Presentation Title

Efficient Numerical Methods for Solving Variable Order Differential Equations

Faculty Mentor

Jeremy Orosco

Start Date

18-11-2017 12:30 PM

End Date

18-11-2017 1:30 PM

Location

BSC-Ursa Minor 104

Session

Poster 2

Type of Presentation

Poster

Subject Area

engineering_computer_science

Abstract

Fractional and variable order calculus deals with integrals and derivatives of arbitrary (i.e. non-integer) order. Additionally, fractional order control theory is a major topic that has been demonstrated to be able to suppress chaotic behaviors in mathematical models. The primary goals of the project are to verify empirically and analytically the efficacy of a hypothetical technique for reducing the computational complexity of a certain class of numerical methods that are used when solving Variable Order Differential Equations (VODE’s). We formulate/derive the VODE model for a hypothetical coupled two-mass system having both viscoelastic (q ∈ (0, 1)) and viscoinertial (q ∈ (1, 2)) elements, then we break this into two separate 4th order models (one for each mass). For clarity, the fourth order equations correspond to the coupled two-mass system where one of the masses is subject to variable viscoelastic damping q ε (0,1), and where q is the continuously variable derivative order. We use standard techniques and the novel proposed technique, simulate the dynamics (i.e. solve the VODE) at differing temporal grid granularities and confirm (by comparison) the utility of the proposed technique over the existing methods in the presence of viscoelastic damping, with and without viscoinertial damping. By comparison with standard methods, the efficacy of the proposed technique is empirically verified.

Summary of research results to be presented

We have found that our proposed numerical technique for solving Variable Order Differential Equations (VODEs) is approximately 50% faster than the conventional trapezoidal method.

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Nov 18th, 12:30 PM Nov 18th, 1:30 PM

Efficient Numerical Methods for Solving Variable Order Differential Equations

BSC-Ursa Minor 104

Fractional and variable order calculus deals with integrals and derivatives of arbitrary (i.e. non-integer) order. Additionally, fractional order control theory is a major topic that has been demonstrated to be able to suppress chaotic behaviors in mathematical models. The primary goals of the project are to verify empirically and analytically the efficacy of a hypothetical technique for reducing the computational complexity of a certain class of numerical methods that are used when solving Variable Order Differential Equations (VODE’s). We formulate/derive the VODE model for a hypothetical coupled two-mass system having both viscoelastic (q ∈ (0, 1)) and viscoinertial (q ∈ (1, 2)) elements, then we break this into two separate 4th order models (one for each mass). For clarity, the fourth order equations correspond to the coupled two-mass system where one of the masses is subject to variable viscoelastic damping q ε (0,1), and where q is the continuously variable derivative order. We use standard techniques and the novel proposed technique, simulate the dynamics (i.e. solve the VODE) at differing temporal grid granularities and confirm (by comparison) the utility of the proposed technique over the existing methods in the presence of viscoelastic damping, with and without viscoinertial damping. By comparison with standard methods, the efficacy of the proposed technique is empirically verified.