Solution Of Fractional Subdiffusion Equation Research Articles

Reduced spline method based on a proper orthogonal decomposition technique for fractional sub-diffusion equations

In this paper, a reduced spline (RS) method based on a proper orthogonal decomposition (POD) technique for numerical solution of time fractional sub-diffusion equations is investigated. Combining POD with polynomial and non-polynomial spline methods yield a new model with lower dimensions and sufficiently high accuracy, so that the amount of computations and the calculation time decrease in comparison with usual spline methods. Although, the classical L1 scheme on uniform meshes is one of the most successful methods for approximation of the Caputo fractional derivative in time, due to the fact that the solution of time fractional sub-diffusion equations typically has a singularity at the origin, a weak convergence will conclude for such scheme. To overcome this difficulty, we use the L1 scheme on graded meshes in time, such that considered time steps are very small near the origin which compensate the singularity of the solution. Also, the error estimates between the POD approximate solution of the RS scheme and the exact solution of fractional sub-diffusion equation are established for two types of uniform and graded meshes in time. Numerical examples are given to illustrate the feasibility and efficiency of the proposed method.

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.

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.

A center Box method for radially symmetric solution of fractional subdiffusion equation

In this paper, a center Box difference method is considered for the radially symmetric solution of fractional subdiffusion equation. By method of order reduction, the derivative boundary condition is transformed into Dirichlet boundary condition and thus the geometrical singularity is successfully removed from the original problem. As a matter of course, a natural discretization scheme is obtained. To investigate the stability and convergence of the method, we define a new norm with a weight rd-1. Thus, the usual Sobolev inequality is not suitable to the new norm. Therefore, we prove three new Sobolev-like embedding inequalities which can also be applied to the other problems in polar coordinates. Then, the scheme is proved to be unconditionally stable and convergent in maximum norm with the help of the new Sobolev-like embedding inequalities. Some illustrative examples are provided to demonstrate the theoretical results. By some comparisons, it can be seen that the natural discretization scheme is accurate and effective in physical simulations. And it can be used to both long time and short time computation.