An investigation of fractional Bagley-Torvik equation

Hossein Fazli,Juan J Nieto

doi:10.1515/math-2019-0040

Abstract

AbstractIn this paper the authors prove the existence as well as approximations of the solutions for the Bagley-Torvik equation admitting only the existence of a lower (coupled lower and upper) solution. Our results rely on an appropriate fixed point theorem in partially ordered normed linear spaces. Illustrative examples are included to demonstrate the validity and applicability of our technique.

Highlights

Fractional di erential equations appear naturally in a number of elds such as physics, geophysics, polymer rheology, regular variation in thermodynamics, biophysics, blood ow phenomena, aerodynamics, electrodynamics of complex medium, viscoelasticity, Bode’s analysis of feedback ampli ers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, tting of experimental data, nonlinear oscillation of earthquake, the uid-dynamic tra c model, etc
Fractional di erential equations serve as an excellent tool for the description of hereditary properties of various materials and processes [15, 16]
Fractional di erential equations are regarded as an alternative model to nonlinear di erential equations [17]

Summary

Introduction

Fractional di erential equations appear naturally in a number of elds such as physics, geophysics, polymer rheology, regular variation in thermodynamics, biophysics, blood ow phenomena, aerodynamics, electrodynamics of complex medium, viscoelasticity, Bode’s analysis of feedback ampli ers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, tting of experimental data, nonlinear oscillation of earthquake, the uid-dynamic tra c model, etc. Fractional di erential equations serve as an excellent tool for the description of hereditary properties of various materials and processes [15, 16]. Fractional di erential equations are regarded as an alternative model to nonlinear di erential equations [17]. The subject of fractional di erential equations is gaining much importance and attention. The Bagley-Torvik equation is a prototype fractional di erential equation which was proposed by Bagley and Torvik as an application of fractional calculus to the theory of viscoelasticity [18,19,20]. The governing equation is given by the fractional di erential equation

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Auxiliary facts and results

Note that Iα tβ

Main results

Observe that k