Two-dimensional Sub-diffusion Equation Research Articles

Analysis of a New NFV Scheme Preserving DMP for Two-Dimensional Sub-diffusion Equation on Distorted Meshes

Analysis of a New NFV Scheme Preserving DMP for Two-Dimensional Sub-diffusion Equation on Distorted Meshes

The Implicit Numerical Method for the Radial Anomalous Subdiffusion Equation

This paper presents a numerical method for solving a two-dimensional subdiffusion equation with a Caputo fractional derivative. The problem considered assumes symmetry in both the equation’s solution domain and the boundary conditions, allowing for a reduction of the two-dimensional equation to a one-dimensional one. The proposed method is an extension of the fractional Crank–Nicolson method, based on the discretization of the equivalent integral-differential equation. To validate the method, the obtained results were compared with a solution obtained through the Laplace transform. The analytical solution in the image of the Laplace transform was inverted using the Gaver–Wynn–Rho algorithm implemented in the specialized mathematical computing environment, Wolfram Mathematica. The results clearly show the mutual convergence of the solutions obtained via the two methods.

A weak Galerkin finite element approximation of two-dimensional sub-diffusion equation with time-fractional derivative

We develop a fully discrete weak Galerkin finite element method for the initial-boundary value problem of two-dimensional sub-diffusion equation with Caputo time-fractional derivative. A traditional $L_1$ discretization for the Caputo time-fractional derivative and a weak Galerkin scheme for the space integer differential operator are employed. We prove the stability of the numerical method and establish the error estimate in $L^2$ and discrete $H^1$ norms, respectively. Some numerical results are reported to confirm the theory.

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.

Factorized difference scheme for two-dimensional subdiffusion equation in nonhomogeneous media

A factorized finite-difference scheme for numerical approximation of initial-boundary value problem for two-dimensional subdiffusion equation in nonhomogeneous media is proposed. Its stability and convergence are investigated. The corresponding error bounds are obtained.

Finite Difference/Collocation Method for Two-Dimensional Sub-Diffusion Equation with Generalized Time Fractional Derivative

Finite Difference/Collocation Method for Two-Dimensional Sub-Diffusion Equation with Generalized Time Fractional Derivative