Analysis of a hidden memory variably distributed-order space-fractional diffusion equation
The wellposedness of a space-fractional diffusion equation governed by a hidden memory variably distributed-order fractional derivative was considered in the work. Recall that according to J. Hadamard, a fractional diffusion equation is wellposed if it satisfies the following conditions: i) a solution to the equation exists (existence problem); ii) the solution is uniqueness (uniqueness problem); and iii) the solution's behaviour changes continuously with respect to the boundary conditions (stability problem). The authors made two assumptions on the probability density function (\(\omega\)): \begin{itemize} \item[(i)] \(\operatorname{supp}\omega (\cdot, x) \in [0,\bar{\gamma}(x)]\) for \(0 \leq \bar{\gamma}(x) \leq \gamma *\) with \(0<\gamma * <1\), \(x\in [0,1]\). \item[(ii)] \(\bar{\gamma} \in C^1[0,\gamma ^*]\) and \(\omega, \ \omega _x \in C([0,\gamma ^*];[0,1])\). \end{itemize} The hidden memory variably distributed-order space-fractional diffusion equation was first transformed into into a Volterra integral equation before integrating in order to its solution. To address the uniqueness and stability problems, the authors stated and proved Theorem 2.1 and Theorem 2.2. In Theorem 2.1, which lays the foundation for the uniqueness of the solution, the authors defined a functional sequence \(\{v_n(x)\}_{n=0}^{\infty}\) on \([0,1]\) by \(v_{n+1}(x):=v_0(x)-d(x) _0I_{x}^{\omega}v_n(x)\), \(\geq 0\), \(v_0(x):=-f(x)\) and showed that this sequence converges uniformly on \([0,1]\). The uniqueness follows from this result, although the proof is skipped by the authors. Theorem 2.2 shows that the solution is uniformly continuous, thereby addressing the stability problem.
variably distributed-order fractional derivative
scientific article; zbMATH DE number 7443282
scientific article