Fractional Order Diffusion Research Articles

A New Kind of Parallel Natural Difference Method for Multi-Term Time Fractional Diffusion Model

Multi-term time fractional diffusion model is not only an important physical subject, but also a practical problem commonly involved in engineering. In this paper, we apply the alternating segment technique to combine the classical explicit and implicit schemes, and propose a parallel nature difference method alternating segment pure explicit–implicit (PASE-I) and alternating segment pure implicit–explicit (PASI-E) difference schemes for multi-term time fractional order diffusion equations. The existence and uniqueness of the solutions are proved, and stability and convergence analysis of the two schemes are also given. Theoretical analyses and numerical experiments show that the PASE-I and PASI-E schemes are unconditionally stable and satisfy second-order accuracy in spatial precision and 2 − α order in time precision. When the computational accuracy is equivalent, the CPU time of the two schemes are reduced by up to 2 / 3 compared with the classical implicit difference method. It indicates that the PASE-I and PASI-E parallel difference methods are efficient and feasible for solving multi-term time fractional diffusion equations.

Numerical Solution of a Class of Time-Fractional Order Diffusion Equations in a New Reproducing Kernel Space

In this paper, we structure some new reproducing kernel spaces based on Jacobi polynomial and give a numerical solution of a class of time fractional order diffusion equations using piecewise reproducing kernel method (RKM). Compared with other methods, numerical results show the reliability of the present method.

The capture of particles on absorbing traps in the medium with anomalous diffusion: The effective fractional order diffusion equation and the slow temporal asymptotic of survival probability

The new asymptotic behavior of the survival probability of particles, which move in the anomalous sub-diffusion way, in the medium with absolutely absorbing traps was established. The new power temporal behavior of the survival probability of diffusing particles at long times was obtained. This result is caused by the anomalous character of diffusion.

Convergence analysis of the space fractional-order diffusion equation based on the compact finite difference scheme

This paper develops a numerical method for approximating the space fractional diffusion equation in Caputo derivative sense. In this discretization process, firstly, the compact finite difference with convergence order $${\mathcal {O}}(\delta \tau ^{2})$$ is used to obtain the semi-discrete in time derivative. Afterward, the spatial fractional derivative is discretized by using the Chebyshev collocation method of the third-kind. This collocation scheme is based on the operational matrix. In addition, time-discrete stability and convergence are theoretically proved in detail. We solve two examples by the proposed method and the obtained results are compared with other numerical methods. The numerical results show that our method is much more accurate than existing methods.

Boundary stabilization and disturbance rejection for a time fractional order diffusion-wave equation

This paper considers boundary control of time fractional order diffusion–wave equation. Both boundary stabilization and disturbance rejection are considered. This paper, for the first time, has confirmed, via hybrid symbolic and numerical simulation studies, that the existing two schemes for boundary stabilization and disturbance rejection for (integer order) wave/beam equations are still valid for time fractional order diffusion–wave equations. The problem definition, the hybrid symbolic and numerical simulation techniques, outlines of the methods for boundary stabilization and disturbance rejection are presented together with extensive simulation results. Different dynamic behaviors are revealed for different fractional orders which may stimulate new future research opportunities.

Fuzzy Active Contour Model Using Fractional-Order Diffusion Based Edge Indicator and Fuzzy Local Fitted Image

Local region-based active contour models (ACMs) can effectively segment images corrupted by intensity inhomogeneity, however, they always converge to local minimum and are sensitive to the initial position of contour. In this paper, a novel fuzzy ACM is proposed to tackle these problems. In order to deal with intensity inhomogeneity, the fuzzy local fitted image is first defined and utilized for constructing a local-region based fuzzy energy term, which is minimized in a variational level set framework to accurately segment inhomogeneous images. Second, the fractional-order diffusion based edge indicator is used to scale the local fuzzy energy term to reduce the effect of intensity inhomogeneity. Third, the fuzzy signed pressure force (FSPF) function defined by local image information is used for constructing the weighted area term to further improve the accuracy of the developed model. Finally, the global FSPF is formulated and used as an adaptive force, which can drive the level set function (LSF) to adaptively move up or down according to image intensity information. Therefore, the initial contour can be initialized as a constant function, which eliminates the problem caused by contour initialization. Moreover, the global FSPF makes the proposed model not easy to fall into local minimum. The results of experiments on synthetic and real images validate the accuracy of the proposed model for inhomogeneous image segmentation.

