---
title: Variable Order Caputo Derivatives
url: https://www.emergentmind.com/topics/variable-order-caputo-derivatives
type: topic
---

# Variable Order Caputo Derivatives

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 [1704.06486].

## 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 $\alpha_n:[a,b]\times[a,b]\to(n-1,n)$ allows for dependence on both the evaluation point $t$ and the integration variable $\tau$, enabling "fully variable" order settings. In all cases, $x$ must be sufficiently smooth (at least $C^n$) and the kernel $(t,\tau)\mapsto\alpha_n(t,\tau)$ continuous on $a\le\tau\le t\le b$ [1704.06486].

A combined Caputo derivative of variable order, important in variational formulations, is defined by
\[
^C_\gamma D_t^{\alpha,\beta} x(t) = \gamma_1\,^C_aD_t^{\alpha(\cdot,\cdot)}x(t) + \gamma_2\,^C_tD_b^{\beta(\cdot,\cdot)}x(t),\quad \gamma=(\gamma_1,\gamma_2)\in[0,1]^2,
\]
which is a convex combination of left and right variable-order Caputo derivatives [1704.06486, 1501.02082].

## 2. Integration by Parts and Adjointness

A crucial analytic result is the higher-order integration by parts formula for variable-order Caputo derivatives [1704.06486, Theorem 2.6]. For $x,y\in C^n([a,b])$:
\[
\int_a^b y(t)\,^C_aD_t^{\alpha_n(\cdot,\cdot)}x(t)\,dt =\int_a^b x(t)\, {_tD_b^{n-\alpha_n(\cdot,\cdot)}\,y(t)} dt + \sum_{k=0}^{n-1}(-1)^k \Bigl[x^{(n-1-k)}(t)\frac{d^k}{dt^k}({_tI_b^{n-\alpha_n(\cdot,\cdot)}\,y(t)})\Bigr]_{t=a}^{t=b},
\]
where
\[
_tI_b^{n-\alpha_n(\tau,t)} y(t) =\int_t^b\frac{1}{\Gamma(n-\alpha_n(\tau,t))}(\tau-t)^{n-1-\alpha_n(\tau,t)} y(\tau) d\tau,
\]
and
\[
_tD_b^{n-\alpha_n(\tau,t)} = (-1)^n \frac{d^n}{dt^n} \, _tI_b^{n-\alpha_n(\tau,t)}.
\]
If $x$ and its derivatives up to order $n-1$ vanish at $a$ and $b$, 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 [1704.06486, 1805.00720].

## 3. Applications in Variational Problems

Consider functionals involving combined Caputo derivatives of variable order:
\[
\mathcal{J}(x,T) = \int_a^T L\bigl(t, x(t), ^C_{\gamma^1}D_t^{\alpha_1,\beta_1}x(t),\dots,^C_{\gamma^n}D_t^{\alpha_n,\beta_n}x(t)\bigr) dt + \phi(T,x(T)),
\]
where the combined derivatives are as above and $\phi$ is a terminal payoff [1704.06486]. Upon variation, and using the integration by parts formula, necessary optimality conditions are obtained:
- A fractional Euler–Lagrange system on $[a,T]$,
\[
\partial_2L + \sum_{i=1}^n D_{\overline{\gamma}^i}^{n-\alpha_i,\beta_i}\bigl(\partial_{i+2}L\bigr) = 0,
\]
where $D_{\overline{\gamma}^i}$ involves the dual combined Riemann–Liouville operators.
- A transmission condition on $[T,b]$,
\[
\sum_{i=1}^n \gamma_2^i \Bigl(_aD_t^{\alpha_i}\partial_{i+2}L - _TD_t^{\beta_i}\partial_{i+2}L\Bigr) = 0.
\]
- Boundary transversality conditions at $t=T$ and $t=b$ ensuring the proper vanishing of boundary terms as dictated by the structure of the nonlocal operators (see equation (3.12) in [1704.06486]).

These structures generalize classical variational calculus and allow for flexible modeling of systems with time- and/or state-dependent memory [1110.4141, 1501.02082, 1805.00720].

## 4. Analytic Properties and Example Computations

For power-law inputs and time-dependent orders, direct computation yields closed-form expressions. If $x(t)=(t-a)^\gamma$, $\gamma>n-1$, and $\alpha_n(t,\tau)=\overline\alpha_n(t)$, then
\[
^C_aD_t^{\overline\alpha_n(t)}(t-a)^\gamma = \frac{\Gamma(\gamma+1)}{\Gamma(\gamma-\overline\alpha_n(t)+1)} (t-a)^{\gamma-\overline\alpha_n(t)}.
\]
For $n=1$, this recovers the standard Caputo result with variable order. Such formulas are instrumental in validating numerical approximations of the operator [1704.06486, 1110.4141].

## 5. Numerical Discretization and Computational Techniques

Effective numerical evaluation of variable-order Caputo derivatives is nontrivial due to their nonlocal and order-dependent kernels. [1704.06486] 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 $\alpha(t,\tau)$ [1704.06486].

## 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 $t$ only, rather than $(t,\tau)$ [1805.00720]. In the general $(t,\tau)$-dependent case, as in [1704.06486], 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 [1704.06486, 1805.00720].

Source: https://www.emergentmind.com/topics/variable-order-caputo-derivatives