Fractional Order Diffusion Equation Research Articles

A high-accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem of variable-coefficient conservative fractional diffusion equations

Fractional diffusion equations were shown to provide an adequate and accurate description of transport processes exhibiting anomalous diffusion behavior. Recently, spectral Galerkin methods were developed for space-fractional diffusion equations aiming at achieving exponential convergence. An optimal order error estimate in the fractional energy norm was proved under the assumption that the true solution to the fractional diffusion equation has the desired regularity. An optimal order error estimate in the L2 norm was proved via the well known Nitsche lifting technique under the assumption that the true solution to the corresponding boundary-value problem of the fractional diffusion equation has the required regularity for each right-hand side.In this paper we show that the true solution to the Dirichlet boundary-value problem of a conservative fractional diffusion equation of order 2−β with 0<β<1 as well as a constant diffusivity coefficient and a constant source term is not in the fractional Sobolev space H3/2−β in general, but is still in the Besov space B∞3/2−β(L2). Hence, the provable convergence rate of a spectral Galerkin method in the L2 norm is at most of the order O(N−(3/2−β)), where N is the degree of the polynomial space in the numerical method. Numerical experiments show that the spectral Galerkin method exhibits a subquadratic convergence in the L2 norm for any 0<β<1.We develop a high-accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem of one-sided variable-coefficient conservative fractional diffusion equations. The method has a proved high-order convergence rate of arbitrary order (i) without requiring the smoothness of the true solution u to the given boundary-value problem, but only assuming that the diffusivity coefficient and the right-hand source term have the desired regularity; (ii) for a variable diffusivity coefficient; and (iii) for an inhomogeneous Dirichlet boundary condition. Numerical experiments substantiate the theoretical analysis and show that the method exhibits exponential convergence provided the diffusivity coefficient and the right-hand source term have the desired regularity.

Leapfrog/finite element method for fractional diffusion equation.

We analyze a fully discrete leapfrog/Galerkin finite element method for the numerical solution of the space fractional order (fractional for simplicity) diffusion equation. The generalized fractional derivative spaces are defined in a bounded interval. And some related properties are further discussed for the following finite element analysis. Then the fractional diffusion equation is discretized in space by the finite element method and in time by the explicit leapfrog scheme. For the resulting fully discrete, conditionally stable scheme, we prove an L 2-error bound of finite element accuracy and of second order in time. Numerical examples are included to confirm our theoretical analysis.

Non-probabilistic Solutions of Uncertain Fractional Order Diffusion Equations

This paper investigates the numerical solution of uncertain fractional order diffusion equation subject to various external forces. Homotopy Perturbation Method (HPM) is used for the analysis. Uncertainties present in the system are modelled through triangular convex normalised fuzzy sets. A new computational technique has been proposed based on double parametric form of fuzzy numbers to handle the fuzzy fractional diffusion equation. Using the single parametric form of fuzzy numbers, the original fuzzy fractional diffusion equation is converted first to an interval fuzzy fractional differential equation. Next this equation is transformed to crisp form by applying double parametric form of fuzzy numbers. Finally the same is solved by HPM symbolically to obtain the uncertain bounds of the solution. Obtained results are depicted in term of plots. Results obtained by the proposed method are compared with existing results in special cases.

Разностные схемы для уравнения диффузии дробного порядка с краевыми условиями третьего рода в многомерной области

Разностные схемы для уравнения диффузии дробного порядка с краевыми условиями третьего рода в многомерной области

A High-Accuracy MOC/FD Method for Solving Fractional Advection-Diffusion Equations

Fractional-order diffusion equations are viewed as generalizations of classical diffusion equations, treating super-diffusive flow processes. In this paper, in order to solve the fractional advection-diffusion equation, the fractional characteristic finite difference method is presented, which is based on the method of characteristics (MOC) and fractional finite difference (FD) procedures. The stability, consistency, convergence, and error estimate of the method are obtained. An example is also given to illustrate the applicability of theoretical results.