Homotopy Analysis Method for a Fractional Order Equation with Dirichlet and Non-Local Integral Conditions

The main purpose of this paper is to obtain some numerical results via the homotopy analysis method for an initial-boundary value problem for a fractional order diffusion equation with a non-local constraint of integral type. Some examples are provided to illustrate the efficiency of the homotopy analysis method (HAM) in solving non-local time-fractional order initial-boundary value problems. We also give some improvements for the proof of the existence and uniqueness of the solution in a fractional Sobolev space.

Numerical solution of the nonlinear diffusion equation by using non-standard/standard finite difference and Fibonacci collocation methods

In this article, the authors have introduced a new nonstandard/standard finite difference scheme with the Fibonacci polynomial. The considered nonlinear fractional diffusion equation is reduced into a system of linear ordinary differential equations which are solved by using nonstandard/standard finite difference method. The developed schemes are unconditionally stable and the effectiveness and efficiency of the method is confirmed by applying it in two linear and one nonlinear problems and through comparison of their numerical results with existing analytical results. After validation of the efficiency of the method, the proposed method is used to solve the highly nonlinear fractional order diffusion equation. The stability of the method is also discussed. The salient feature of the article is the graphical exhibitions of the effects on the solute profiles due to increasing in non-linearity of the model and of the order of the spatial derivative for different particular cases.

Phase asymmetry ultrasound despeckling with fractional anisotropic diffusion and total variation.

We propose an ultrasound speckle filtering method for not only preserving various edge features but also filtering tissue-dependent complex speckle noises in ultrasound images. The key idea is to detect these various edges using a phase congruence-based edge significance measure called phase asymmetry (PAS), which is invariant to the intensity amplitude of edges and takes 0 in non-edge smooth regions and 1 at the idea step edge, while also taking intermediate values at slowly varying ramp edges. By leveraging the PAS metric in designing weighting coefficients to maintain a balance between fractional-order anisotropic diffusion and total variation (TV) filters in TV cost function, we propose a new fractional TV framework to not only achieve the best despeckling performance with ramp edge preservation but also reduce the staircase effect produced by integral-order filters. Then, we exploit the PAS metric in designing a new fractional-order diffusion coefficient to properly preserve low-contrast edges in diffusion filtering. Finally, different from fixed fractional-order diffusion filters, an adaptive fractional order is introduced based on the PAS metric to enhance various weak edges in the spatially transitional areas between objects. The proposed fractional TV model is minimized using the gradient descent method to obtain the final denoised image. The experimental results and real application of ultrasound breast image segmentation show that the proposed method outperforms other state-of-the-art ultrasound despeckling filters for both speckle reduction and feature preservation in terms of visual evaluation and quantitative indices. The best scores on feature similarity indices have achieved 0.867, 0.844 and 0.834 under three different levels of noise, while the best breast ultrasound segmentation accuracy in terms of the mean and median dice similarity coefficient are 96.25% and 96.15%, respectively.

Analytical Solutions of Fractional-Order Diffusion Equations by Natural Transform Decomposition Method.

In the present article, fractional-order diffusion equations are solved using the Natural transform decomposition method. The series form solutions are obtained for fractional-order diffusion equations using the proposed method. Some numerical examples are presented to understand the procedure of the Natural transform decomposition method. The Natural transform decomposition method has shown the least volume of calculations and a high rate of convergence compared to other analytical techniques, the proposed method can also be easily applied to other non-linear problems. Therefore, the Natural transform decomposition method is considered to be one of the best analytical technique, to solve fractional-order linear and non-linear partial deferential equations, particularly fractional-order diffusion equation.

