Anomalous Sub-diffusion Equation Research Articles

Numerical simulation using the non-standard weighted average FDM for 2Dim variable-order Cable equation

The fractional variable-order (VO) two-dimensional (2Dim) Cable equation is one of the most significant types of anomalous subdiffusion equations that emerge strongly in spiny neural dendrites and is solved by using an accurate numerical technique in this study. The non-standard weighted average finite difference approach is a simple proposed technique (NSWAFDM). The proposed method’s stability is investigated in depth by using the John von Neumann stability analysis approach. The arbitrary weight factor and multiple discretization schemes of the VO derivative are used to develop a specific stability criterion. To demonstrate the usefulness and accuracy of the suggested technique, numerical examples are offered.

NUMERICAL SIMULATIONS FOR THE VARIABLE ORDER TWO-DIMENSIONAL REACTION SUB-DIFFUSION EQUATION: LINEAR AND NONLINEAR

The applications and the fields that use the anomalous sub-diffusion equations cannot be easily listed due to their wide area. Sure, one of the main physical reasons for using and researching fractional order diffusion equations is to explain anomalous diffusion that occurs in transport processes through complex and/or disordered structures, such as fractal media. One of the important applications is their use in chemical reactions, where a single material continues to shift from a high concentration area to a low concentration area until the concentration across the space is equal. The mathematical model that describes these physical-chemical phenomena is called the reaction sub-diffusion equation. In our study, we try to solve the 2D variable order version of these equations (2DVORSE) (linear and nonlinear) by using an accurate numerical technique which is the variable weighted average finite difference method (WAFDM). We will analyze the stability of the resulting scheme by using a modified suitable version of the John Von Neumann procedure. Specific stability conditions that occur or some parameters in the resulting schemes are derived and checked. At the end of the study, numerical examples are simulated to check the stability and the accuracy of the proposed technique.

The implicit numerical method for the one-dimensional anomalous subdiffusion equation with a nonlinear source term

The implicit numerical method for the one-dimensional anomalous subdiffusion equation with a nonlinear source term

A New Variant of B-Spline for the Solution of Modified Fractional Anomalous Subdiffusion Equation

The objective of this paper is to present an efficient numerical technique for solving time fractional modified anomalous subdiffusion equation. Anomalous diffusion equation has its role in various branches of biological sciences. B-spline is a piecewise function to draw curves and surfaces, which maintain its degree of smoothness at the connecting points. B-spline provides an active process of approximation to the limit curve. In current attempt, B-spline curve is used to approximate the solution curve of time fractional modified anomalous subdiffusion equation. The process is kept simple involving collocation procedure to the data points. The time fractional derivative is approximated with the discretized form of the Riemann-Liouville derivative. The process results in the form of system of algebraic equations, which is solved using a variant of Thomas algorithm. In order to ensure the convergence of the procedure, a valid method named Von Neumann stability analysis is attempted. The graphical and tabular display of results for the illustrated examples is presented, which stamped the efficiency of the proposed algorithm.

Fast High-Order Difference Scheme for the Modified Anomalous Subdiffusion Equation Based on Fast Discrete Sine Transform

The modified anomalous subdiffusion equation plays an important role in the modeling of the processes that become less anomalous as time evolves. In this paper, we consider the efficient difference scheme for solving such time-fractional equation in two space dimensions. By using the modified L1 method and the compact difference operator with fast discrete sine transform technique, we develop a fast Crank-Nicolson compact difference scheme which is proved to be stable with the accuracy of O τ min 1 + α , 1 + β + h 4 . Here, α and β are the fractional orders which both range from 0 to 1, and τ and h are, respectively, the temporal and spatial stepsizes. We also consider the method of adding correction terms to efficiently deal with the nonsmooth problems. Numerical examples are provided to verify the effectiveness of the proposed scheme.

Numerical investigation of reproducing kernel particle Galerkin method for solving fractional modified distributed-order anomalous sub-diffusion equation with error estimation

In the Galerkin weak form technique based on various kernels that they do not have δ-Kronecker property, in order to apply the essential boundary condition, there are two straight strategies that one of them is the Lagrange multiplier method and another one is the penalty method. In the penalty method the main boundary value problem (BVP) is converted to a new BVP with Robin boundary condition. So, we obtain a new BVP that it must be solved. The main purpose of this paper is to propose an error analysis to verify that the solutions of penalty method obtained by applying the essential boundary condition are convergent to the solution of main BVP with essential boundary condition. For this aim, we select fractional modified distributed-order anomalous sub-diffusion equation. At the first stage, we propose a second-order difference scheme for the temporal variable. The convergence and stability analysis for the time-discrete scheme are proposed. At the second stage, we derive the full-discrete scheme based on the Galerkin weak form and shape functions of reproducing kernel particle method (RKPM) as the mentioned shape functions do not have the δ-Kronecker property. Furthermore, it is shown that when the penalty parameter goes to infinity then the solutions of BVP with Robin boundary condition are convergent to the solutions of BVP based on the essential boundary condition. The proposed examples verify that the present error estimate is true.

