Fractional Diffusion Processes Research Articles

A hybrid yang transform adomian decomposition method for solving time-fractional nonlinear partial differential equation

Nonlinear time-fractional partial differential equations (NTFPDEs) play a great role in the mathematical modeling of real-world phenomena like traffic models, the design of earthquakes, fractional stochastic systems, diffusion processes, and control processing. Solving such problems is reasonably challenging, and the nonlinear part and fractional operator make them more problematic. Thus, developing suitable numerical methods is an active area of research. In this paper, we develop a new numerical method called Yang transform Adomian decomposition method (YTADM) by mixing the Yang transform and the Adomian decomposition method for solving NTFPDEs. The derivative of the problem is considered in sense of Caputo fractional order. The stability and convergence of the developed method are discussed in the Banach space sense. The effectiveness, validity, and practicability of the method are demonstrated by solving four examples of NTFPEs. The findings suggest that the proposed method gives a better solution than other compared numerical methods. Additionally, the proposed scheme achieves an accurate solution with a few numbers of iteration, and thus the method is suitable for handling a wide class of NTFPDEs arising in the application of nonlinear phenomena.

Inferring nonlinear fractional diffusion processes from single trajectories

We present a method to infer the arbitrary space-dependent drift and diffusion of a nonlinear stochastic model driven by multiplicative fractional Gaussian noise from a single trajectory. Our method, fractional Onsager-Machlup optimisation (fOMo), introduces a maximum likelihood estimator by minimising a field-theoretic action which we construct from the observed time series. We successfully test fOMo for a wide range of Hurst exponents using artificial data with strong nonlinearities, and apply it to a data set of daily mean temperatures. We further highlight the significant systematic estimation errors when ignoring non-Markovianity, underlining the need for nonlinear fractional inference methods when studying real-world long-range (anti-)correlated systems.

On new aspects of Chebyshev polynomials for space-time fractional diffusion process

Abstract Chebyshev collocation scheme and Finite difference method plays central roles for solving fractional differential equations (FDE). Therefore purpose of this paper is to solve fractional mathematical problem of diffusion by Chebyshev collocation method which turns the original problems into the system of fractional ordinary differential and algebraic equations by imposing orthogonality property. This system is solved by implementing Finite difference method. The numerical illustrations confirm that the combination of these two methods allow us to establish one of the best truncated solution in the series form.

MLE Evolution Equation for Fractional Diffusions and Berry-Esseen Inequality of Stochastic Gradient Descent Algorithm for American Option

We study recursive parameter estimation in fractional diffusion processes. First, stability and asymptotic properties of the global maximum likelihood estimator (MLE) of the drift parameter are obtained under some regularity conditions. Then we obtain an evolution equation for the MLE of the drift parameter in nonhomogeneous fractional stochastic differential equation (fSDE) driven by fractional Brownian motion. This equation is then modified to yield an algorithm which is consistent, asymptotically efficient and converges to the MLE. The gradient and Newton type algorithm are firstorder approximations. Finally we study the Berry-Esseen inequality for stochastic gradient descent in continuous time (SGDCT) algorithm for American option. We compare it with Longstaff-Schwartz regression based method.

Wright functions of the second kind and Whittaker functions

In the framework of higher transcendental functions, the Wright functions of the second kind have increased their relevance resulting from their applications in probability theory and, in particular, in fractional diffusion processes. Here, these functions are compared with the well-known Whittaker functions in some special cases of fractional order. In addition, we point out two erroneous representations in the literature.

Hydrodynamics and transport in the long-range-interacting φ 4 chain

We present a simulation study of the one-dimensional φ 4 lattice theory with long-range interactions decaying as an inverse power r −(1+σ) of the intersite distance r, σ > 0. We consider the cases of single and double-well local potentials with both attractive and repulsive couplings. The double-well, attractive case displays a phase transition for 0 < σ ⩽ 1 analogous to the Ising model with long-range ferromagnetic interactions. A dynamical scaling analysis of both energy structure factors and excess energy correlations shows that the effective hydrodynamics is diffusive for σ > 1 and anomalous for 0 < σ < 1, where fluctuations propagate superdiffusively. We argue that this is accounted for by a fractional diffusion process and we compare the results with an effective model of energy transport based on Lévy flights. Remarkably, this result is fairly insensitive on the phase transition. Nonequilibrium simulations with an applied thermal gradient are in quantitative agreement with the above scenario.

Averaged dynamics and control for heat equations with random diffusion

This paper deals with the averaged dynamics for heat equations in the degenerate case where the diffusivity coefficient, assumed to be constant, is allowed to take the null value. First we prove that the averaged dynamics is analytic. This allows to show that, most often, the averaged dynamics enjoys the property of unique continuation and is approximately controllable. We then determine if the averaged dynamics is actually null controllable or not depending on how the density of averaging behaves when the diffusivity vanishes. In the critical density threshold the dynamics of the average is similar to the 1 2 -fractional Laplacian, which is well-known to be critical in the context of the controllability of fractional diffusion processes. Null controllability then fails (resp. holds) when the density weights more (resp. less) in the null diffusivity regime than in this critical regime.

