Fractional Sensitivity Equation Method: Applications to Fractional Model Construction

TL;DR

This paper introduces a fractional sensitivity equation method for analyzing and constructing fractional models, enabling accurate parameter estimation through adjoint equations and a spectral numerical approach.

Contribution

It develops a novel fractional sensitivity analysis framework with adjoint equations and a spectral solver for fractional model construction.

Findings

Derived adjoint fractional sensitivity equations with a new fractional operator.

Implemented a gradient-based algorithm for precise parameter estimation.

Developed a stable Petrov-Galerkin spectral method for solving coupled fractional systems.

Abstract

Fractional differential equations provide a tractable mathematical framework to describe anomalous behavior in complex physical systems, yet they introduce new sensitive model parameters, i.e. derivative orders, in addition to model coefficients. We formulate a sensitivity analysis of fractional models by developing a fractional sensitivity equation method. We obtain the adjoint fractional sensitivity equations, in which we present a fractional operator associated with logarithmic-power law kernel. We further construct a gradient-based optimization algorithm to compute an accurate parameter estimation in fractional model construction. We develop a fast, stable, and convergent Petrov-Galerkin spectral method to numerically solve the coupled system of original fractional model and its corresponding adjoint fractional sensitivity equations.