A semi-analytical approach to Caputo type time-fractional modified anomalous sub-diffusion equations

This article is devoted to a new semi-analytical algorithm for solving time-fractional modified anomalous sub-diffusion equations (FMASDEs). In this method first, the main problem is reduced to a system of fractional-order ordinary differential equations (FODEs) under known initial value conditions by using the Chebyshev collocation procedure. After that, to solve this system, some auxiliary initial value problems are defined. Next, we find an optimal linear combination of some particular solutions for these problems and finally we use this linear combination to construct a semi-analytical approximate solution for the main problem. To demonstrate the convergence property of the new method, a residual error analysis is performed in details. Some test problems are investigated to show reliability and accuracy of the proposed method. Besides, convergence order's indicators are evaluated for all test problems and are compared with ones of the other methods. Moreover, a comparison between our computed numerical results and the reported results of the other numerical schemes in the literature exhibits that the proposed technique is more precise and reliable.In summary advantages of the proposed method are: high accuracy, easy programming, high experimental convergence order, and solving another types of fractional differential equations.

New group iterative schemes for solving the two-dimensional anomalous fractional sub-diffusion equation

New group iterative schemes for solving the two-dimensional anomalous fractional sub-diffusion equation

Discrete-time orthogonal spline collocation method for a modified anomalous diffusion equation

The modified anomalous sub-diffusion equation (MASDE) describes processes that become less anomalous as time progresses. In this paper, based on Crank-Nicolson and weighted and shifted Grünwald difference (WSGD) formula, we develop orthogonal spline collocation (OSC) method with second-order temporal accuracy and fourth-order spatial accuracy for MASDE. The stability and convergence of our numerical schemes are analysed in detail. Also, our numerical results are consistent with our theoretical analysis.

A Galerkin finite element method for the modified distributed-order anomalous sub-diffusion equation

In this paper, we consider a modified time distributed-order anomalous sub-diffusion equation for the description of processes becoming less anomalous over the course of time. First, we discretize the integral term using a mid-point quadrature rule and transform the time distributed-order fractional diffusion equation into a multi-term equation. Then applying the backward difference method in time and Galerkin finite element method in space, we present a fully discrete scheme to solve the time distributed-order fractional diffusion equation. Moreover, the stability and convergence of the proposed method are also discussed. Numerical examples are given to verify the accuracy and stability of our scheme.

An efficient iterative approach for three-dimensional modified anomalous fractional sub-diffusion equations on a large domain

The fractional sub-diffusion equation, which is obtained by replacing the time derivative in ordinary diffusion by a fractional derivative of order ϑ with 0 < vartheta < 1, has appeared in numerous complex system. In this paper, we suggest an efficient and accurate iterative method based on coupling the variational iteration method (VIM) with an auxiliary parameter for solving three-dimensional FDEs described in the Riemann–Liouville sense. However, though the standard VIM is often invalid on large domains, the VIM with an auxiliary parameter is highly efficient in approximating the solution of complex systems even on large domains. The procedure of obtaining an optimal auxiliary parameter is illustrated through some examples, while the theoretical analysis confirms the convergence of the proposed method. Comparing the results of standard VIM and modified VIM by the auxiliary parameter confirms the effectiveness of using the new technique on the magnitude of the convergence region.

Alternating direction implicit-spectral element method (ADI-SEM) for solving multi-dimensional generalized modified anomalous sub-diffusion equation

The main aim of the current paper is to solve the multi-dimensional generalized modified anomalous sub-diffusion equation by using a new spectral element method. At first, the time variable has been discretized by a finite difference scheme with second-order accuracy. The stability and convergence of the time-discrete scheme have been investigated. We show that the time-discrete scheme is unconditionally stable and the convergence order is O(τ2) in the temporal direction. Secondly, the Galerkin spectral element method has been combined with alternating direction implicit idea to discrete the space variable. The unconditional stability and convergence of the full-discrete scheme have been proved. By developing the proposed scheme, we need to calculate one-dimensional integrals for two-dimensional problems and two-dimensional integrals for three-dimensional problems. Thus, the used CPU time for the presented numerical procedure is lower than the two- and three-dimensional Galerkin spectral element methods. Also, the proposed method is suitable for computational domains obtained from the tensor product. Finally, two examples are analyzed to check the theoretical results.

