Fractional Relaxation Equation Research Articles

Hadamard fractional discrete-time relaxation equation’s solutions and asymptotic stability

This paper suggests two numerical schemes for the famous fractional logistic equations by use of the time scale theory. First, the Hadamard discrete fractional calculus is introduced. Leibniz integral law is used to obtain the fractional sum equation. Then, Hadamard fractional derivative and integral’s numerical schemes are obtained and employed for numerical solutions. It can be concluded the fractional differential equation on time scales is very suitable for both continuous and discrete–time modeling.

The Limited Validity of the Fractional Euler Finite Difference Method and an Alternative Definition of the Caputo Fractional Derivative to Justify Modification of the Method

A method, advanced as the fractional Euler finite difference method (FEFDM), a general method for the finite difference discretization of fractional initial value problems (IVPs) for 0<α≤1 for the Caputo derivative, is shown to be valid only for α=1. This is accomplished by establishing, through a recently proposed generalized difference quotient representation of the fractional derivative, that the FEFDM is valid only if a property of the Mittag-Leffler function holds that has only been shown to be valid only for α=1. It is also shown that the FEFDM is inconsistent with the exact discretization of the IVP for the Caputo fractional relaxation equation. The generalized derivative representation is also used to derive a modified generalized Euler’s method, its nonstandard finite difference alternative, their improved Euler versions, and to recover a recent result by Mainardi relating the Caputo and conformable derivatives.

Finite-dimensional exact controllability of an abstract semilinear fractional composite relaxation equation

In Hilbert space, the finite-dimensional exact controllability of an abstract semilinear fractional composite relaxation equation is researched. We make assumptions about the parameters in the equation and suppose that the linear equation associated with the abstract semilinear fractional relaxation equation is approximately controllable. We apply the variational method, the resolvent theory and the fixed point trick to demonstrate the finite-dimensional exact controllability of the abstract semilinear equation. An application is given in the last paper to testify our results.

Fractional relaxation noises, motions and the fractional energy balance equation

Abstract. We consider the statistical properties of solutions of the stochastic fractional relaxation equation and its fractionally integrated extensions that are models for the Earth's energy balance. In these equations, the highest-order derivative term is fractional, and it models the energy storage processes that are scaling over a wide range. When driven stochastically, the system is a fractional Langevin equation (FLE) that has been considered in the context of random walks where it yields highly nonstationary behaviour. An important difference with the usual applications is that we instead consider the stationary solutions of the Weyl fractional relaxation equations whose domain is −∞ to t rather than 0 to t. An additional key difference is that, unlike the (usual) FLEs – where the highest-order term is of integer order and the fractional term represents a scaling damping – in the fractional relaxation equation, the fractional term is of the highest order. When its order is less than 1/2 (this is the main empirically relevant range), the solutions are noises (generalized functions) whose high-frequency limits are fractional Gaussian noises (fGn). In order to yield physical processes, they must be smoothed, and this is conveniently done by considering their integrals. Whereas the basic processes are (stationary) fractional relaxation noises (fRn), their integrals are (nonstationary) fractional relaxation motions (fRm) that generalize both fractional Brownian motion (fBm) as well as Ornstein–Uhlenbeck processes. Since these processes are Gaussian, their properties are determined by their second-order statistics; using Fourier and Laplace techniques, we analytically develop corresponding power series expansions for fRn and fRm and their fractionally integrated extensions needed to model energy storage processes. We show extensive analytic and numerical results on the autocorrelation functions, Haar fluctuations and spectra. We display sample realizations. Finally, we discuss the predictability of these processes which – due to long memories – is a past value problem, not an initial value problem (that is used for example in highly skillful monthly and seasonal temperature forecasts). We develop an analytic formula for the fRn forecast skills and compare it to fGn skill. The large-scale white noise and fGn limits are attained in a slow power law manner so that when the temporal resolution of the series is small compared to the relaxation time (of the order of a few years on the Earth), fRn and its extensions can mimic a long memory process with a range of exponents wider than possible with fGn or fBm. We discuss the implications for monthly, seasonal, and annual forecasts of the Earth's temperature as well as for projecting the temperature to 2050 and 2100.

