Fractional Sub-diffusion Equation Research Articles

A second order box-type scheme for fractional sub-diffusion equation with spatially variable coefficient under Neumann boundary conditions

In the present work, a box-type difference scheme with convergence order O(tau^{2}+h^{2}) is proposed for the fractional sub-diffusion equation with spatially variable coefficient under Neumann boundary conditions. Here h, τ are space and temporal step length, respectively. The method is based on applying the L2-1_{sigma} formula to approximate the time Caputo fractional derivative and introducing the auxiliary variable. By virtue of the special properties of the L2-1_{sigma} formula and the mathematical induction method, the unconditional stability and convergence for our scheme are proved by the discrete energy method. Numerical examples are given to verify the theoretical analysis and efficiency of the box-type scheme.

Grain boundary diffusion in terms of the tempered fractional calculus

Mathematical treatment of grain-boundary diffusion based on the model first proposed by Fisher is usually formulated in terms of normal diffusion equations in a two-component nonhomogeneous medium. On the other hand, fractional equations of anomalous diffusion proved themselves to be useful in description of grain-boundary diffusion phenomena. Moreover, the most important propagation regime predicted by Fisher's model demonstrates subdiffusive behavior. However, the direct link between fractional approach and the Fisher model and its modifications has not found yet. Here, we fill this gap and show that solution of fractional subdiffusion equation offers general properties of classical solutions obtained by Whipple and Suzuoka. The tempered fractional approach is a convenient tool for studying precipitation in granular materials as the tempered subdiffusion limited process.

TWO-SCALE DISTRIBUTION OF PRIME NUMBER AND ITS DIFFUSION EQUATION MODEL

This study reveals the two-scale characteristics in prime number distribution. It is observed a sub-diffusion process of power law decay at the small scale of the natural number [Formula: see text], but is found to obey the classical Brownian motion of an exponential decay at the large scale [Formula: see text]. Such two-scale mechanism gives rise to the multi-fractal scaling from the power law to the exponential law distributions in a transition region of the natural number [Formula: see text]. In the small range, the sub-diffusion of prime number distribution is well depicted by the fractional derivative equation model, and in the large scale, exponential decay distribution can accurately be described by a classical diffusion equation model. The Riemann diffusion equation proposed recently by the present authors can accurately model the prime distribution from small to moderate to large scales and is reduced to the fractional derivative sub-diffusion equation at small scale and the classical Brownian motion diffusion equation at large scale, respectively.

The Temporal Second Order Difference Schemes Based on the Interpolation Approximation for Solving the Time Multi-term and Distributed-Order Fractional Sub-diffusion Equations

In this article, a special point is found for the interpolation approximation of the linear combination of multi-term fractional derivatives. The derived numerical differentiation formula can achieve at least second order accuracy. Then the formula is used to numerically solve the time multi-term and distributed-order fractional sub-diffusion equations. Several unconditionally stable and convergent difference schemes are presented. The stability and convergence of the difference schemes are discussed. Some numerical examples are provided to show the efficiency of the proposed difference schemes.

A second-order compact difference scheme for the fourth-order fractional sub-diffusion equation

In the present work, a compact difference scheme with convergence order O(τ2 + h4) is proposed for the fourth-order fractional sub-diffusion equation, where h and τ are space and temporal step length, respectively. The method is based on applying the L2−1σ formula to approximate the time Caputo fractional derivative and employing compact operator to approximate the spatial fourth-order derivative. Using the special properties of L2−1σ formula and mathematical induction method, we obtain the unconditional stability and convergence for our scheme by discrete energy method. Furthermore, the extension to the two-dimensional case is also considered. Numerical examples are given to verify the theoretical analysis and efficiency of the new developed scheme.

The error analysis of Crank-Nicolson-type difference scheme for fractional subdiffusion equation with spatially variable coefficient

A Crank-Nicolson-type difference scheme is presented for the spatial variable coefficient subdiffusion equation with Riemann-Liouville fractional derivative. The truncation errors in temporal and spatial directions are analyzed rigorously. At each time level, it results in a linear system in which the coefficient matrix is tridiagonal and strictly diagonally dominant, so it can be solved by the Thomas algorithm. The unconditional stability and convergence of the scheme are proved in the discrete L_{2} norm by the energy method. The convergence order is min {2-frac{alpha}{2}, 1+alpha } in the temporal direction and two in the spatial one. Finally, numerical examples are presented to verify the efficiency of our method.

Crank-Nicolson finite difference method for two-dimensional fractional sub-diffusion equation

A Crank-Nicolson finite difference method is presented to solve the time fractional two-dimensional sub-diffusion equation in the case where the Grunwald-Letnikov definition is used for the time-fractional derivative. The stability and convergence of the proposed Crank-Nicolson scheme are also analyzed. Finally, numerical examples are presented to test that the numerical scheme is accurate and feasible.

