Analysis and numerical approximation to time-fractional diffusion equation with a general time-dependent variable order

2021 ◽  
Author(s):  
Xiangcheng Zheng ◽  
Hong Wang
Keyword(s):  
Diffusion Equation ◽  
Numerical Approximation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Time Dependent ◽  
Variable Order ◽  
Time Fractional Diffusion Equation ◽  
General Time

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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