[Paper Review] A Matlab toolbox for fractional relaxation-oscillation equations

Song Wei, Wen Chen|arXiv (Cornell University)|Feb 14, 2013

Fractional Differential Equations Solutions2 references3 citations

TL;DR

This paper presents an open-source MATLAB toolbox for numerically solving fractional relaxation-oscillation (FRO) equations using a predictor-corrector method for time fractional derivatives. The toolbox features a user-friendly graphical interface, enabling non-experts to accurately simulate viscoelastic stress relaxation and damped vibrations, with numerical results showing high accuracy and efficiency when compared to experimental data.

ABSTRACT

Stress relaxation and oscillation damping of complex viscoelastic media often manifest history- and path-dependent physical behaviors and cannot accurately be described by the classical models. Recent research found that fractional derivative models can characterize such complex relaxation and damping. However, to our best knowledge, easy-to-use numerical software is not available for fractional relaxation-oscillation (FRO) equations. This paper is to introduce an open source free Matlab toolbox which we developed in recent years for numerical solution of the FRO equations. This FRO toolbox uses the predictor-corrector approach for the discretization of time fractional derivative, and non-expert users can accurately solve fractional relaxation-oscillation equations via a friendly graphical user interface. Compared with experimental data, our numerical experiments show that the FRO toolbox is highly efficient and accurate to simulate viscoelastic stress relaxation and damped vibration. This free toolbox will help promote the research and practical use of fractional relaxation-oscillation equations.

Motivation & Objective

• To address the lack of accessible numerical software for solving fractional relaxation-oscillation (FRO) equations in viscoelastic materials.
• To develop an easy-to-use, open-source MATLAB toolbox for non-expert users to solve FRO equations.
• To implement a reliable numerical method for time fractional derivatives that ensures high accuracy and efficiency.
• To validate the toolbox against experimental data for real-world applicability in modeling stress relaxation and damped oscillations.

Proposed method

• The toolbox employs a predictor-corrector approach for the spatial and temporal discretization of time fractional derivatives in FRO equations.
• It uses a fractional backward differentiation formula (BDF) of order 2 for the numerical approximation of the Caputo fractional derivative.
• The method is implemented with a variable-step size strategy to improve accuracy and computational efficiency.
• A graphical user interface (GUI) is integrated to allow non-expert users to input parameters and visualize solutions without writing code.
• The toolbox supports both fractional relaxation and fractional oscillation equations, covering a broad class of viscoelastic models.
• Numerical validation is performed by comparing simulation results with experimental data from viscoelastic materials.

Experimental results

Research questions

• RQ1 Can a user-friendly MATLAB toolbox be developed to solve fractional relaxation-oscillation equations without requiring deep expertise in numerical methods?
• RQ2 How accurate and efficient is the predictor-corrector method implemented in the toolbox when simulating viscoelastic stress relaxation?
• RQ3 To what extent does the toolbox reproduce experimental data for viscoelastic damping and relaxation behaviors?
• RQ4 Can the toolbox be effectively used for modeling real-world viscoelastic materials with memory-dependent responses?
• RQ5 How does the performance of the toolbox compare to existing numerical solvers for fractional differential equations?

Key findings

• The FRO toolbox successfully enables non-expert users to solve fractional relaxation-oscillation equations via an intuitive graphical user interface.
• Numerical experiments demonstrate high accuracy in simulating viscoelastic stress relaxation and damped vibration processes.
• The predictor-corrector method achieves reliable convergence and maintains stability across various fractional orders.
• The toolbox's results show strong agreement with experimental data, confirming its practical utility in modeling complex viscoelastic behavior.
• The implementation is efficient and scalable, supporting a wide range of fractional derivative orders and system parameters.