Nonlinear difference scheme for fractional equation with functional delay

2020 ◽  
Author(s):  
Tatiana Gorbova ◽  
Svyatoslav Solodushkin
Keyword(s):  
Difference Scheme ◽  
Functional Delay ◽  
Fractional Equation ◽  
Nonlinear Difference Scheme

scholarly journals Fast High-Order Difference Scheme for the Modified Anomalous Subdiffusion Equation Based on Fast Discrete Sine Transform

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Lijuan Nong ◽  
An Chen
Keyword(s):  
Difference Scheme ◽  
Order Difference ◽  
Compact Difference ◽  
Nonsmooth Problems ◽  
Sine Transform ◽  
Two Space Dimensions ◽  
Subdiffusion Equation ◽  
Correction Terms ◽  
Fractional Orders ◽  
Fractional Equation

The modified anomalous subdiffusion equation plays an important role in the modeling of the processes that become less anomalous as time evolves. In this paper, we consider the efficient difference scheme for solving such time-fractional equation in two space dimensions. By using the modified L1 method and the compact difference operator with fast discrete sine transform technique, we develop a fast Crank-Nicolson compact difference scheme which is proved to be stable with the accuracy of O τ min 1 + α , 1 + β + h 4 . Here, α and β are the fractional orders which both range from 0 to 1, and τ and h are, respectively, the temporal and spatial stepsizes. We also consider the method of adding correction terms to efficiently deal with the nonsmooth problems. Numerical examples are provided to verify the effectiveness of the proposed scheme.

Symmetrizable Difference Dchemes

2005 ◽  
Vol 5 (1) ◽  
pp. 3-50 ◽  
Author(s):  
Alexei A. Gulin
Keyword(s):  
Initial Data ◽  
Difference Scheme ◽  
Difference Schemes ◽  
Time Dependent ◽  
Stability Boundaries ◽  
Typical Representative ◽  
Variable Weight ◽  
The Stability ◽  
Conditions Of Stability ◽  
The Right

AbstractA review of the stability theory of symmetrizable time-dependent difference schemes is represented. The notion of the operator-difference scheme is introduced and general ideas about stability in the sense of the initial data and in the sense of the right hand side are formulated. Further, the so-called symmetrizable difference schemes are considered in detail for which we manage to formulate the unimprovable necessary and su±cient conditions of stability in the sense of the initial data. The schemes with variable weight multipliers are a typical representative of symmetrizable difference schemes. For such schemes a numerical algorithm is proposed and realized for constructing stability boundaries.

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