Dr. Xu's research interest begins with fractional calculus and ordinary differential equation, inspired by his master and doctoral advisor, Professor Zhimin He, at the first year of graduate school. Later on, he was fortunately supported by China Scholarship Council to study and work with O.P. Agrawal (distinguished scholar, SIUC, USA) in several fields of fractional derivatives and their applications, such as theoretical aspects of generalized fractional calculus, numerical approximation of generalized fractional differential equations, fractional variational problems in convex domain, dynamics of fractional systems. After studying abroad, he came back and joined Department of Applied Mathematics, Central South University, the place where he deeply loves.
Since 2012, he gradually realized that his main interest moves to numerical methods for differential equations, as well as approximation theory, matrix theory and variational calculus. Innumerable ODEs and PDEs may not be solved by analytical methods like integral transforms and series expansions. Therefore, numerical tools including finite differences, finite elements and spectral methods will play significant role in studying differential equations. Moreover, nonlinear ones with certain singularity always exhibit more attractive phenomena. The research of numerical methods for finding blowup, quenching and multiple solutions seems to be not completed yet.
Thanks to China Postdoctoral Science Foundation, National Natural Science Foundation of China and University of Macau, he continued his research, theoretical and numerical fractional differential equations, with the help and guidance from Professors H.W. Sun and Q. Sheng (Baylor University, USA) since the end of 2015. Encouraged by them, he found that there are many mathematical models from combustion process, whose properties were not well understood in mathematics. The amazing quenching and blowup phenomena appear in several non-men-made problems, which are very important in industry also.
Now Dr. Xu and his colleagues put much attention on the reaction diffusion equation arising from combustion and explosion phenomena but rebuilt with temporal fractional derivatives and fractional Laplacian, as well as other possible nonlocal operators. It is challenge in general to find either theoretical or numerical techniques to deal with these mathematical models.
“How many secretes still hiding in fractional version of Stefan problem?” “Are we capable to build a conservation law for quenching and blowup problems?” He keeps searching, and expects some questions can be tackled in the near future.