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Fractional Order Mathematical Model for the Cell Cycle of a Tumour Cell
Published in Devendra Kumar, Jagdev Singh, Fractional Calculus in Medical and Health Science, 2020
Ritu Agarwal, Kritika, Sunil Dutt Purohit
For tumour growth, a number of models have been already studied, which are governed by ordinary differential equations. But the mathematical formulation of these models was lacking in non-locality effect and hence has declined to model the real-world problems. The concept of fractional calculus, which is a natural generalisation of classical calculus and the operators of differentiation, is a non-local operator. It is only in the past decade or so that fractional calculus has drawn the attention of main-stream science as a way to describe the dynamics of complex phenomena. Since, in modelling real-world problems [9–25] fractional operators play important roles. Bagley and Torvik [26] showed that the models of fractional order are more convenient than the integer order models. Fractional calculus provides a concise model for the description of the dynamic events that occur in biological tissues. Such a description is important for gaining an understanding of the underlying multiscale processes that occur. Many researchers have studied biological models possessing derivatives of arbitrary order and also found the numerical solution for them. Using a fractional advection diffusion equation, Agarwal et al. [27] studied the concentration profile of calcium. Arshad et al. [28] proposed a fractional model for human immunodeficiency virus (HIV) infection.
Nonlinear vibrations of fractional nonlocal viscoelastic nanotube resting on a Kelvin–Voigt foundation
Published in Mechanics of Advanced Materials and Structures, 2022
In the last decades, the concept of the fractional order calculus has been developed for the dynamic analysis of the viscoelastic nanobeams [18, 19]. The most important advantage of the fractional derivative approach in applications is their nonlocal property [20–23]. It is well known that the integer order differential operator is a local operator, but the fractional order differential operator is a nonlocal operator. This means that the next state of the system depends not only upon its current state, but also upon all of its historical states. This is more realistic and it is one reason why the fractional calculus has become more and more popular. A more extensive review of the fractional differential equations can be found in Podlubny’s work [24]. It is noted that the CNTs are embedded in a medium in many of their applications in nanotechnology. Such a system can be reasonably treated as a nanobeam resting on a foundation is presented by Cajic et al. [25].
3D static bending analysis of functionally graded piezoelectric microplates resting on an elastic medium subjected to electro-mechanical loads using a size-dependent Hermitian C 2 finite layer method based on the consistent couple stress theory
Published in Mechanics Based Design of Structures and Machines, 2023
Some potential computational methods in mechanics developed recently are worth mentioning here. Ren, Zhuang, and Rabczuk (2020) proposed a nonlocal operator method (NOM) for solving partial differential equations (PDEs) of mechanical problems. They indicated that the nonlocal operator could be regarded as the integral form, equivalent to the differential form in solving a nonlocal interaction model of an unknown field, such that the local PDEs can be converted into a nonlocal integral form to solve. Rabczuk, Ren, and Zhuang (2019) applied the NOM to analyze the electromagnetic waveguide problem, for which an hourglass energy functional was introduced to eliminate the zero-energy modes. Samaniego et al. (2020) explored a deep energy method (DEM), for which the artificial neural networks (ANNs) were regarded as function approximation machines to minimize an energy functional corresponding to the PDE of interest. Afterward, the DEM was successfully applied to some representative mechanical problems, including static and dynamic linear elasticity problems, phase field modeling of fracture, piezoelectricity problems, and Kirchhoff plate bending problems. Guo, Rabczuk, and Zhuang (2019) explored a deep collocation method (DCM), for which the ANNs were regarded as function approximation machines to minimize a loss function characterizing the error forms of the PDE and its associated boundary conditions. Afterward, the DCM was illustrated as suitable for the Kirchhoff plate’s bending analysis with various geometries. Finally, Zhuang et al. (2021) developed a deep autoencoder-based energy method (DAEM) for static bending, free vibration, and elastobuckling analyses of Kirchhoff plates.
A generalized thermoelastic diffusion problem of thin plate heated by the ultrashort laser pulses with memory-dependent and spatial nonlocal effect
Published in Journal of Thermal Stresses, 2021
Yan Li, Tianhu He, Pengfei Luo, Xiaogeng Tian
The introduction of nonlocal operator will lead to the differential-integral governing equation, which is difficult to solve. In order to simplify it, Eringen [24] introduced new kernel function and nonlocal operator, and rewrote the integral constitutive equation into differential form