دانلود کتاب مقدمه ای بر مدل سازی ریاضی چند مقیاسی

This book is based on the courses given for the master’s and Ph.D. students of 2015–2021 batch at Skoltech (Moscow), University of Chile (Santiago), University Jean Monnet (Saint-Etienne, France), University of Benevento (Italy), and Politecnico di Torino. The third chapter was given as a mini-course at the 9th International Moscow Winter School in Physics. The book introduces main mathematical models describing mechanical behavior at microscopic level of heterogeneous media and for blood flow in a network of vessels. Homogenization technique is applied for multiscale analysis of heterogeneous media. For the network of vessels, asymptotic methods (matching, boundary layers) are presented. The method of asymptotic partial decomposition of the domain defines hybrid dimension models combining one-dimensional (1-D) description obtained from the dimension reduction with three-dimensional (3-D) zooms. It justifies the special exponentially precise junction conditions at the interface of 1-D and 3-D parts. It can be applied to model the blood flow in vessels with a thrombus or a stent. The course contains an important introductory part recalling the necessary mathematical background so that it is accessible not only for the master’s and Ph.D. students in mathematics but also for those in engineering and biophysics.