Non-equidistant partition predictor–corrector method for fractional differential equations with exponential memory

Abstract A new class of fractional differential equations with exponential memory was recently defined in the space A C δ n [ a , b ] $A{C}_{\delta }^{n}\left[a,b\right]$ . In order to use the famous predictor–corrector method, a new quasi-linear interpolation with a non-equidistant partition is suggested in this study. New Euler and Adams–Moulton methods are proposed for the fractional integral equation. Error estimates of the generalized fractional integral and numerical solutions are provided. The predictor–corrector method for the new fractional differential equation is developed and numerical solutions of fractional nonlinear relaxation equation are given. It can be concluded that the non-equidistant partition is needed for non-standard fractional differential equations.

Existence and uniqueness of positive solutions for fractional relaxation equation in terms of ψ-Caputo fractional derivative

Abstract This manuscript is committed to deal with the existence and uniqueness of positive solutions for fractional relaxation equation involving ψ-Caputo fractional derivative. The existence of solution is carried out with the help of Schauder’s fixed point theorem, while the uniqueness of the solution is obtained by applying the Banach contraction principle, along with Bielecki type norm. Moreover, two explicit monotone iterative sequences are constructed for the approximation of the extreme positive solutions to the proposed problem. Lastly, two examples are presented to support the obtained results.

Existence, uniqueness, and approximate solutions for the general nonlinear distributed-order fractional differential equations in a Banach space

The purpose of this paper is to provide sufficient conditions for the local and global existence of solutions for the general nonlinear distributed-order fractional differential equations in the time domain. Also, we provide sufficient conditions for the uniqueness of the solutions. Furthermore, we use operational matrices for the fractional integral operator of the second kind Chebyshev wavelets and shifted fractional-order Jacobi polynomials via Gauss–Legendre quadrature formula and collocation methods to reduce the proposed equations into systems of nonlinear equations. Also, error bounds and convergence of the presented methods are investigated. In addition, the presented methods are implemented for two test problems and some famous distributed-order models, such as the model that describes the motion of the oscillator, the distributed-order fractional relaxation equation, and the Bagley–Torvik equation, to demonstrate the desired efficiency and accuracy of the proposed approaches. Comparisons between the methods proposed in this paper and the existing methods are given, which show that our numerical schemes exhibit better performances than the existing ones.

Black swans, extreme risks, and the e-pile model of self-organized criticality

A new model of self-organized criticality (SOC) is described which differs from ordinary sand-pile SOC models in that it uses electrically charged particles of different kinds to propagate activities and to generate the critical state. The model, arguably called the e-pile model (as an alternative to sand-pile, with “e-” standing for “electric charge”), is motivated by the problem of dielectric relaxation in self-asembling random lattices with disorder under the action of electrostatic forces, but in principle it may be applied to different SOC processes with the random injection scheme and charitable lattice-redistribution rule. We show that the critical state is that of self-evolving random percolation clusters at the edge of percolation and is also different from known “self-organized” versions of the percolation problem based on the directed percolation. A set of critical exponents is obtained based on the random walks, using the Kramers-Kronig relation and the formalism of frequency-dependent complex conductivity. The relaxation of a supercritical system to SOC is shown to obey the Mittag-Leffler pattern and fractional relaxation equation, with a broad distribution of durations of relaxation events. A peculiar feature of the e-pile system is that it is characterized by an intrinsic bias disregarding a vast majority of incipient relaxation events (by killing them “in their egg”), while it also favors other events that may, then, grow into extremely large sizes. We use this observation to explain the phenomenon of the “black swan,” by which one means a family of rare, large events, whose sudden occurrence is very difficult to predict, despite the important impact this type of event may have over the entire system. The event-size distribution of the black swans is found to be a power law, with the drop-off exponent being sensitive to the complexity features of the underlying percolation clusters. On the basis of this observation, we show that the black swans are less probable in more complex environments, and we use this argument to explain why driven, dissipative systems would develop complex structures as a result of dynamical evolution. The e-pile model reveals the upper critical dimension dc=6 for which a crossover to mean-field SOC is found. In six and higher dimensions, the probability density to observe a black swan of a given size behaves as the inverse cube of the size. Below dc the decay is always slower than the inverse-cube and is flatter at lower dimensions. The implications of the mean-field law in country risk assessment, business, and finance are discussed. Finally, we show that the occurrence of the power-law tails is limited to the rate of the driving, and that for too high a rate a different branch of extreme events may result, with the defining features enabling to associate this branch with the phenomenon of dragon kings.

