Int. J. Appl. Comput. Math (2020) 6:119 https://doi.org/10.1007/s40819-020-00872-9 ORIGINAL PAPER Fractional Order Models for Viscoelasticity in Lung Tissues with Power, Exponential and Mittag–Leffler Memories Priyanka Harjule 1 · Manish Kumar Bansal 2 © Springer Nature India Private Limited 2020 Abstract Various models on Viscoelasticity have been used to comprehend mechanics of lung tissues in a better way. In this paper we present efficient mathematical framework using new and mod- ified fractional derivatives to model the viscoelasticity in lung tissues. We demonstrate that replacing the time derivatives by fractionary-order derivatives in the constitutional expression of classical spring-dashpot system instinctively gives rise to power-law relaxation function and continuous-period impedance. Application of fractionary-order time derivative involving non-local as well as non-singular kernel is presented. Results obtained in this paper can be closely compared to the results obtained by Craiem et al. (Phys Med Biol 53:4543–4554, 2008). Keywords Stress relaxation · Tissue viscoelasticity · Fractional derivatives · Lung tissues impedance · Non-local kernel, Non-singular kernel · Mittag–Leffler function Introduction Generalization of integer-order calculus is Fractional calculus. The description of one-half order derivative was first observed by Leibniz in his letter to L’Hospital [2]. Inspite of such a long history, its only in the past decades that researchers across the globe have observed and described the applications of fractional derivatives and fractional differential equations in many areas of science and engineering. Use of fractional order derivatives to model a nonlinear phenomena encountered in several scientific areas includes biology, physics, chemistry, fuzzy control, automatic control, signal processing, and robotics (see for e.g. [3–10]) have shown accurate and better results than their integer-order counterparts because of more accurate representation of systems and physical phenomenon than integer order calculus. B Manish Kumar Bansal bansalmanish443@gmail.com Priyanka Harjule priyanka.maths@iiitkota.ac.in 1 Department of Mathematics, IIIT, Kota, MNIT Campus, Jaipur, Rajasthan 302017, India 2 Department of Applied Sciences, Engineering College, Banswara, Rajasthan 327001, India 0123456789().: V,-vol 123