Fuzzy fractional diffusion processes and drug release

In this paper, the fuzzy fundamental triangular solution of the fractional diffusion equation under Caputo generalized Hukuhara partial differentiability by using the fuzzy Laplace transform and the fuzzy Fourier transform is obtained. As a practical model of the fractional diffusion equations, the fuzzy mathematical model of anti-cancer drug release in the tumor is investigated. Finally, given to the various examples represented, the efficacy and accuracy of the method are examined.

Capturing Subdiffusive Solute Dynamics and Predicting Selectivity in Nanoscale Pores with Time Series Modeling.

Fitting mathematical models with a direct connection to experimental observables to the outputs of molecular simulations can be a powerful tool for extracting important physical information from them. In this study, we present two new approaches that use stochastic time series modeling to predict long-time-scale behavior and macroscopic properties from molecular simulation, which can be generalized to other molecular systems where complex diffusion occurs. In our previous work, we studied long molecular dynamics (MD) simulation trajectories of a cross-linked HII phase lyotropic liquid crystal (LLC) membrane, where we observed subdiffusive solute transport behavior characterized by intermittent hops separated by periods of entrapment. In this work, we use our models to parameterize the behavior of the same systems, so we can generate characteristic trajectory realizations that can be used to predict solute mean-squared displacements (MSDs), solute flux, and solute selectivity in macroscopic length pores. First, using anomalous diffusion theory, we show how solute dynamics can be modeled as a fractional diffusion process subordinate to a continuous time random walk. From the MD simulations, we parameterize the distribution of dwell times, hop lengths between dwells, and correlation between hops. We explore two variations of the anomalous diffusion modeling approach. The first variation applies a single set of parameters to the solute displacements and the second applies two sets of parameters based on the solute's radial distance from the closest pore center. Next, we present an approach that generalizes Markov state models, treating the configurational states of the system as a Markov process where each state has distinct transport properties. For each state and transition between states, we parameterize the distribution and temporal correlation structure of positional fluctuations as a means of characterization and to allow us to predict solute MSDs. We show that both stochastic models reasonably reproduce the MSDs calculated from MD simulations. However, qualitative differences between MD and Markov state-dependent model-generated trajectories may in some cases limit their usefulness. With these parameterized stochastic models, we demonstrate how one can estimate the flux of a solute across a macroscopic length pore and, based on these quantities, the membrane's selectivity toward each solute. This work therefore helps to connect microscopic, chemically dependent solute motions that do not follow simple diffusive behavior with long-time-scale behavior, in an approach generalizable to many types of molecular systems with complex dynamics.

Nonparametric estimation for small fractional diffusion processes with random effects

We propose a nonparametric estimation of random effects from the following small fractional diffusions βi is random variable and is are standard fractional Brownian motions with a common known Hurst index The asymptotic behavior of the proposed estimators is established when ε and N are, respectively, sufficiently small and large.

Neumann boundary geometric control of a fractional diffusion process

In this paper, a boundary controller is developed for the fractional diffusion equation in the framework of geometric control. The case of Neumann actuation with a spatial weighted average output is addressed. This non-collocated configuration is characterised by an infinite characteristic index. To overcome this difficulty, the notion of the extended operator is exploited to derive an equivalent distributed control problem. The equivalent model, obtained by the Laplace transform in space domain, is used both for controller design and stability analysis of the resulting closed-loop. Thus, based on the notion of the characteristic index, a state feedback control that enforces an output tracking is designed. Then, the exponential stability of the closed-loop is demonstrated based on the semigroup theory. The effectiveness of the developed controller is shown, through numerical simulation, in the case of a heated aluminium rod that exhibits an anomalous diffusion phenomenon.

Neumann boundary geometric control of a fractional diffusion process

Neumann boundary geometric control of a fractional diffusion process

Remaining useful life prediction for fractional degradation processes under varying modes

AbstractDegradation processes of practical chemical engineering systems are difficult to model accurately because of complicated nonlinearities, non‐stationarities, and non‐Markovian properties. Traditional prognostics techniques tend to neglect the multi‐mode switching issue, and therefore cannot be used for predicting the remaining useful life (RUL) of a piece of long‐life equipment, especially when it is continuously operated under varying modes. Specifically, the stationarity of differential data also plays an important role in depicting the evolutionary trend, and further affects the extrapolation procedure for RUL prediction. In this paper, we construct a multi‐mode degradation model with regard to fractional diffusion processes by considering both stationary and non‐stationary increments. The drift term is represented as a time‐related nonlinear function, while the diffusion term is driven by fractional Brownian motion (FBM) or sub‐fractional Brownian motion (sub‐FBM), according to the results of the stationary test. Based on the monitored data, the model is identified by combining change‐point detection and parameter estimation. The closed‐form distribution of RUL is then derived through a weak convergence transformation. A numerical example and a case study of a large blast furnace are provided to evaluate the performance of the proposed prognostics framework.

