Problem Of Anomalous Diffusion Research Articles

A novel distributed order time fractional model for heat conduction, anomalous diffusion, and viscoelastic flow problems

A novel distributed order time fractional model is constructed to solve heat conduction, anomalous diffusion and viscoelastic flow problems. Solutions of the formulated governing equation are obtained by the numerical discretization method that the mid-point quadrature rule is applied to approach the distributed order integrals and the L1-formula and L2-formula are used to discrete the time fractional derivatives. The stability and convergence of difference scheme with zero boundary conditions are studied and proven. In the end, three numerical examples are given. By employing an optimization algorithm, we obtain a good fitting result between the computational data with our model and heat conduction experimental data, which validates the effectiveness of the proposed model. Based on the background of the anomalous diffusion of volatile organic compound (VOC) adsorbed on building materials, the second example analyses the error and the convergence order by comparing the discrete solution with the exact solution, which validates the accuracy of the numerical difference method. Based on the solutions by using the difference method, the last example studies the unidirectional flows of generalized Oldroyd-B fluid driven by the moving plate or the pressure gradient and the effects of involved parameters on the velocity distribution are discussed graphically.

An ε-Approximate Approach for Solving Variable-Order Fractional Differential Equations

As a mathematical tool, variable-order (VO) fractional calculus (FC) was developed rapidly in the engineering field due to it better describing the anomalous diffusion problems in engineering; thus, the research of the solutions of VO fractional differential equations (FDEs) has become a hot topic for the FC community. In this paper, we propose an effective numerical method, named as the ε-approximate approach, based on the least squares theory and the idea of residuals, for the solutions of VO-FDEs and VO fractional integro-differential equations (VO-FIDEs). First, the VO-FDEs and VO-FIDEs are considered to be analyzed in appropriate Sobolev spaces H2n[0,1] and the corresponding orthonormal bases are constructed based on scale functions. Then, the space H2,02[0,1] is chosen which is just suitable for one of the models the authors want to solve to demonstrate the algorithm. Next, the numerical scheme is given, and the stability and convergence are discussed. Finally, four examples with different characteristics are shown, which reflect the accuracy, effectiveness, and wide application of the algorithm.

On stability and regularity for semilinear anomalous diffusion equations perturbed by weak-valued nonlinearities

Various diffusion processes in media with memory are depicted by anomalous diffusion equations. We are concerned with a class of anomalous diffusion equations with the nonlinearity taking values in Hilbert scales of negative order. The fundamental questions on global solvability, stability and regularity of solutions to the Cauchy problem governed by the mentioned equations are taken into account. We obtain some solvability results under different assumptions on the regularity of the nonlinear function. When the nonlinearity becomes less singular, the asymptotic stability of solutions is proved. Provided that the kernel function is 2-regular and sectorial, we show the Hölder continuity of the obtained solutions. Our approach is based on the resolvent theory, fixed point argument and embeddings of fractional Sobolev spaces. The applicability of our results is presented for some anomalous diffusion problems, where the nonlinear function is of polynomial or convection type.

Efficient numerical techniques for computing the Riesz fractional-order reaction-diffusion models arising in biology

In this work, the solution of Riesz space fractional partial differential equations of parabolic type is considered. Since fractional-in-space operators have been applied to model anomalous diffusion or dispersion problems in the area of mathematical physics with success, we are motivated in this paper to model the standard Brownian motion with the fractional order operator in the sense of the Riesz derivative. We formulate two viable, efficient and reliable high-order approximation schemes for the Riesz derivative which incorporated both the left- and right-hand sides of the Riemann-Liouville derivatives. The proposed methods are analyzed for both stability and convergence. Finally, the methods are used to explore the dynamic richness of pattern formation in two important fractional reaction-diffusion equations that are still of recurring interest. Experimental results for different values of the fractional parameters are reported.

Space–time fractional diffusion equations in d-dimensions

We analyze a new space–time fractional diffusion equation encompassing different diffusion processes in d-dimensions. The first-order time derivative is replaced with a time derivative of the Caputo type of arbitrary order β; the spatial-fractional operator of Riesz–Feller or Riesz–Weyl type is replaced with its extension to d-dimensions, defined by means of an extended Fourier transform. The mathematical problem with the spatial-fractional operator proposed here is formulated to tackle anomalous diffusion in heterogeneous media (fractal structures) and incorporating power-law distributions. A formal solution is proposed using the Green function method which, for appropriate initial and boundary conditions, can be expressed in terms of the generalized H-function of Fox—a typical track of anomalous diffusive processes. These mathematical tools provide a new powerful framework to model anomalous diffusion and relaxation problems in heterogeneous media.

