Efficient compact finite difference method for variable coefficient fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions in conservative form

2018 ◽  
Vol 37 (5) ◽  
pp. 6252-6269 ◽  
Author(s):  
Lei Ren ◽  
Lei Liu
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Conditions ◽  
Difference Method ◽  
Variable Coefficient ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Diffusion Equations ◽  
Compact Finite Difference Method ◽  
Conservative Form

scholarly journals Eighth-Order Compact Finite Difference Scheme for 1D Heat Conduction Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Asma Yosaf ◽  
Shafiq Ur Rehman ◽  
Fayyaz Ahmad ◽  
Malik Zaka Ullah ◽  
Ali Saleh Alshomrani
Keyword(s):  
Heat Conduction ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Conditions ◽  
Difference Method ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Eighth Order ◽  
Compact Finite Difference ◽  
Compact Finite Difference Method

The purpose of this paper is to develop a high-order compact finite difference method for solving one-dimensional (1D) heat conduction equation with Dirichlet and Neumann boundary conditions, respectively. A parameter is used for the direct implementation of Dirichlet and Neumann boundary conditions. The introduced parameter adjusts the position of the neighboring nodes very next to the boundary. In the case of Dirichlet boundary condition, we developed eighth-order compact finite difference method for the entire domain and fourth-order accurate proposal is presented for the Neumann boundary conditions. In the case of Dirichlet boundary conditions, the introduced parameter behaves like a free parameter and could take any value from its defined domain but for the Neumann boundary condition we obtained a particular value of the parameter. In both proposed compact finite difference methods, the order of accuracy is the same for all nodes. The time discretization is performed by using Crank-Nicholson finite difference method. The unconditional convergence of the proposed methods is presented. Finally, a set of 1D heat conduction equations is solved to show the validity and accuracy of our proposed methods.

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