A new semi-analytical collocation method for solving multi-term fractional partial differential equations with time variable coefficients

The aim of this paper is to present a new numerical method for solving a wide class of fractional partial differential equations (FPDEs) such as wave-diffusion equations, modified anomalous fractional sub-diffusion equations, time-fractional telegraph equations. The proposed method is based on the Fourier series expansion along the spatial coordinate which transforms the original equation into a sequence of multi-term fractional ordinary differential equations (ODEs). These fractional equations are solved by the use of a new efficient numerical technique – the backward substitution method. The numerical examples confirm the high accuracy and efficiency of the proposed numerical scheme in solving FPDEs with variable in time coefficients.

A cubic trigonometric B-spline collocation approach for the fractional sub-diffusion equations

A cubic trigonometric B-spline collocation approach for the numerical solution of fractional sub-diffusion equation is presented in this paper. The approach is based on the usual finite difference scheme to discretize the time derivative while the approximation of the second-order derivative with respect to space is obtained by the cubic trigonometric B-spline functions with the help of Grünwald–Letnikov discretization of the Riemann–Liouville derivative. The scheme is shown to be stable using the Fourier method and the accuracy of the scheme is tested by application to a test problem. The results of the numerical test verify the accuracy and efficiency of the proposed algorithm.

Sturm-Liouville problem and numerical method of fractional diffusion equation on fractals

Classical diffusion theory is widely applied in natural science and has made a great achievement. However, the phenomenon of anomalous diffusion in discontinuous media (fractal, porous, etc.) shows that classical diffusion theory is no longer suitable. The differential equations with fractional order have recently been proved to be powerful tools for describing anomalous diffusion. Nevertheless, the analysis methods and numerical methods for fractional differential equations are still in the stage of exploration. In the paper, we consider the Sturm-Liouville problem and the numerical method of a fractional sub-diffusion equation with Dirichlet condition, respectively. We have given the series solution of equation and proved the stability and the convergence of the implicit numerical scheme. It is found that the numerical results are in satisfactory agreement with the analytical solution. Through the robustness analysis, it is also found that the diffusion processes on fractals are more sensitive to the spectral dimension than to the anomalous diffusion exponent.

Wavelets Galerkin Method for the Fractional Subdiffusion Equation

The time fractional subdiffusion equation (FSDE) as a class of anomalous diffusive systems has obtained by replacing the time derivative in ordinary diffusion by a fractional derivative of order 0<α<1. Since analytically solving this problem is often impossible, proposing numerical methods for its solution has practical importance. In this paper, an efficient and accurate Galerkin method based on the Legendre wavelets (LWs) is proposed for solving this equation. The time fractional derivatives are described in the Riemann–Liouville sense. To do this, we first transform the original subdiffusion problem into an equivalent problem with fractional derivatives in the Caputo sense. The LWs and their fractional operational matrix (FOM) of integration together with the Galerkin method are used to transform the problem under consideration into the corresponding linear system of algebraic equations, which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account, automatically. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional subdiffusion equation

In this paper, a high-order local discontinuous Galerkin (LDG) method combined with weighted and shifted Grünwald difference (WSGD) approximation is developed and discussed for a Caputo time-fractional subdiffusion equation. The time fractional derivative of order α, 0<α<1, is approximated by a third-order method based on the idea of WSGD operator, while the spatial operator is approximated by the LDG method. Some useful lemmas are first introduced and proved, then the analysis of stability and optimal error estimate O(Δt3+hk+1) are obtained for the LDG method. Extensive numerical results using Pk,k=0,1,2,3, elements are presented to validate our theoretical analysis.

Numerical solution of fractional sub-diffusion and time-fractional diffusion-wave equations via fractional-order Legendre functions

Fractional calculus has been used to model physical and engineering processes that are best described by fractional differential equations. Therefore designing efficient and reliable techniques for the solution of such equations is an important task. In this paper, we propose an efficient and accurate Galerkin method based on the fractional-order Legendre functions (FLFs) for solving the fractional sub-diffusion equation (FSDE) and the time-fractional diffusion-wave equation (FDWE). The time-fractional derivatives for FSDE are described in the Riemann-Liouville sense, while for FDWE are described in the Caputo sense. To this end, we first derive a new operational matrix of fractional integration (OMFI) in the Riemann-Liouville sense for FLFs. Next, we transform the original FSDE into an equivalent problem with fractional derivatives in the Caputo sense. Then the FLFs and their OMFI together with the Galerkin method are used to transform the problems under consideration into the corresponding linear systems of algebraic equations, which can be simply solved to achieve the numerical solutions of the problems. The proposed method is very convenient for solving such kind of problems, since the initial and boundary conditions are taken into account automatically. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

A fast accurate approximation method with multigrid solver for two-dimensional fractional sub-diffusion equation

A fast accurate approximation method with multigrid solver is proposed to solve a two-dimensional fractional sub-diffusion equation. Using the finite difference discretization of fractional time derivative, a block lower triangular Toeplitz matrix is obtained where each main diagonal block contains a two-dimensional matrix for the Laplacian operator. Our idea is to make use of the block ϵ-circulant approximation via fast Fourier transforms, so that the resulting task is to solve a block diagonal system, where each diagonal block matrix is the sum of a complex scalar times the identity matrix and a Laplacian matrix. We show that the accuracy of the approximation scheme is of O(ϵ). Because of the special diagonal block structure, we employ the multigrid method to solve the resulting linear systems. The convergence of the multigrid method is studied. Numerical examples are presented to illustrate the accuracy of the proposed approximation scheme and the efficiency of the proposed solver.