Modeling multiple anomalous diffusion behaviors on comb-like structures

In this work, a generalized comb model which includes the memory kernels and linear reactions with irreversible and reversible parts are introduced to describe complex anomalous diffusion behavior. The probability density function (PDF) and the mean squared displacement (MSD) are obtained by analytical methods. Three different physical models are studied according to different reaction processes. When no reactions take place, we extend the diffusion process in 1-D under stochastic resetting to the N-D comb-like structures with backbone resetting and global resetting by using physically derived memory kernels. We find that the two different resetting ways only affect the asymptotic behavior of MSD in the long time. For the irreversible reaction, we obtain memory kernels based on experimental evidence of the transport of inert particles in spiny dendrites and explore the front propagation of CaMKII along dendrites. The reversible reaction plays an important role in the intermediate time, but the asymptotic behavior of MSD is the same with that in case of no reaction terms. The proposed reaction-diffusion model on the comb structure provides a generalized method for further study of various anomalous diffusion problems.

The y function applied in the study of an anomalous diffusion

In this article, we propose a new family of the extended analogues to the Y function for the first time. The relationships among the Y function, Fox H function, Meijer G function, Wright generalized hypergeometric function, and Clausen hypergeometric function are discussed in detail. This result is used to represent the solutions for the anomalous diffusion problems.

Investigation on the generalized thermoelastic-diffusive problem with variable properties in three different memory-dependent effect theories

The applicability of the generalized thermoelastic theories with integer-order derivative in solving anomalous or transient thermoelastic diffusion problems is questionable. Due to the feature of memory-dependence and heredity, the fractional-order derivative and the memory-dependent derivative are introduced to modify the generalized thermoelastic theories for extending their applicability. To demonstrate the features of such theories, the thermoelastic-diffusive dynamic response of an infinite elastic medium containing a spherical cavity with variable material properties is investigated in three different generalized theories with memory-dependent effect, which are incorporated in a unified form. Of them, two are based on fractional-order derivative and the other one is based on the memory-dependent derivative. The corresponding governing equations are formulated and then solved by Laplace transform together with its numerical inversion. The distributions of the non-dimensional temperature, displacement, stress, concentration and chemical potential under the three theories are obtained and illustrated graphically for comparing with those obtained from one of the integer-order theories. The effects of the fractional-order parameter, the time-delay factor, the variable thermal conductivity and diffusivity on the variations of the considered variables are considered and discussed in detail.

A boundary element method formulation based on the Caputo derivative for the solution of the anomalous diffusion problem

This work presents a boundary element method formulation for the solution of the anomalous diffusion problem. By keeping the fractional time derivative as it appears in the governing differential equation of the problem, and by employing a Weighted Residuals Method approach with the steady state fundamental solution for anisotropic media playing the role of the weighting function, one obtains the boundary integral equation of the proposed formulation. The presence of a domain integral with the fractional time derivative as part of its integrand, and the evaluation of this fractional time derivative as a Caputo derivative, constitute the main feature of the formulation. The analyses of some examples, in which the numerical results are always compared with the corresponding analytical solutions, show the robustness of the formulation, as accurate results are obtained even for small values of the order of the time derivative.

A meshless computational approach for solving two-dimensional inverse time-fractional diffusion problem with non-local boundary condition

This paper is devoted to investigating a two-dimensional inverse anomalous diffusion problem. The missing solely time-dependent Dirichlet boundary condition is recovered by imposing an additional integral measurement over the domain. An efficient computational technique based on a combination of a time integration scheme and local meshless Petrov–Galerkin method is implemented to solve the governing inverse problem. Firstly, an implicit time integration scheme is used to discretize the model in the temporal direction. To fully discretize the model, the primary spatial domain is represented by a set of distributed nodes and data-dependent basis functions are constructed by using the radial point interpolation method. Then, the local meshless Petrov–Galerkin method is used to discretize the problem in the spatial direction. Numerical examples are presented to verify the accuracy and efficiency of the proposed technique. The stability of the method is examined when the input data are contaminated with noise.

The boundary element method applied to the solution of the anomalous diffusion problem

A Boundary Element Method formulation is developed for the solution of the two-dimensional anomalous diffusion equation. Initially, the Riemann–Liouville Fractional derivative is applied on both sides of the partial differential equation (PDE), thus transferring the fractional derivatives to the Laplacian. The boundary integral equation is then obtained through a Weighted Residual formulation that employs the fundamental solution of the steady-state problem as the weighting function. The integral contained in the Riemann–Liouville formula is evaluated assuming that both the variable of interest and its normal derivative are constant in each time interval. Five examples are presented and discussed, in which the results from the proposed formulation are compared with the analytical solution, where available, otherwise with those furnished by a Finite Difference Method formulation. The analysis shows that the new method is capable of producing accurate results for a variety of problems, but small time steps are needed to capture the large temporal gradients that arise in the solution to problems governed by PDEs containing the fractional derivative ∂αu/∂tα with α < 0.5. The use of the steady-state fundamental solution presents no hindrance to the ability of the new formulation to provide accurate solutions to time-dependent problems, and the method is shown to outperform a finite difference scheme in providing highly accurate solutions, even for problems dominated by conditions within the material and remote from the domain boundary.

