Numerical Approximation and Error Estimation of a Time Fractional Order Diffusion Equation

2009 ◽  
Author(s):  
Changpin Li ◽  
Zhengang Zhao ◽  
YangQuan Chen
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Fractional Order ◽  
Numerical Approximation ◽  
Linear Time ◽  
Long Tail ◽  
Linear Time Invariant ◽  
Matter Flux ◽  
Element Method

Finite element method is used to approximately solve a class of linear time-invariant, time-fractional-order diffusion equation formulated by the non-classical Fick law and a “long-tail” power kernel. In our derivation, “long-tail” power kernel relates the matter flux vector to the concentration gradient while the power-law relates the mean-squared displacement to the Gauss white noise. This work contributes a numerical analysis of a fully discrete numerical approximation using the space Galerkin finite element method and the approximation property of the Caputo time fractional derivative of an efficient fractional finite difference scheme. Both approximate schemes and error estimates are presented in details. Numerical examples are included to validate the theoretical predictions for various values of order of fractional derivatives.

Dynamic Analysis of Anisotropic Asymmetric Rotor-Bearing System Based on Three-Dimensional Finite Element Method

2014 ◽  
Author(s):  
Weimeng Ma ◽  
Jianjun Wang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Dynamic Analysis ◽  
Linear Time ◽  
Time Varying ◽  
3D Finite Element ◽  
Linear Time Invariant ◽  
Rotor Bearing ◽  
Asymmetric Rotor ◽  
Element Method

3D finite element modeling is a promising method, but it is not well extended to the dynamic analysis of anisotropic asymmetric rotor bearing systems. This paper is aimed at proposing a dynamic analysis method of anisotropic asymmetric rotor bearing systems based on 3D finite element method. In the proposed approach, the advantages of 3D finite element modeling are used to accurately simulate the dynamic characteristics of the asymmetric rotor. The asymmetric characteristics of the stator are considered and simplified as time-varying bearing stiffness parameters in the rotating frame. The time-varying coefficient differential equation of motion of the rotor bearing system is established in a rotating frame by taking consideration of the bearing characteristics. Floquet theory and Hill expansion method are adopted to obtain the set of equivalent linear time-invariant equations of the original time-varying coefficient differential equations. Frequency characteristics and stability of the system were obtained by solving the equivalent linear time-invariant equations. Two numerical examples are analyzed. The results of the two examples show that the proposed 3D finite element method is a good tool for dynamic analysis of anisotropic asymmetric rotor bearing systems.

Hybrid Nodal Integral / Finite Element Method for Time-Dependent Convection Diffusion Equation

2021 ◽  
Author(s):  
Sundar Namala ◽  
Rizwan Uddin
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Numerical Solutions ◽  
Second Order ◽  
Time Dependent ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Integral Methods ◽  
Element Method

Abstract Nodal integral methods (NIM) are a class of efficient coarse mesh methods that use transverse averaging to reduce the governing partial differential equation(s) (PDE) into a set of ordinary differential equations (ODE). The standard application of NIM is restricted to domains that have boundaries parallel to one of the coordinate axes/palnes (in 2D/3D). The hybrid nodal-integral/finite-element method (NI-FEM) reported here has been developed to extend the application of NIM to arbitrary domains. NI-FEM is based on the idea that the interior region and the regions with boundaries parallel to the coordinate axes (2D) or coordinate planes (3D) can be solved using NIM, and the rest of the domain can be discretized and solved using FEM. The crux of the hybrid NI-FEM is in developing interfacial conditions at the common interfaces between the NIM regions and FEM regions. We here report the development of hybrid NI-FEM for the time-dependent convection-diffusion equation (CDE) in arbitrary domains. Resulting hybrid numerical scheme is implemented in a parallel framework in Fortran and solved using PETSc. The preliminary approach to domain decomposition is also discussed. Numerical solutions are compared with exact solutions, and the scheme is shown to be second order accurate in both space and time. The order of approximations used for the development of the scheme are also shown to be second order. The hybrid method is more efficient compared to standalone conventional numerical schemes like FEM.

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