Fractional differential equations and related exact mechanical models (original) (raw)
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Generalized fractional derivatives and their applications to mechanical systems
Archive of Applied Mechanics, 2014
New fractional derivatives, termed henceforth generalized fractional derivatives (GFDs), are introduced. Their definition is based on the concept that fractional derivatives (FDs) interpolate the integer-order derivatives. This idea generates infinite classes of FDs. The new FDs provide, beside the fractional order, any number of free parameters to better calibrate the response of a physical system or procedure. Their usefulness and consequences are subject of further investigation. Like the Caputo FD, the GFDs allow the application of initial conditions having direct physical significance. A numerical method is also developed for the solution of differential equations involving GFDs. Mechanical systems including fractional oscillators, viscoelastic plane bodies and plates described by such equations are analyzed.
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In this paper, the variational iteration method and the Adomian decomposition method are implemented to give approximate solutions for linear and nonlinear systems of differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper presents a numerical comparison between the two methods for solving systems of fractional differential equations. Numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order. x
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This paper deals with the application of a novel variable-and constant-order fractional derivative with no singular kernel in the modeling of a mass-spring-damper system. The variable-order fractional derivative can be set as a smooth function, bounded on (0; 1], while the constant-order fractional derivative can be set as a fractional equation, bounded on (0; 1]. Our results show that the mechanical components exhibit viscoelastic behaviors producing temporal fractality at different scales. In the variable-order model, in contrast to the constant-order fractional mass-spring-damper system, the displacement changes with time. This means that the memory rate of the system changes with time and is determined by the current time instant. For different time periods we have different memory abilities. The integer-order classical model is recovered when the order of the fractional derivative is equal to 1.
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Mechanics Research Communications, 2016
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FRACTIONAL ORDER DYNAMICAL SYSTEMS AND ITS APPLICATIONS
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Fractal and Fractional
Modelling, simulation, and applications of Fractional Calculus have recently become increasingly popular subjects, with impressive growth concerning applications. The founding and limited ideas on fractional derivatives have achieved an incredibly valuable status. The variety of applications in mathematics, physics, engineering, economics, biology, and medicine, have opened new, challenging fields of research. For instance, in soil mechanics, a suitable definition of the fractional operator has shed some light on viscoelasticity, explaining memory effects on materials. Needless to say, these applications require the development of practical mathematical tools in order to extract quantitative information from models, newly reformulated in terms of fractional differential equations. Even confining ourselves to the field of ordinary differential equations, the well-known Bagley-Torvik model showed that fractional derivatives may actually arise naturally within certain physical models, and are not merely fanciful mathematical generalizations. This Special Issue focuses on the most recent advances in fractional calculus, applied to dynamic problems, linear and nonlinear fractional ordinary and partial differential equations, integral fractional differential equations, and stochastic integral problems arising in all fields of science, engineering, and other applied fields. In this issue, we have collected several significant papers devoted to applications of fractional methods with a focus on dynamical aspects. The applications range from theoretical mathematical-numerical aspects [1,2] to bio-medical subjects [3-7]. Applications to complex materials are investigated in [8], aiming at proposing a generalized definition of fractional operators. Special diffusion models are studied in [9-11].