Boundary value problems for the diffusion equation of the variable order in differential and difference settings

Solutions of boundary value problems for a diffusion equation of fractional and variable order in differential and difference settings are studied. It is shown that the method of the energy inequalities is applicable to obtaining a priori estimates for these problems exactly as in the classical case. The credibility of the obtained results is verified by performing numerical calculations for a test problem.

Grounds for the application of fractional-order diffusion equations in channeling theory using the Langevin approach

Transverse-energy distribution densities in the case of ultrarelativistic proton channeling in the (100) planes of diamond-like crystals have been numerically calculated within the framework of the Langevin approach. Based on the self-similarity principle, the subdiffusion character of the motion under consideration has been established, which can be regarded as grounds for using fractional-order diffusion equations to adequately describe channeling and dechanneling processes mathematically.

Entropy and Information in a Fractional Order Model of Anomalous Diffusion

Fractional order dynamic models (e.g., systems of ordinary and partial differential equations of non-integer order in time and space) are becoming more popular for characterizing the behavior of complex systems. Justification for such models is typically based on improved fits to experimental data or a reduced mean squared error for models with the same number of fitting parameters. This rationale, however, is relative to the form of the selected fitting function, and is dependent on the order of the derivatives. Nevertheless, there seems to be a recognition that fractional order models work better than integer order models in describing the electrical and mechanical properties of multi-scale, heterogeneous materials. In order to address this issue and to offer a new approach for establishing the utility of fractional order models, we calculate the total Shannon spectral entropy for the case of anomalous diffusion governed by a fractional order diffusion equation generalized in space and in time. This fractional order representation of the continuous time, random walk model of diffusion gives a spectral entropy minimum for normal (i.e., Gaussian) diffusion with surrounding values leading to greater values of spectral entropy.

Application of Adomian’s Decomposition Method for the Analytical Solution of Space Fractional Diffusion Equation

Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are increasingly used in modeling practical super diffusive problems in fluid flow, finance and others areas of application. This paper presents the analytical solutions of the space fractional diffusion equations by Adomian’s decomposition method (ADM). By using initial conditions, the explicit solutions of the equations have been presented in the closed form. Two examples, the first one is one-dimensional and the second one is two-dimensional fractional diffusion equation, are presented to show the application of the present techniques. The present method performs extremely well in terms of efficiency and simplicity.

Application of He’s Variational Iteration Method for the Analytical Solution of Space Fractional Diffusion Equation

Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are increasingly used in modeling practical super diffusive problems in fluid flow, finance and others areas of application. This paper presents the analytical solutions of the space fractional diffusion equations by variational iteration method (VIM). By using initial conditions, the explicit solutions of the equations have been presented in the closed form. Two examples, the first one is one-dimensional and the second one is two-dimensional fractional diffusion equation, are presented to show the application of the present techniques. The present method performs extremely well in terms of efficiency and simplicity.

Fractal controls and quasi-analytic classes of functions in the Cauchy problem for a fractional-order diffusion equation

Fractal controls and quasi-analytic classes of functions in the Cauchy problem for a fractional-order diffusion equation

Numerical solution via Laplace transforms of a fractional order evolution equation

We consider the discretization in time if a fractional order diffusion equation. The approximation is based on a further development of the approach of using Laplace transformation to represent the solution as a contour integral, evaluated to high accuracy by quadrature. This technique reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. Three different methods, using 2N + 1 quadrature points, are discussed. The first has an error of order O (e(-cN)) away from t = 0, whereas the second and third methods are uniformly accurate of order O (e(-C root N)). Unlike the first and second methods, the third method does not use the Laplace transform of the forcing term. The basic analysis of the time discretizaiton takes place in a Banach space setting and uses a resolvent esitmate for the associated elliptic operator. The methods are combined with finite element discretization in the spatial variable to yield fully discrete methods.

Finite difference approximations for the fractional advection–diffusion equation

Fractional order diffusion equations are viewed as generalizations of classical diffusion equations, treating super-diffusive flow processes. In this Letter, in order to solve the two-sided fractional advection–diffusion equation, the fractional Crank–Nicholson method (FCN) is given, which is based on shifted Grünwald–Letnikov formula. It is shown that this method is unconditionally stable, consistent and convergent. The accuracy with respect to the time step is of order ( Δ t ) 2 . A numerical example is presented to confirm the conclusions.

