A Compact Difference Scheme for Fractional Sub-diffusion Equations with the Spatially Variable Coefficient Under Neumann Boundary Conditions

2015 ◽  
Vol 66 (2) ◽  
pp. 725-739 ◽  
Author(s):  
Seakweng Vong ◽  
Pin Lyu ◽  
Zhibo Wang
Keyword(s):  
Boundary Conditions ◽  
Difference Scheme ◽  
Variable Coefficient ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Diffusion Equations ◽  
Compact Difference Scheme ◽  
Compact Difference

scholarly journals A fast compact difference scheme for the fourth-order multi-term fractional sub-diffusion equation with non-smooth solution

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1495-1509
Author(s):  
Dakang Cen ◽  
Zhibo Wang ◽  
Yan Mo
Keyword(s):  
Diffusion Equation ◽  
Boundary Conditions ◽  
Difference Scheme ◽  
Fourth Order ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Stability And Convergence ◽  
Compact Difference Scheme ◽  
Compact Difference ◽  
Graded Meshes

In this paper, we develop a fast compact difference scheme for the fourth-order multi-term fractional sub-diffusion equation with Neumann boundary conditions. Combining L1 formula on graded meshes and the efficient sum-of-exponentials approximation to the kernels, the proposed scheme recovers the losing temporal convergence accuracy and spares the computational costs. Meanwhile, difficulty caused by the Neumann boundary conditions and fourth-order derivative is also carefully handled. The unique solvability, unconditional stability and convergence of the proposed scheme are analyzed by the energy method. At last, the theoretical results are verified by numerical experiments.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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