Solution of inverse anomalous diffusion problems using empirical and phenomenological models

In recent years, the anomalous diffusion phenomenon has attracted attention from the scientific community due to the number of applications and development of experimental procedures to observe this phenomenon. From a mathematical point of view, anomalous diffusion can be described by empirical (algebraic models) and phenomenological models (differential models with integer or fractional order). Considering the empirical models, one of the ways to observe the presence of anomalous diffusion is to check if the particle concentration profiles collapse around the reference profile, defined in terms of a new independent variable that depends on spatial and temporal variables. This new variable (scale factor) is used to characterize the diffusion type (classical or anomalous). For phenomenological models, diffusivity is usually considered as a function of particle concentration to characterize anomalous diffusion. For both cases, an inverse problem needs to be formulated and solved to obtain the parameters for each methodology. In this context, the present contribution aims to formulate and solve two inverse anomalous diffusion problems related to empirical and phenomenological models. For this purpose, two experimental data sets and Differential Evolution (DE) are considered. The results obtained demonstrate that the DE strategy was able to find good estimates for the scale factor and for the parameters related to the phenomenological model.

Lattice Boltzmann model for time sub-diffusion equation in Caputo sense

Anomalous diffusions, including subdiffusion and superdiffusion, are usually encountered in many diverse applications in science and engineering. Although many numerical methods have been proposed to study anomalous diffusion problems that are modeled by fractional advection-diffusion equations, in this paper, a fresh lattice Boltzmann (LB) model for time sub-diffusion equation in Caputo sense is proposed. Through the Chapman-Enskog analysis, the time-fractional diffusion equation can be recovered from the developed LB model. In addition, we also test the present LB model through some problems, and find that the numerical results agree well with the analytical solutions to these problems.

Fundamental solutions of anomalous diffusion equations with the decay exponential kernel

In the letter, we consider the anomalous diffusion equations involving the general derivatives with the decay exponential kernel for the first time. With the use of the Laplace transform, the analytical solutions with graphs are discussed in detail. The results are accurate and efficient in the description of the behaviors of the anomalous diffusion problems.

A new fractional derivative model for the anomalous diffusion problem

In this paper, a new fractional derivative within the exponential decay kernel is addressed for the first time. A new anomalous diffusion model is proposed to describe the heat-conduction problem. With the use of the Laplace transform, the analytical solution is discussed in detail. The presented result is as an accurate and efficient approach proposed for the heat-conduction problem in the complex phenomena.

Efficient Implementation of Rational Approximations to Fractional Differential Operators

This paper deals with some numerical issues about the rational approximation to fractional differential operators provided by the Pade approximants. In particular, the attention is focused on the fractional Laplacian and on the Caputo’s derivative which, in this context, occur into the definition of anomalous diffusion problems and of time fractional differential equations (FDEs), respectively. The paper provides the algorithms for an efficient implementation of the IMEX schemes for semi-discrete anomalous diffusion problems and of the short-memory-FBDF methods for Caputo’s FDEs.

Asymptotic Green’s functions for time-fractional diffusion equation and their application for anomalous diffusion problem

Asymptotic Green’s functions for short and long times for time-fractional diffusion equation, derived by simple and heuristic method, are provided in case if fractional derivative is presented in Caputo sense. The applicability of the asymptotic Green’s functions for solving the anomalous diffusion problem on a semi-infinite rod is demonstrated. The initial value problem for longtime solution of the time-fractional diffusion equation by Green’s function approach is resolved.

A classification of structural dissipations for evolution operators

In this paper, we study the asymptotic profile of the solution for a σ‐evolution equation with a time‐dependent structural damping. We introduce a classification of the damping term, which clarifies whether the solution behaves like the solution to an anomalous diffusion problem. We call this damping effective, whereas we say that the damping is noneffective when the solution shows oscillations in its asymptotic profile that cannot be neglected. Our classification shows a completely new interplay between the strength of the damping and the long time behavior of its coefficient. Copyright © 2015 John Wiley &amp; Sons, Ltd.

First passage time distributions of anomalous biased diffusion with double absorbing barriers

We investigate the first passage time (FPT) problem of anomalous diffusion governed by the Galilei variant fractional diffusion–advection equation in the semi-infinite and finite domains subject to an absorbing boundary condition. We obtain explicit solutions for the FPT distributions and the corresponding Laplace transforms for both zero and constant drift cases by using the method of separation of variables as well as the properties of the Fox H function. An important relation between the FPT distributions corresponding to one and two absorbing barriers is revealed to determine the conditional FPT distributions. It shows that the proportion between the conditional FPT distributions only depends on the general Péclet number. We further discuss the asymptotic behavior of the FPT distributions and confirm our theoretical analysis by numerical results.

Anomalous diffusion with transient subordinators: A link to compound relaxation laws

This paper deals with a problem of transient anomalous diffusion which is currently found to emerge from a wide range of complex processes. The nonscaling behavior of such phenomena reflects changes in time-scaling exponents of the mean-squared displacement through time domain - a more general picture of the anomalous diffusion observed in nature. Our study is based on the identification of some transient subordinators responsible for transient anomalous diffusion. We derive the corresponding fractional diffusion equation and provide links to the corresponding compound relaxation laws supported by this case generalizing many empirical dependencies well-known in relaxation investigations.