Evolution fractional differential problems with impulses and nonlocal conditions

We obtain existence results for mild solutions of a fractional differential inclusion subjected to impulses and nonlocal initial conditions. By means of a technique based on the weak topology in connection with the Glicksberg-Ky Fan Fixed Point Theorem we are able to avoid any hypotheses of compactness on the semigroup and on the nonlinear term and at the same time we do not need to assume hypotheses of monotonicity or Lipschitz regularity neither on the nonlinear term, nor on the impulse functions, nor on the nonlocal condition. An application to a fractional diffusion process complete the discussion of the studied problem. 200 words.

Entropy Production Rates of the Multi-Dimensional Fractional Diffusion Processes

Our starting point is the n-dimensional time-space-fractional partial differential equation (PDE) with the Caputo time-fractional derivative of order and the fractional spatial derivative (fractional Laplacian) of order . For this equation, we first derive some integral representations of the fundamental solution and then discuss its important properties including scaling invariants and non-negativity. The time-space-fractional PDE governs a fractional diffusion process if and only if its fundamental solution is non-negative and can be interpreted as a spatial probability density function evolving in time. These conditions are satisfied for an arbitrary dimension if and additionally for in the one-dimensional case. In all these cases, we derive the explicit formulas for the Shannon entropy and for the entropy production rate of a fractional diffusion process governed by the corresponding time-space-fractional PDE. The entropy production rate depends on the orders and of the time and spatial derivatives and on the space dimension n and is given by the expression , t being the time variable. Even if it is an increasing function in , one cannot speak about any entropy production paradoxes related to these processes (as stated in some publications) because the time-space-fractional PDE governs a fractional diffusion process in all dimensions only under the condition , i.e., only the slow and the conventional diffusion can be described by this equation.

Novel technique of Atangana and Baleanu for heat dissipation in transmission line of electrical circuit

Abstract The new idea of Atangana and Baleanu introduced recently for fractional derivatives has received tremendous attention from researchers of various fields. However, this idea has not been applied for heat dissipation in transmission line of electrical circuit. Although the phenomenon of heat dissipation potentially damages the electrical devices which leads to catastrophic failure and shortens the life expectancy of costly electrical components. This manuscript signifies an electro-analog model of fractional diffusion process which enables to simulate heat dissipation in circuit board. The modeling of the problem has been established through modern operators of fractional calculus via the governing differential equation. By invoking Laplace transform on fractional governing equation of heat dissipation in transmission line of circuit, the analytic solutions are investigated through the exponential and Mittage Leffler kernel of non-integer order differentiations. The duality of analytic solutions has been compared with Caputo–Fabrizio and Atangana–Baleanu fractional differentiations graphically using embedded pertinent parameters. The comparison of both approaches of fractional differentiations has disclosed the heat transfer of circuit board and diffusion process with dissipation vividly.

Convergence of fractional diffusion processes in extension domains

We study the asymptotic behavior of anomalous fractional diffusion processes in bad domains via the convergence of the associated energy forms. We introduce the associated Robin–Venttsel’ problems for the regional fractional Laplacian. We provide a suitable notion of fractional normal derivative on irregular sets via a fractional Green formula as well as existence and uniqueness results for the solution of the Robin–Venttsel’ problem by a semigroup approach. Submarkovianity and ultracontractivity properties of the associated semigroup are proved.

From Power Laws to Fractional Diffusion Processes with and Without External Forces, the Non Direct Way

Abstract In this paper, a wide view on the theory of the continuous time random walk (CTRW) and its relations to the space–time fractional diffusion process is given. We begin from the basic model of CTRW (Montroll and Weiss [19], 1965) that also can be considered as a compound renewal process. We are interested in studying the random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. We prove the relation between the integral equation of the CTRW having the above fat tails waiting and jump width distributions and the space–time fractional diffusion equations in the Laplace–Fourier domain. The space–time fractional Fokker–Planck equation could also be driven from the discrete Ehren–Fest model and is represented by the theory of CTRW. These space–time fractional diffusion processes are getting increasing popularity in applications in physics, chemistry, finance, biology, medicine and many other fields. The asymptotic behavior of the Mittag–Leffler function plays a significant rule on simulating these models. The behaviors of the studied CTRW models are well approximated and visualized by simulating various types of random walks by using the Monte Carlo method.

Fractional powers of the noncommutative Fourier's law by the S‐spectrum approach

Let eℓ, for ℓ = 1,2,3, be orthogonal unit vectors in and let be a bounded open set with smooth boundary ∂Ω. Denoting by a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: urn:x-wiley:mma:media:mma5466:mma5466-math-0004 into the conservation of energy law, here a, b, are given functions. With the S‐spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, the fractional powers of T exist in the sense of the S‐spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory.

Parameter Estimation for Fractional Diffusion Process with Discrete Observations

This paper deals with the problem of estimating the parameters for fractional diffusion process from discrete observations when the Hurst parameter H is unknown. With combination of several methods, such as the Donsker type approximate formula of fractional Brownian motion, quadratic variation method, and the maximum likelihood approach, we give the parameter estimations of the Hurst index, diffusion coefficients, and volatility and then prove their strong consistency. Finally, an extension for generalized fractional diffusion process and further work are briefly discussed.