scholarly journals The numerical solution of high dimensional variable-order time fractional diffusion equation via the singular boundary method

2021 ◽  
Author(s):  
Vahid Reza Hosseini ◽  
Farzaneh Yousefi ◽  
W.-N. Zou
Keyword(s):  
Diffusion Equation ◽  
Numerical Solution ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
High Dimensional ◽  
Variable Order ◽  
Singular Boundary ◽  
Time Fractional Diffusion Equation ◽  
Dimensional Variable ◽  
Singular Boundary Method

Wellposedness and regularity of a variable-order space-time fractional diffusion equation

2020 ◽  
Vol 18 (04) ◽  
pp. 615-638 ◽  
Author(s):  
Xiangcheng Zheng ◽  
Hong Wang
Keyword(s):  
Diffusion Equation ◽  
Linear Space ◽  
Fractional Diffusion ◽  
Space Time ◽  
Fractional Diffusion Equation ◽  
Variable Order ◽  
Order Linear ◽  
Integer Order ◽  
Time Fractional Diffusion Equation ◽  
Full Regularity

We prove wellposedness of a variable-order linear space-time fractional diffusion equation in multiple space dimensions. In addition we prove that the regularity of its solutions depends on the behavior of the variable order (and its derivatives) at time [Formula: see text], in addition to the usual smoothness assumptions. More precisely, we prove that its solutions have full regularity like its integer-order analogue if the variable order has an integer limit at [Formula: see text] or have certain singularity at [Formula: see text] like its constant-order fractional analogue if the variable order has a non-integer value at time [Formula: see text].

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