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Caputo derivatives of fractional variable order: numerical approximations
Dina Tavares, Ricardo Almeida, Delfim F. M. Torres
TL;DR
Variable-order Caputo equations are difficult to solve analytically, motivating numerical tools. The paper develops integer-derivative decompositions for three Caputo-type operators, derives error estimates, and applies the method to partial differential equations. The approximation error can be decreased as much as desired by increasing N.
Problem
Variable-order fractional differential equations often lack analytically known solutions, so numerical approximations are needed.
Method
The paper develops approximation formulas for three variable-order Caputo operators using standard integer-order derivatives, with error estimates and applications to partial differential equations.
Results
The approximation error can be decreased as much as desired by increasing the value of N.
Takeaways & Limitations
The method rewrites variable-order fractional differential equations as systems of ordinary differential equations solvable with standard techniques.
Takeaways & Limitations
The paper reports no similar variable-fractional-calculus methods for direct performance comparison and therefore compares approximations with exact solutions.
Abstract
from arXiv · showhide
We present a new numerical tool to solve partial differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them an approximation formula is obtained in terms of standard (integer-order) derivatives only. Estimations for the error of the approximations are also provided. We then compare the numerical approximation of some test function with its exact fractional derivative. We end with an exemplification of how the presented methods can be used to solve partial fractional differential equations of variable order.
1 Introduction
Variable-order fractional derivatives model systems whose order changes with time, but their analytical solutions are often difficult to obtain. The paper extends integer-derivative decompositions to Caputo problems of variable order and applies them to partial differential equations.
- Variable-order fractional calculus allows the derivative order to depend on time and has applications in physics, control, and signal processing.
- Analytical solutions for variable-order fractional problems are usually difficult to obtain, motivating numerical approximation.
- Existing numerical approaches typically discretize time or replace fractional operators with suitable decompositions.
- The paper extends decompositions based only on integer-order derivatives to Caputo fractional problems of variable order.
- The study derives approximations and error bounds, tests them against exact fractional derivatives, and applies them to diffusion and fractional Burgers equations.
2 Fractional calculus of variable order
The paper defines three types of Caputo fractional derivatives whose order varies with time, then extends them to several variables. It relates these operators, derives special-function formulas, and compares variable- and constant-order derivatives.
- The Caputo derivative is recalled as a fractional operator whose constant-function derivative is zero, unlike the Riemann–Liouville derivative.
- The variable order α(t) takes values in (0, 1) and is motivated by processes such as diffusion in inhomogeneous media.
- Variable order Caputo derivatives for functions of one variable: Three Caputo derivative types— I, II, and III—are introduced with left and right variants for functions of one variable.
- Variable order Caputo derivatives for functions of one variable: When the order is variable, the different derivative types do not coincide, unlike the constant-order case.
- The definitions are related through identities involving Riemann–Liouville and Caputo operators, including endpoint behavior and constant-order special cases.
- For power functions, the paper derives explicit variable-order Caputo formulas and compares them with constant orders α = 0.1 and α = 0.6 in Figure 1.
- Variable order Caputo derivatives for functions of several variables: Partial Caputo derivatives extend the construction to several variables, with each coordinate tk assigned an order αk(tk) in (0, 1).
3 Approximation of variable order Caputo derivatives
The paper approximates variable-order Caputo derivatives using expansions involving standard derivatives and derives upper bounds for the approximation errors. The formulas are developed for left and right operators and extended to partial derivatives.
- Upper-bound formulas are derived for the approximation errors under smoothness assumptions and for truncation orders N ≥ n.
- The derivations repeatedly use integration by parts to obtain the expansion formulas.
- The results include particular-case expansion formulas for the variable-order Caputo operators.
- Corresponding expansion and error formulas are provided for the right fractional operators as well as the left operators.
4 An example
The example tests variable-order Caputo derivative approximations using x(t)=t^2, comparing analytic derivatives with numerical results from the proposed theorems.
- The test function is x(t)=t^2 on t ∈[0, 1].
- The numerical tests use Theorems 10, 12, and 13 with fixed n=1 and N ∈ {2, 4, 6}.
- Figure 2 compares the analytic and numerical Type III left Caputo derivatives of order α(t) using Theorem 10.
- Figure 3 compares the analytic and numerical Type I left Caputo derivatives of order α(t) using Theorem 12.
5 Applications
The paper applies its variable-order Caputo approximations to time-fractional diffusion and fractional Burgers’ equations. The diffusion example compares analytic and numerical derivatives, while approximation accuracy improves as N increases.
- 5.1 A time-fractional diffusion equation: The proposed technique extends a one-dimensional time-fractional diffusion equation to variable order α(t) on [0,1]^2.The problem is posed with homogeneous boundary conditions u(0,t)=u(1,t)=0.
- 5.1 A time-fractional diffusion equation: Replacing the fractional derivative with the Theorem 10 approximation yields a system of second-order partial differential equations.The approximation uses n=1 and an arbitrary N≥1.
- 5.1 A time-fractional diffusion equation: As N increases, the approximation error decreases and the formula converges to the fractional derivative.The authors stopped at N=6 in plots because larger values made the approximation and exact solution visually indistinguishable.
- 5.2 A fractional partial differential equation in fluid mechanics: The methods are also applied to a variable-order linear inhomogeneous fractional Burgers’ equation used in gas dynamics, traffic flow, turbulence, and fluid mechanics.The example has exact solution u(x,t)=x^2+t^2, and its approximation error can be reduced by increasing N.