Analytical solution and nonconforming finite element approximation for the 2D multi-term fractional subdiffusion equation

The current paper focuses on analytical solution and numerical approximation for the two-dimensional(2D) multi-term time fractional subdiffusion equation. In order to derive an effective numerical solution of the equation, an unconditionally stable fully-discrete approximate scheme is established by using a modified L1 approximation and spatial nonconforming finite element method, which has many advantages comparing with conforming one. Moreover, by employing the Crouzeix–Raviart type EQ1rot nonconforming element and dealing with fractional derivatives skillfully, temporal optimal order error estimates and spatial optimal convergence rates in both L2-norm and broken energy norm are proposed without restrictions between time step and mesh size. Then, several numerical results have been provided to give an insight into the efficiency and reliability of the theoretical analysis. Finally, the derivation process of the analytical solution is presented by means of a method of separating variables.

Actuator characterisations to achieve approximate controllability for a class of fractional sub-diffusion equations

ABSTRACTThis paper is devoted to analysing the actuator characterisations for the fractional sub-diffusion equation under consideration to become approximately controllable. Two different cases are considered, where the control inputs emerge in the differential equation as distributed inputs and as boundary inputs in the boundary conditions. The dual system for fractional sub-diffusion equation is solved and the necessary and sufficient conditions for the approximate controllability of the system are established. Several examples are worked out in the end to illustrate our results.

Fourth order finite difference schemes for time–space fractional sub-diffusion equations

In this paper, we devote to the study of high order finite difference schemes for one- and two-dimensional time–space fractional sub-diffusion equations. A fourth order finite difference scheme is invoked for the spatial fractional derivatives, and the L1 approximation is applied to the temporal fractional parts. For the two-dimensional case, an alternating direction implicit scheme based on L1 approximation is proposed. The stability and convergence of the proposed methods are studied. Numerical experiments are performed to verify the effectiveness and accuracy of the proposed difference schemes.

Quadratic spline collocation method for the time fractional subdiffusion equation

In this paper, exploiting the quadratic spline collocation (QSC) method, we numerically solve the time fractional subdiffusion equation with Dirichelt boundary value conditions. The coefficient matrix of the discretized linear system is investigated in detail. Theoretical analyses and numerical examples demonstrate the proposed technique can enjoy the global error bound with O(τ3+h3) under the L∞ norm provided that the solution v(x, t) has four-order continual derivative with respects to x and t, and it can achieve the accuracy of O(τ4+h4) at collocation points, where τ, h are the step sizes in time and space, respectively.

Double integral-balance method to the fractional subdiffusion equation: Approximate solutions, optimization problems to be resolved and numerical simulations

An approximate integral-balance solution of the fractional subdiffusion equation by a double-integration technique has been conceived. The time-fractional linear subdiffusion equation with Dirichlet boundary condition (and zero initial condition) has been chosen as a test example. Approximations of time-fractional Riemann-Liouville and Caputo derivatives when the distribution is assumed as a parabolic profile with unspecified exponent have been developed. Problems pertinent to determination of the optimal exponent of the parabolic profile and approximations of the time-fractional derivative of by different approaches have been formulated. Solved and unresolved problems in determination of the optimal exponents have been demonstrated. Examples with predetermined quadratic and cubic assumed profiles are analyzed, too. Comparative numerical studies with exact solutions expressed by the Mainardi function in terms of a similarity variable have been performed.

A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations

This article adapts an operational matrix formulation of the collocation method for the one- and two-dimensional nonlinear fractional sub-diffusion equations (FSDEs). In the proposed collocation approach, the double and triple shifted Jacobi polynomials are used as base functions for approximate solutions of the one- and two-dimensional cases. The space and time fractional derivatives given in the underline problems are expressed by means of Jacobi operational matrices. This investigates spectral collocation schemes for both temporal and spatial discretizations. Thereby, the expansion coefficients are then determined by reducing the FSDEs, with their initial and boundary conditions, into systems of nonlinear algebraic equations which are far easier to be solved. Furthermore, the error of the approximate solution is estimated theoretically along with graphical analysis to confirm the exponential convergence rate of the proposed method in both spatial and temporal discretizations. In order to show the high accuracy of our algorithms, we report the numerical results of some numerical examples and compare our numerical results with those reported in the literature.