At the beginning of 2023, a research paper entitled "Symmetry and Free Boundary Points in a Class of Linear Ordinary Differential Equations" was published in the SIAM Undergraduate Research Online (SIURO). This research report was written by four Mathematics with Applied Mathematics (4+0) students, recording their research on ordinary differential equations under the supervision of Professor Behrouz Emamizadeh and Dr Amin Farjudian in the summer of 2021.

The Society for Industrial and Applied Mathematics (SIAM) is famous for its publication of quality, peer-reviewed journals that are respected and highly-cited throughout the applied mathematics and computational science community. SIURO is an open-access web-based publication for undergraduates, publishing reviewed undergraduate manuscripts since 2008.

Ordinary differential equations (ODEs) arise in many contexts of mathematics and social and natural sciences. Scientific fields include much of physics and astronomy, meteorology, chemistry, biology, ecology and population modeling, economics.

This paper is concerned with the qualitative properties of the solutions of a class of linear ordinary differential equations. The existence and uniqueness of solutions are addressed, and properties of the graph of the solution when imposing some restrictions are derived. A new notion of derivative, called the force derivative, is introduced and an orthogonality result, between the force derivative of the solution and the force function, is obtained. All the important results are verified by numerical examples using MATLAB. Finally, an inequality result reminiscent of the famous G. Talenti's inequality is proved.

Two supervisors of the project have brought great support to students. Professor Behrouz Emamizadeh is the Head of the Partial Differential Equations Research Group. Dr Amin Farjudian, from the Department of Computer Science, has expertise in many interdisciplinary areas such as ordinary differential equations and partial differential equations, algorithmic randomness, data mining, set theory and philosophy of language.

The students of the project team said that the experience of the summer research made them gain a profound understanding of the process of research. Feng Jiang, who is currently studying for MSc in Quantitative Finance at the ETH Zürich, said that “I really enjoy the process of losing track of time and entering a state of flow while solving mathematical problems.”

This research experience has no doubt benefiting students in their further studies. Zhengyang Guo mentioned that “the acceptance of this paper by SIURO is so important to my scientific research experience. I think thanks to that, I can be admitted to Columbia University for MSc in Applied Statistics."

Another two members of the team, Yanghao Wang and Dong’ang Liu are now studying Fintech and Data Science at the Imperial College London and the University of Macau respectively.

According to preliminary statistics, there are 67 Mathematics with Applied Mathematics graduates in 2022 choose to study a postgraduate programme, of which 40.3% are admitted to the top 10 universities in the world.