An accurate numerical technique for solving two-dimensional time fractional order diffusion equation

ABSTRACTIn this paper, we consider a numerical technique for solving the two-dimensional fractional order diffusion equation with a time fractional derivative. The proposed technique depends basically on the fact that an expansion of the required approximate solution in a series of shifted Chebyshev polynomials of the first kind in the time and the Sinc function in the space. Then, the expansion coefficients are determined by reducing the problem with the initial conditions to a system of algebraic equations. The fractional derivatives are expressed in the Caputo sense. The numerical results of the proposed technique are compared with other published results to show the efficiency of the presented technique.

An alternating direction implicit scheme of a fractional-order diffusion tensor image registration model

In this paper, we propose an alternating direction implicit (ADI) scheme of a fractional-order diffusion tensor image (DTI) registration model. Furthermore, compatibility, stability and convergence of ADI scheme are proved. Moreover, five numerical tests are also performed to show the efficiency of this scheme.

Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations

In this paper we consider a $ d $-dimensional ($ d = 1, 2 $) parabolic-elliptic Keller-Segel equation with a logistic forcing and a fractional diffusion of order $ \alpha \in (0, 2) $. We prove uniform in time boundedness of its solution in the supercritical range $ \alpha>d\left(1-c\right) $, where $ c $ is an explicit constant depending on parameters of our problem. Furthermore, we establish sufficient conditions for $ \|u(t)-u_\infty\|_{L^\infty}\rightarrow0 $, where $ u_\infty\equiv 1 $ is the only nontrivial homogeneous solution. Finally, we provide a uniqueness result.

Analytical solution of the time fractional diffusion equation and fractional convection-diffusion equation

In this paper, we obtain analytical solutions for the time-fractional diffusion and time-fractional convection-diffusion equations. These equations are obtained from the standard equations by replacing the time derivative with a fractional derivative of order $\alpha$. Fractional operators of type Liouville-Caputo, Atangana-Baleanu-Caputo, fractional conformable derivative in Liouville-Caputo sense and Atangana-Koca-Caputo were used to model diffusion and convection-diffusion equation. The Laplace and Fourier transforms were applied to obtain the analytical solutions for the fractional order diffusion and convection-diffusion equations. The solutions obtained can be useful to understand the modeling of anomalous diffusive, subdiffusive systems and super-diffusive systems, transport processes, random walk and wave propagation phenomenon.

Radial basis functions method for solving the fractional diffusion equations

Fractional order diffusion equations are generalizations of classical diffusion equations, treating super-diffusive flow processes. The paper presents a meshless method based on spatial trial spaces spanned by the radial basis functions (RBFs) for the numerical solution of a class of initial-boundary value fractional diffusion equations with variable coefficients on a finite domain. The space fractional derivatives are defined by using Riemann–Liouville fractional derivative. We first provide Riemann–Liouville fractional derivatives for the five kinds of RBFs, including the Powers, Gaussian, Multiquadric, Matérn and Thin-plate splines, in one dimension. The time-dependent fractional diffusion equation is discretized in space with the RBF collocation method and the remaining system of ordinary differential equations (ODEs) is advanced in time with an ODE method using a method of lines approach. Some numerical results are given in order to demonstrate the efficiency and accuracy of the method. Additionally, some physical properties of this fractional diffusion system are simulated, which further confirm the effectiveness of our method. The stability of the linear systems arising from discretizing Riemann–Liouville fractional differential operator with RBFs is also analysed.

Commentary: Atrial Rotor Dynamics Under Complex Fractional Order Diffusion.

Commentary: Atrial Rotor Dynamics Under Complex Fractional Order Diffusion.

Atrial Rotor Dynamics Under Complex Fractional Order Diffusion.

