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Fractional Calculus and Fractional Differential Equations pp 59–75Cite as

Birkhäuser

Analysis of 2-Term Fractional-Order Delay Differential Equations

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Abstract

The value of a state variable at past time usually affects its rate of change at the present time. So, it is very natural to consider the delay while modeling the real-life systems. Further, the nonlocal fractional derivative operator is also useful in modeling memory in the system. Hence, the models involving delay as well as fractional derivative are very important. In this chapter, we review the basic results regarding the dynamical systems, fractional calculus, and delay differential equations. Further, we analyze 2-term nonlinear fractional-order delay differential equation \(D^\alpha x + c D^\beta x = f\left( x,x_\tau \right) \), with constant delay \(\tau >0\) and fractional orders \(0<\alpha<\beta <1\). We present a numerical method for solving such equations and present an example exhibiting chaotic oscillations.

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Acknowledgements

The author acknowledges the Council of Scientific and Industrial Research, New Delhi, India for funding through Research Project [25(0245)/15/EMR-II] and the Science and Engineering Research Board (SERB), New Delhi, India for the Research Grant [Ref. MTR/2017/000068] under Mathematical Research Impact Centric Support (MATRICS) Scheme.

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  1. Department of Mathematics, Shivaji University, Vidyanagar, Kolhapur, 416004, India

    Sachin Bhalekar

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  1. Sachin Bhalekar
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Correspondence to Sachin Bhalekar .

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  1. Department of Mathematics, Savitribai Phule Pune University, Pune, Maharashtra, India

    Varsha Daftardar-Gejji

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Bhalekar, S. (2019). Analysis of 2-Term Fractional-Order Delay Differential Equations. In: Daftardar-Gejji, V. (eds) Fractional Calculus and Fractional Differential Equations. Trends in Mathematics. Birkhäuser, Singapore. https://doi.org/10.1007/978-981-13-9227-6_4

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