부산대학교 도서관

In this work, the authors consider the $d$-dimensional time-space fractional partial differential equations, with $d\ge 3$, where the Caputo fractional derivative of order $\gamma$ is considered in a time variable, with $0 < \gamma < 1$, and the $\alpha_k$-order left and right Riemann-Liouville fractional derivatives are considered in the spatial variables, with ${1 < \alpha_k < 2}$. \par To solve the continuous problem, an alternating direction implicit method (ADI) is used to discretize in time on a uniform mesh. The Caputo derivative is approximated by a $L_1$ approximation, and the linear combination of Riemann-Liouville fractional derivatives are approximated by shifted Grünwald approximations. These types of approximations, for the fractional derivatives, in practice require a huge computational cost and storage. \par To reduce those difficulties, a tensor-train format is used, which is a good choice to obtain a faster solver in the context of the numerical resolution of fractional partial differential equations. In order to apply this technique, the ADI scheme is reformulated in an appropriate way, and therefore the divide-and-conquer solver is better defined. \par In my opinion, the main interest of the article is that an adequate method is used to considerably reduce the computational cost and the storage need of the numerical method used to solve an important type of partial differential equation. These problems are more difficult than classical problems, because fractional derivatives appear in both the time and the spatial variables. The new algorithm permits one to obtain good and efficient solutions for the type of problems considered, which are good models for many problems that appear in different areas related to applied mathematics.