Upper and Lower Solution method for Positive solution of generalized Caputo fractional differential equations

In this research paper, the nonlinear fractional relaxation equation involving the generalized Caputo derivative is reduced to an equivalent integral equation via the generalized Laplace transform. Moreover, the upper and lower solutions method combined with some fixed point theorems, and the properties of the Mittag-Leffler function are applied to investigate the existence and uniqueness of positive solutions for the problem at hand. At the end, to illustrate our results, we give an example.

On Complete Monotonicity of Solution to the Fractional Relaxation Equation with the nth Level Fractional Derivative

In this paper, we first deduce the explicit formulas for the projector of the nth level fractional derivative and for its Laplace transform. Afterwards, the fractional relaxation equation with the nth level fractional derivative is discussed. It turns out that, under some conditions, the solutions to the initial-value problems for this equation are completely monotone functions that can be represented in form of the linear combinations of the Mittag–Leffler functions with some power law weights. Special attention is given to the case of the relaxation equation with the second level derivative.

Qualitative analysis of fractional relaxation equation and coupled system with Ψ-Caputo fractional derivative in Banach spaces

<abstract> <p>The aim of the reported results in this manuscript is to handle the existence, uniqueness, extremal solutions, and Ulam-Hyers stability of solutions for a class of $ \Psi $-Caputo fractional relaxation differential equations and a coupled system of $ \Psi $-Caputo fractional relaxation differential equations in Banach spaces. The obtained results are derived by different methods of nonlinear analysis like the method of upper and lower solutions along with monotone iterative technique, Banach contraction principle, and Mönch's fixed point theorem concerted with the measures of noncompactness. Furthermore, the Ulam-Hyers stability of the proposed system is studied. Finally, two examples are presented to illustrate our theoretical findings. Our acquired results are recent in the frame of a $ \Psi $-Caputo derivative with initial conditions in Banach spaces via the monotone iterative technique. As a results, we aim to fill this gap in the literature and contribute to enriching this academic area.</p> </abstract>

A fractional derivative perspective on transient pulse test for determining the permeability of rocks

This work presents a fractional relaxation equation that, in the full respect of the time-dependent behavior of flow transport and under appropriate simplifying assumptions, leads to an accurate description of the differential pressure decay in transient pulse test by the Mittag-Leffler function. The permeability of granite at different water pressures is measured. Moreover, the results of experimental data interpretation illustrate that the transient pulse test based on the Mittag-Leffler law fitting provides a much better determination of permeability than the conventional exponential decay model. The relationship between the presented fractional relaxation equation and fractional diffusion equation is also discussed in the framework of continuous time random walk.

Estimates for a general fractional relaxation equation and application to an inverse source problem

A general fractional relaxation equation is considered with a convolutional derivative in time introduced by A. Kochubei (Integr. Equ. Oper. Theory 71 (2011), 583–600). This equation generalizes the single‐term, multiterm, and distributed‐order fractional relaxation equations. The fundamental and the impulse‐response solutions are studied in detail. Properties such as analyticity and subordination identities are established and employed in the proof of an upper and a lower bound. The obtained results extend some known properties of the Mittag‐Leffler functions. As an application of the estimates, uniqueness and conditional stability are established for an inverse source problem for the general time‐fractional diffusion equation on a bounded domain.