A note on fractional diffusion equations

In the present paper, the solutions of the fractional diffusion equation of order α ( 0 < α ⩽ 1 ) with the initial condition x n , n is a positive integer are obtained with the help of a powerful method, i.e., the variational iteration method. The method performs extremely well in terms of efficiency and simplicity. Numerical results are presented graphically.

Application of Fractional Differential Equations for Modeling the Anomalous Diffusion of Contaminant from Fracture into Porous Rock Matrix with Bordering Alteration Zone

Solute diffusion from a fracture into a porous rock with an altered zone bordering the fracture is modeled by a system of two diffusion equations (one for the altered zone and another for the intact porous matrix) with different coefficients of effective diffusivity. Since experimental studies of diffusion into rock samples with altered zones indicate that mathematical models of diffusion based on Fick’s law do not adequately describe the concentration field in a sample, fractional order diffusion equations are chosen in this study for modeling the anomalous mass transport in the rocks. In the case of significantly higher porosity of the altered zone (e.g., this is typical for carbonates) the effective diffusivity here can be much higher than the effective diffusivity of non-altered rocks. By introducing a small parameter that is the ratio of effective diffusivities in the non-altered and altered regions and applying the technique of perturbations, approximate analytical solutions for concentrations in the altered zone bordering the fracture and in the intact surrounding rocks are obtained. Based on these solutions, different regimes of diffusion into the rocks with different physical properties are modeled and analyzed. It is shown that, using experimentally obtained data, the orders of the fractional derivatives in the differential equations can be readily calibrated for the every specific rock.

Stochastic Models of Solute Transport in Highly Heterogeneous Geologic Media

A stochastic model of anomalous diffusion was developed in which transport occurs by random motion of Brownian particles, described by distribution functions of random displacements with heavy (power‐law) tails. One variant of an effective algorithm for random function generation with a power‐law asymptotic and arbitrary factor of asymmetry is proposed that is based on the Gnedenko–Levy limit theorem and makes it possible to reproduce all known Levy α‐stable fractal processes. A two‐dimensional stochastic random walk algorithm has been developed that approximates anomalous diffusion with streamline‐dependent and space‐dependent parameters. The motivation for introducing such a type of dispersion model is the observed fact that tracers in natural aquifers spread at different super‐Fickian rates in different directions. For this and other important cases, stochastic random walk models are the only known way to solve the so‐called multiscaling fractional order diffusion equation with space‐dependent parameters. Some comparisons of model results and field experiments are presented.

Locally one-dimensional difference schemes for the fractional order diffusion equation

Locally-one-dimensional difference schemes for the fractional diffusion equation in multidimensional domains are considered. Stability and convergence of locally one-dimensional schemes for this equation are proved.

WITHDRAWN: Analytical approximate solution of diffusion equation of fractional order by variational iteration method

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Analytical solution for the space fractional diffusion equation by two-step Adomian Decomposition Method

Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are increasingly used in modeling practical superdiffusive problems in fluid flow, finance and other areas of application. This paper presents the analytical solutions of the space fractional diffusion equations by two-step Adomian Decomposition Method (TSADM). By using initial conditions, the explicit solutions of the equations have been presented in the closed form and then their solutions have been represented graphically. Two examples, the first one is one-dimensional and the second one is two-dimensional fractional diffusion equation, are presented to show the application of the present technique. The solutions obtained by the standard decomposition method have been numerically evaluated and presented in the form of tables and then compared with those obtained by TSADM. The present TSADM performs extremely well in terms of efficiency and simplicity.

A SPLITTING TYPE ALGORITHM FOR NUMERICAL SOLUTION OF PDES OF FRACTIONAL ORDER

Fractional order diffusion equations are generalizations of classical diffusion equations, treating super‐diffusive flow processes. In this paper, we examine a splitting type numerical methods to solve a class of two‐dimensional initial‐boundary value fractional diffusive equations. Stability, consistency and convergence of the methods are investigated. It is shown that both schemes are unconditionally stable. A numerical example is presented.