Search results for " Fractional derivative"
showing 10 items of 11 documents
On the critical behavior for time-fractional pseudo-parabolic type equations with combined nonlinearities
2022
AbstractWe are concerned with the existence and nonexistence of global weak solutions for a certain class of time-fractional inhomogeneous pseudo-parabolic-type equations involving a nonlinearity of the form $|u|^{p}+\iota |\nabla u|^{q}$ | u | p + ι | ∇ u | q , where $p,q>1$ p , q > 1 , and $\iota \geq 0$ ι ≥ 0 is a constant. The cases $\iota =0$ ι = 0 and $\iota >0$ ι > 0 are discussed separately. For each case, the critical exponent in the Fujita sense is obtained. We point out two interesting phenomena. First, the obtained critical exponents are independent of the fractional orders of the time derivative. Secondly, in the case $\iota >0$ ι > 0 , we show that the gradie…
Infinitely many solutions to boundary value problem for fractional differential equations
2018
Variational methods and critical point theorems are used to discuss existence of infinitely many solutions to boundary value problem for fractional order differential equations where Riemann-Liouville fractional derivatives and Caputo fractional derivatives are used. An example is given to illustrate our result.
A fractional order theory of poroelasticity
2019
Abstract We introduce a time memory formalism in the flux-pressure constitutive relation, ruling the fluid diffusion phenomenon occurring in several classes of porous media. The resulting flux-pressure law is adopted into the Biot’s formulation of the poroelasticity problem. The time memory formalism, useful to capture non-Darcy behavior, is modeled by the Caputo’s fractional derivative. We show that the time-evolution of both the degree of settlement and the pressure field is strongly influenced by the order of Caputo’s fractional derivative. Also a numerical experiment aiming at simulating the confined compression test poroelasticity problem of a sand sample is performed. In such a case, …
Deterministic and Random Vibration of Linear Systems with Singular Parameter Matrices and Fractional Derivative Terms
2021
Both time- and frequency-domain solution techniques are developed for determining the response of linear multi-degree-of-freedom systems exhibiting singular parameter matrices and endowed with derivative terms of noninteger orders modeled as rational numbers. This is done based on the Moore-Penrose matrix inverse theory, in conjunction with a state variable formulation and with a complex modal analysis treatment. It is worth noting that, for the class of systems considered herein, this treatment also yields decoupled governing equations, thus facilitating further their numerical solution. Next, a generalization of the standard frequency-domain input-output (excitation-response) relationship…
Finite element method on fractional visco-elastic frames
2016
Viscoelastic behavior is defined by fractional operators.Quasi static FEM analysis of frames with fractional constitutive law is performed.FEM solution is decoupled into a set of fractional Kelvin Voigt elements.Proposed approach could be easily integrated in existing FEM codes. In this study the Finite Element Method (FEM) on viscoelastic frames is presented. It is assumed that the Creep function of the constituent material is of power law type, as a consequence the local constitutive law is ruled by fractional operators. The Euler Bernoulli beam and the FEM for the frames are introduced. It is shown that the whole system is ruled by a set of coupled fractional differential equations. In q…
Fractional differential calculus for 3D mechanically based non-local elasticity
2011
This paper aims to formulate the three-dimensional (3D) problem of non-local elasticity in terms of fractional differential operators. The non-local continuum is framed in the context of the mechanically based non-local elasticity established by the authors in a previous study; Non-local interactions are expressed in terms of central body forces depending on the relative displacement between non-adjacent volume elements as well as on the product of interacting volumes. The non-local, long-range interactions are assumed to be proportional to a power-law decaying function of the interaction distance. It is shown that, as far as an unbounded domain is considered, the elastic equilibrium proble…
A mechanical picture of fractional-order Darcy equation
2015
Abstract In this paper the authors show that fractional-order force-flux relations are obtained considering the flux of a viscous fluid across an elastic porous media. Indeed the one-dimensional fluid mass transport in an unbounded porous media with power-law variation of geometrical and physical properties yields a fractional-order relation among the ingoing flux and the applied pressure to the control section. As a power-law decay of the physical properties from the control section is considered, then the flux is related to a Caputo fractional derivative of the pressure of order 0 ⩽ β ≤ 1 . If, instead, the physical properties of the media show a power-law increase from the control sectio…
Response of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitations: A Path Integral approach based on La…
2023
In this paper, an approximate analytical technique is developed for determining the non-stationary response amplitude probability density function (PDF) of nonlinear/hysteretic oscillators endowed with fractional element and subjected to evolutionary excitations. This is achieved by a novel formulation of the Path Integral (PI) approach. Specifically, a stochastic averaging/linearization treatment of the original fractional order governing equation of motion yields a first-order stochastic differential equation (SDE) for the oscillator response amplitude. Associated with this first-order SDE is the Chapman–Kolmogorov (CK) equation governing the evolution in time of the non-stationary respon…
An exact thermodynamical model of power-law temperature time scaling
2016
In this paper a physical model for the anomalous temperature time evolution (decay) observed in complex thermodynamical system in presence of uniform heat source is provided. Measures involving temperatures T with power-law variation in time as T(t)∝tβ with β∈R shows a different evolution of the temperature time rate T(t) with respect to the temperature time-dependence T(t). Indeed the temperature evolution is a power-law increasing function whereas the temperature time rate is a power-law decreasing function of time. Such a behavior may be captured by a physical model that allows for a fast thermal energy diffusion close to the insulated location but must offer more resistance to the therm…
Free energy and states of fractional-order hereditariness
2014
AbstractComplex materials, often encountered in recent engineering and material sciences applications, show no complete separations between solid and fluid phases. This aspect is reflected in the continuous relaxation time spectra recorded in cyclic load tests. As a consequence the material free energy cannot be defined in a unique manner yielding a significative lack of knowledge of the maximum recoverable work that can extracted from the material. The non-uniqueness of the free energy function is removed in the paper for power-laws relaxation/creep function by using a recently proposed mechanical analogue to fractional-order hereditariness.