Improvements in a method for solving fractional integral equations with some links with fractional differential equations

Abstract In this work, we apply and extend our ideas presented in [4] for solving fractional integral equations with Riemann-Liouville definition. The approach made in [4] turned any linear fractional integral equation with constant coefficients and rational orders into a similar one, but with integer orders. If the right hand side was smooth enough we could differentiate at both sides to arrive to a linear ODE with constant coefficients and some initial conditions, that can be solved via an standard procedure. In this procedure, there were two major obstacles that did not allow to obtain a full result. These were the assumptions over the smoothness of the source term and the assumption about the rationality of the orders. So, one of the main topics of this document is to describe a modification of the procedure presented in [4], when the source term is not smooth enough to differentiate the required amount of times. Furthermore, we will also study the fractional integral equations with non-rational orders by a limit process of fractional integral equations with rational orders. Finally, we will connect the previous material with some fractional differential equations with Caputo derivatives described in [7]. For instance, we will deal with the fractional oscillation equation, the fractional relaxation equation and, specially, its particular case of the Basset problem. We also expose how to compute these solutions for the Riemann-Liouville case.

A Note on the Equivalence of Fractional Relaxation Equations to Differential Equations with Varying Coefficients

In this note, we show how an initial value problem for a relaxation process governed by a differential equation of a non-integer order with a constant coefficient may be equivalent to that of a differential equation of the first order with a varying coefficient. This equivalence is shown for the simple fractional relaxation equation that points out the relevance of the Mittag–Leffler function in fractional calculus. This simple argument may lead to the equivalence of more general processes governed by evolution equations of fractional order with constant coefficients to processes governed by differential equations of integer order but with varying coefficients. Our main motivation is to solicit the researchers to extend this approach to other areas of applied science in order to have a deeper knowledge of certain phenomena, both deterministic and stochastic ones, investigated nowadays with the techniques of the fractional calculus.

About a composite fractional relaxation equation via regularized families

This work deals with asymptotic periodicity and compactness for a class of composite fractional relaxationequation. Some difficulties arises when the effect of different kinds of nonhomogeneous terms are taken intoconsideration. To overcome these we use methods coming from regularized families and fixed point techniques,which are an important tool to study of nonlinear phenomena. We can cover a large class of nonlinearities.

Probabilistic solutions to nonlinear fractional differential equations of generalized Caputo and Riemann–Liouville type

This paper provides well-posedness and integral representations of the solutions to nonlinear equations involving generalized Caputo and Riemann–Liouville type fractional derivatives. As particular cases, we study the linear equation with non constant coefficients and the generalized composite fractional relaxation equation. Our approach relies on the probabilistic representation of the solution to the generalized linear problem recently obtained by the authors. These results encompass some known cases in the context of classical fractional derivatives, as well as their far reaching extensions including various mixed derivatives.

Positive Solutions of the Fractional Relaxation Equation Using Lower and Upper Solutions

In this paper, by means of some fixed point theorems, we establish the existence and uniqueness of positive solution of the fractional relaxation equation. The analysis is based on the method of upper and lower solutions. The results are illustrated by a numerical example using MATLAB.

Approximate controllability for semilinear composite fractional relaxation equations

Abstract We consider a control system governed by a semilinear composite fractional relaxation equation in Hilbert space. We first prove that the system has a mild solution. Then, we investigate the approximate controllability of the relaxation equation under the assumption that the corresponding linear system is approximately controllable. An example is also given to illustrate our results.

Toward an innovative stochastic modeling of electric charges loss through dielectric

This paper deals with new stochastic modeling of very low tunneling currents in Non-Volatile Memories. For this purpose, we first develop current measurement method based on Floating Gate technique. In order to reach the long time behavior of electrical dynamic, we aim at using very basic tools (power supply, multimeter...) but still having a very good current resolution. Also, our measurement is led in a very particular low-noise environment (underground laboratory) allowing to keep the electrical contacts on the device under test as long as possible. After showing the feasibility of such measurements, we present a modeling approach of the charge loss process inside the Non-volatile Memories by using mathematical tool involving long memory effect. The model is based on stochastic counting process with memory effect yielding to a fractional relaxation equation for the charge loss over time. The main interest of the present model lies in the fact that the corresponding inversion problem involves only two parameters that can be carried out efficiently.