The mechanisms of atrial fibrillation (AF) are a challenging research topic. The rotor hypothesis states that the AF is sustained by a reentrant wave that propagates around an unexcited core. Cardiac tissue heterogeneities, both structural and cellular, play an important role during fibrillatory dynamics, so that the ionic characteristics of the currents, their spatial distribution and their structural heterogeneity determine the meandering of the rotor. Several studies about rotor dynamics implement the standard diffusion equation. However, this mathematical scheme carries some limitations. It assumes the myocardium as a continuous medium, ignoring, therefore, its discrete and heterogeneous aspects. A computational model integrating both, electrical and structural properties could complement experimental and clinical results. A new mathematical model of the action potential propagation, based on complex fractional order derivatives is presented. The complex derivative order appears of considering the myocardium as discrete-scale invariant fractal. The main aim is to study the role of a myocardial, with fractal characteristics, on atrial fibrillatory dynamics. For this purpose, the degree of structural heterogeneity is described through derivatives of complex order γ = α + jβ. A set of variations for γ is tested. The real part α takes values ranging from 1.1 to 2 and the imaginary part β from 0 to 0.28. Under this scheme, the standard diffusion is recovered when α = 2 and β = 0. The effect of γ on the action potential propagation over an atrial strand is investigated. Rotors are generated in a 2D model of atrial tissue under electrical remodeling due to chronic AF. The results show that the degree of structural heterogeneity, given by γ, modulates the electrophysiological properties and the dynamics of rotor-type reentrant mechanisms. The spatial stability of the rotor and the area of its unexcited core are modulated. As the real part decreases and the imaginary part increases, simulating a higher structural heterogeneity, the vulnerable window to reentrant is increased, as the total meandering of the rotor tip. This in silico study suggests that structural heterogeneity, described by means of complex order derivatives, modulates the stability of rotors and that a wide range of rotor dynamics can be generated.

Fractional elliptic quasi-variational inequalities: Theory and numerics

This paper introduces an elliptic quasi-variational inequality (QVI) problem class with fractional diffusion of order $s \in (0, 1)$, studies existence and uniqueness of solutions and develops a solution algorithm. As the fractional diffusion prohibits the use of standard tools to approximate the QVI, instead we realize it as a Dirichlet-to-Neumann map for a problem posed on a semi-infinite cylinder. We first study existence and uniqueness of solutions for this extended QVI and then transfer the results to the fractional QVI: This introduces a new paradigm in the field of fractional QVIs. Further, we truncate the semi-infinite cylinder and show that the solution to the truncated problem converges to the solution of the extended problem, under fairly mild assumptions, as the truncation parameter tends to infinity. Since the constraint set changes with the solution, we develop an argument using Mosco convergence. We state an algorithm to solve the truncated problem and show its convergence in function space. Finally, we conclude with several illustrative numerical examples.

Time-Optimal Boundary Control for Systems Defined by a Fractional Order Diffusion Equation

We consider the optimal control problem for a system defined by a one-dimensional diffusion equation with a fractional time derivative. We consider the case when the controls occur only in the boundary conditions. The optimal control problem is posed as the problem of transferring an object from the initial state to a given final state in minimal possible time with a restriction on the norm of the controls. We assume that admissible controls belong to the class of functions L∞[0, T ]. The optimal control problem is reduced to an infinite-dimensional problem of moments. We also consider the approximation of the problem constructed on the basis of approximating the exact solution of the diffusion equation and leading to a finitedimensional problem of moments. We study an example of boundary control computation and dependencies of the control time and the form of how temporal dependencies in the control dependent on the fractional derivative index.

A spectral Galerkin method for the fractional order diffusion and wave equation

We are going to present a suitable bases to treat the space- and timefractional diffusion equation with the Galerkin method to obtain spectral convergence in both, time and space. Furthermore, by carefully choosing a Fourier ansatz in space, we can guarantee the resulting matrices to be sparse, even though fractional order differential equations are global operator. This is due to the fact that the chosen basis consists of eigenfunctions of the given fractional differential operator. Numerical experiments validate the theoretically predicted spectral convergence for smooth problems. Additionally, we show that this method is also capable of computing approximation of the solution of the wave equation by letting the order of the spatial and temporal derivative approach two arbitrarily close.