Analytical and Numerical Solutions of Fractional Differential Equations

Abstract

Differential equations are often used to explain the behaviours of real-life phenomena, and those are usually modelled by various differential equations with integer orders. Sometimes the behaviours of the physical problems may be advantageous to understand using non-integer order derivatives. In this regard, fractional calculus (FC) was introduced. Due to its hereditary and the description of memory properties, fractional-order models are more realistic and best suited in real phenomena than the integer-order models. 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. It deals with the differential and integral operators with non-integral powers. The fractional derivative has been used in various physical problems, such as frequency-dependent damping behaviour of structures, motion of a plate in a Newtonian fluid, controller for the control of dynamical systems, etc. 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.   D PI One of the most notable features of fractional derivatives is their distinctive nonlocal properties. This property allows to forecast the behaviours of phenomena by looking at their progress from the past to the present. Mostly used definitions of fractional calculus are Riemann-Liouville (RL) and Caputo fractional operators, defined by the convolution and the Power decay functions as kernel. Several researchers have extended the principles of fractional differential and integral operators to the fields related to various science and engineering problems using power-law distribution. However, when the fractional order is less than 1, this power-law distribution has no statistical significance. The fractional differential operators based on the power-law kernel meet certain classical conditions, such as index law, classical mechanical law, and singular kernels. It suggests that those operators based on the power-law kernel are physically weak and may not deal with more complex phenomena. Another problem is singularity which is challenging to explain in different natural phenomena. In order to address these problems, two important fractional derivatives, namely Caputo-Fabrizio and Atangana-Baleanu are developed. Although, these modern derivatives do not work under a power distribution, but they have nonsingular kernels. A generalized Mittag-Leffler function is used for Atangana–Baleanu derivative, and the Caputo-Fabrizio operator relies on the exponential law. Such operators have been able to model several scientific processes. Further, a new kind of operator was developed to represent two forms of fractional order, which reflect the fractional-order and the fractal dimension. The definitions of fractal-fractional differential and integral operators seem superior to the present fractional operators. One may obtain the fractal differential and integral operators when the fractional order is removed in the fractal-fractional differential and integral operators. Further, when the fractal dimension is neglected, then fractional derivatives and integrals are obtained. Hence, these fractal-fractional operators may catch more complexities than current operators as they have both exact and self-similar properties. Further, the uncertainties or randomness of the parameters and variables involved in the fractional systems are of serious concern. Investigations on a variety of fractional models are usually done by taking deterministic or crisp parameters, but the truth is quite diverse. The primary causes of the spread of uncertainty or randomness are defects in measurement, observations, environmental conditions, etc., which hinder the behaviour of models. As a matter of fact, these investigation anomalies indicate that the fractional models may not have the capability to demonstrate their normal behaviours. The influence of uncertainties becomes much more profound in the case of physical and structural problems due to the possibility of errors in the experiments or observations. In fact, several fractional physical and structural dynamics studies also support the claim of the possible inclusion of uncertainties in various parameters and initial conditions. In view of the above, the objective of this thesis has been to investigate a variety of fractional and fractal-fractional order models arise in i) wave dynamics, ii) fluid dynamics, iii) structural dynamics, iv) biology, v) economics, and vi) interpersonal relationship. In some of the problems, initial conditions and involved parameters are also considered as uncertain. Various computationally efficient analytical or numerical methods (where appropriate) are used/developed to investigate the models accordingly. Although a few methods have been developed by other researchers to analyse the above problems, but often those are problem dependent and are not efficient.