Solving integro-differential equations arising in material science

My name is Szymon Sobieszek and currently I am a fourth-year Applied Mathematics student at the Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Poland.

My scientific interests are mostly aimed at the deterministic methods in industrial mathematics. In connection to these, I have recently (July 17) participated in 30th edition of the ECMI Modelling Week which took place in Sofia, Bulgaria. During this event, students from all over the Europe were working in small groups (under a supervision of the instructor) on projects that were based on the real-life problems. The task of my group was to find the optimal shape of an electric motor, that is the shape which produced the highest torque. To solve this task we had to use our knowledge gained during mathematical studies on such courses as partial differential equations, numerical methods and optimization. I believe that participating in such projects and seeing how one can use (sometimes quite abstract) university knowledge to overcome industrial problems can be highly motivating for future study.

Apart from that, I am currently working on my undergraduate thesis under supervision of Łukasz Płociniczak, PhD. It is closely connected to the publication that we’ve recently submitted. It concerns numerical schemes in integro-differential equations with Erdelyi-Kober (E-K) fractional operator. We have there proposed two different ways of discretizing it and proved their errors asymptotic behaviour. Investigating them is very important in every numerical-connected work. It gives us some criterion based on which we can choose one method over the others. In the picture below you can see error’s asymptotic behaviour of the two kind of discretizing operators, denoted by L and K (with superscripts corresponding with variants of operators – “r” stands for rectangular rule and “t” for trapezoidal rule). For fixed evaluation interval, the “ratio” denotes the ratio of numerical evaluation of the E-K operator on a test function and the analytical value which was found by us. As we can see with increasing number of interval divisions n, the ratio converges to the unity what confirms our theoretical studies.

As you can see, the precision of the algorithms vary, which was one of the properties we wanted to show.

The significance of the acquired results is particularly important from the applied point of view. As some recent experiments have shown, certain building materials exhibit different than usual pace of moisture infiltration. This phenomenon is known as an anomalous diffusion and the equation I am investigating models its particular case (subdiffusion). Knowing how water percolates porous media of great complexity is very important from the point of view of construction engineering. Having an efficient numerical method of solving these complicated problems can greatly aid the experiment. In my opinion, it is an interesting and rewarding field of study.