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Computation and Modeling for Fractional Order Systems provides
readers with problem-solving techniques for obtaining exact and/or
approximate solutions of governing equations arising in fractional
dynamical systems presented using various analytical,
semi-analytical, and numerical methods. In this regard, this book
brings together contemporary and computationally efficient methods
for investigating real-world fractional order systems in one
volume. Fractional calculus has gained increasing popularity and
relevance over the last few decades, due to its well-established
applications in various fields of science and engineering. It deals
with the differential and integral operators with non-integral
powers. Fractional differential equations are the pillar of various
systems occurring in a wide range of science and engineering
disciplines, namely physics, chemical engineering, mathematical
biology, financial mathematics, structural mechanics, control
theory, circuit analysis, and biomechanics, among others. The
fractional derivative has also been used in various other physical
problems, such as frequency-dependent damping behavior of
structures, motion of a plate in a Newtonian fluid, PID controller
for the control of dynamical systems, and many others. The
mathematical models in electromagnetics, rheology, viscoelasticity,
electrochemistry, control theory, Brownian motion, signal and image
processing, fluid dynamics, financial mathematics, and material
science are well defined by fractional-order differential
equations. Generally, these physical models are demonstrated either
by ordinary or partial differential equations. However, modeling
these problems by fractional differential equations, on the other
hand, can make the physics of the systems more feasible and
practical in some cases. In order to know the behavior of these
systems, we need to study the solutions of the governing fractional
models. The exact solution of fractional differential equations may
not always be possible using known classical methods. Generally,
the physical models occurring in nature comprise complex phenomena,
and it is sometimes challenging to obtain the solution (both
analytical and numerical) of nonlinear differential equations of
fractional order. Various aspects of mathematical modeling that may
include deterministic or uncertain (viz. fuzzy or interval or
stochastic) scenarios along with fractional order
(singular/non-singular kernels) are important to understand the
dynamical systems. Computation and Modeling for Fractional Order
Systems covers various types of fractional order models in
deterministic and non-deterministic scenarios. Various
analytical/semi-analytical/numerical methods are applied for
solving real-life fractional order problems. The comprehensive
descriptions of different recently developed fractional singular,
non-singular, fractal-fractional, and discrete fractional
operators, along with computationally efficient methods, are
included for the reader to understand how these may be applied to
real-world systems, and a wide variety of dynamical systems such as
deterministic, stochastic, continuous, and discrete are addressed
by the authors of the book.
The subject of fractional calculus has gained considerable
popularity and importance during the past three decades, mainly due
to its validated applications in various fields of science and
engineering. It is a generalization of ordinary differentiation and
integration to arbitrary (non-integer) order. The fractional
derivative has been used in various physical problems, such as
frequency-dependent damping behavior of structures, biological
systems, motion of a plate in a Newtonian fluid, controller for the
control of dynamical systems, and so on. It is challenging to
obtain the solution (both analytical and numerical) of related
nonlinear partial differential equations of fractional order. So
for the last few decades, a great deal of attention has been
directed towards the solution for these kind of problems. Different
methods have been developed by other researchers to analyze the
above problems with respect to crisp (exact) parameters. However,
in real-life applications such as for biological problems, it is
not always possible to get exact values of the associated
parameters due to errors in measurements/experiments, observations,
and many other errors. Therefore, the associated parameters and
variables may be considered uncertain. Here, the uncertainties are
considered interval/fuzzy. Therefore, the development of
appropriate efficient methods and their use in solving the
mentioned uncertain problems are the recent challenge. In view of
the above, this book is a new attempt to rigorously present a
variety of fuzzy (and interval) time-fractional dynamical models
with respect to different biological systems using computationally
efficient method. The authors believe this book will be helpful to
undergraduates, graduates, researchers, industry, faculties, and
others throughout the globe.
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