New in Wolfram Language | the Analytical solution of partial equations2 years ago
Transfer of a post of Devendra Kapadia "New in the Wolfram Language: Symbolic PDEs".
The code given in article can be downloaded here.
I express huge gratitude to Kirill Guzenko of KirillGuzenko for the help in transfer and preparation of the publication.
Partial equations (UrChP) play very important role in mathematics and its applications. They can be used for modeling of the real phenomena, such as fluctuations of the tense string, distribution of a heat flow in a rod, in financial areas. The purpose of this article — to slightly open a veil to the world of UrChP (who else is not familiar to those with it) and to acquaint the reader with how it is possible to solve effectively UrChP in Wolfram Language, using new functionality for a solution of boundary value problems in DSolve, and also new DEigensystem function which appeared in version 10.3.
History UrChP goes back to works of the famous mathematicians of the eighteenth century — Euler,Dalambera,Laplace, however development of this area in the last three centuries did not stop. And therefore I will provide both classical, and modern examples of UrChP in article that will allow to consider this knowledge domain under different corners.
Let's begin with consideration of fluctuations of the tense string with a length π, fixed on both ends. Fluctuations of a string can be simulated by means of the one-dimensional wave equation given below. Here u (x, t) — vertical shift of a point of a string with coordinate x in t timepoint: