Abstract
A time-fractional substantial diffusion equation of variable order is investigated, in which the variable-order fractional substantial derivative accommodates the memory effects and the structure change of the surroundings of the physical processes with respect to time. The existence and uniqueness of the solutions to the proposed model are proved, based on which the weighted high-order regularity of the solutions, in which the weight function characterizes the singularity of the solutions, are analyzed.
Highlights
The Caputo fractional derivative is widely used in various applications that exhibit the power law type memory effects due to the power function integral α(·,t)
The Caputo variable-order fractional substantial derivative operator σ ∂t is defined in the following for some σ ≥ 0 and 0 ≤ α(s, t) < 1 [1–3]
A weight function is introduced in the estimate of high-order smoothing properties to characterize the initial singularity of the solutions
Summary
There are several investigations in mathematical analysis and numerical methods to such kinds of problems [3–9], while the corresponding studies for the variable-order fractional substantial derivative models, in which the variable fractional order accommodates, e.g., the structure change of the surroundings with respect to time, are rarely found in the literature. The following Caputo variable-order time-fractional diffusion equations are investigated mathematically and numerically [10–16]. The Caputo variable-order fractional derivative operator is defined as [17–19]. The Caputo (variable-order) fractional derivative is widely used in various applications that exhibit the power law type memory effects due to the power function integral α(·,t). The Caputo variable-order fractional substantial derivative operator σ ∂t is defined in the following for some σ ≥ 0 and 0 ≤ α(s, t) < 1 [1–3]. Q denotes a generic constant that may assume different values at different cases
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