling of systems with time- and/or state-dependent memory (Odzijewicz et al., 2011, Tavares et al., 2015, Almeida et al., 2018).

4. Analytic Properties and Example Computations

For power-law inputs and time-dependent orders, direct computation yields closed-form expressions. If $\alpha_n:[a,b]\times[a,b]\to(n-1,n)$5, $\alpha_n:[a,b]\times[a,b]\to(n-1,n)$6, and $\alpha_n:[a,b]\times[a,b]\to(n-1,n)$7, then

$\alpha_n:[a,b]\times[a,b]\to(n-1,n)$8

For $\alpha_n:[a,b]\times[a,b]\to(n-1,n)$9, this recovers the standard Caputo result with variable order. Such formulas are instrumental in validating numerical approximations of the operator (Tavares et al., 2017, Odzijewicz et al., 2011).

5. Numerical Discretization and Computational Techniques

Effective numerical evaluation of variable-order Caputo derivatives is nontrivial due to their nonlocal and order-dependent kernels. (Tavares et al., 2017) implements all such operators using the MATLAB Chebfun package, leveraging symbolic differentiation and adaptive quadrature. The core routines (leftCaputo.m, rightCaputo.m, and combinedCaputo.m) translate the analytic definitions directly into numerical integration over Chebyshev grids, enabling:

• High-accuracy quadratic or spectral collocation schemes for associated differential equations and variational problems.
• Verification of expansion and boundary terms to machine precision for analytic solutions.

The approach applies equally to higher-order, right-sided, and combined forms, and is extendable to state-dependent order functions $x\in C^n([a,b])$0 (Tavares et al., 2017).

6. Relation to Other Variable-Order Fractional Operators

Multiple definitions of variable-order Caputo derivatives occur in the literature, notably three types (type I, II, III) distinguished by placement of the integer differentiation and restriction to functions of $x\in C^n([a,b])$1 only, rather than $x\in C^n([a,b])$2 (Almeida et al., 2018). In the general $x\in C^n([a,b])$3-dependent case, as in (Tavares et al., 2017), the operators do not coincide except in the constant-order or constant-function limit. The distinctions arise in analytic properties, especially for nonconstant order functions, and must be respected in both theoretical and numerical applications.

A key conclusion is that the variable-order Caputo derivative always reduces to the classical Caputo derivative when the order function is constant. The various expansion and decomposition results for variable-order fractional operators enable efficient and accurate simulation of nonlocal, variable-memory systems in science and engineering (Tavares et al., 2017, Almeida et al., 2018).