The Monte Carlo Markov chain method for solving the modified anomalous fractional sub-diffusion equation

In this paper, the Monte Carlo Markov chain method for solving the modified anomalous fractional sub-diffusion equation is studied. Most of the previous methods are low in temporal and spatial accuracy order. Based on the idea of Monte Carlo Markov chain method and compact finite difference schemes, a probability model for solving the modified anomalous fractional sub-diffusion equation is established. Numerical examples are given to show the feasibility of the proposed scheme. Compared with the compact finite difference method, the present method is truly meshless and is easy to be implemented with high temporal and spatial accuracy order. And it is also applied to solve partial differential equation in irregular domains.

Optimal control problem of the two-dimensional modified anomalous subdiffusion equation with discontinuous Galerkin approximation

In this paper, we consider an optimal control problem (OCP) governed by a modified anomalous subdiffusion equation with box constraints on the control. We apply space–time discontinuous Galerkin method to discretize the problem. Specifically, symmetric interior penalty Galerkin method and piecewise discontinuous constant temporal approximations are used. Then, we discuss the equivalence of optimize then discretize and discretize then optimize approaches in a general setting. We derive stability and convergence analysis for the fully discrete problem. At the end, we present some numerical results to measure the numerical rate of convergence.

Time fractional modified anomalous sub-diffusion equation with a nonlinear source term through locally applied meshless radial point interpolation

In this paper, an alternative approach of spectral meshless radial point interpolation (SMRPI) is applied to the modified anomalous fractional sub-diffusion equation with a nonlinear source term in one and two dimensions. The time fractional derivative is described in the Riemann–Liouville sense. The applied approach is based on a combination of meshless methods and the spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct the shape functions which act as basis functions in the frame of the SMRPI. It is proved that the scheme is unconditionally stable with respect to the time variable in [Formula: see text] and convergent with the order of convergence [Formula: see text], [Formula: see text]. In this work, the thin plate splines (TPS) are used as the radial basis functions. In order to eliminate the nonlinearity, a simple predictor–corrector (P–C) scheme is used. The results of numerical experiments are compared to the analytical solutions in order to confirm the accuracy and the efficiency of the presented scheme.

Local RBF-FD technique for solving the two-dimensional modified anomalous sub-diffusion equation

The main aim of this paper is to propose an efficient and suitable numerical procedure based on the local meshless collocation method for solving the two-dimensional modified anomalous sub-diffusion equation. The fractional derivative is based on the Riemann–Liouville fractional integral. Firstly, a finite difference scheme with O(τ) has been employed to discrete the time variable and also the local radial basis-finite difference (LRBF-FD) method is used to discrete the spatial direction. For the presented numerical technique, we prove the unconditional stability and also obtain an error bound. We employ a test problem to show the accuracy of the proposed technique. Also, we solve the mentioned model on irregular domain to show the efficincy of the developed technique.

Effective numerical treatment of sub-diffusion equation with non-smooth solution

ABSTRACTIn this paper we investigate a sub-diffusion equation for simulating the anomalous diffusion phenomenon in real physical environment. Based on an equivalent transformation of the original sub-diffusion equation followed by the use of a smooth operator, we devise a high-order numerical scheme by combining the Nyström method in temporal direction with the compact finite difference method and the spectral method in spatial direction. The distinct advantage of this approach in comparison with most current methods is its high convergence rate even though the solution of the anomalous sub-diffusion equation usually has lower regularity on the starting point. The effectiveness and efficiency of our proposed method are verified by several numerical experiments.

Modified implicit fractional difference scheme for 2D modified anomalous fractional sub-diffusion equation

In this paper, we solve two-dimensional modified anomalous fractional sub-diffusion equation using modified implicit finite difference approximation. The stability and convergence of the proposed scheme are analyzed by the Fourier series method. We show that the scheme is unconditionally stable and the approximate solution converges to the exact solution. A numerical example is given to show the application and feasibility of the proposed scheme.

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.

The implicit midpoint method for the modified anomalous sub-diffusion equation with a nonlinear source term

In this paper, the implicit midpoint method is used to solve the semi-discrete modified anomalous sub-diffusion equation with a nonlinear source term, and the weighted and shifted Grünwald–Letnikov difference operator and the compact difference operator are applied to approximate the Riemann–Liouville fractional derivative and space partial derivative respectively, then the new numerical scheme is constructed. The stability and the convergence of this method are analyzed. Numerical experiment demonstrates the high accuracy of this method and confirm our theoretical results.