Application Examples for Distributions
摘要
The chapter is devoted to practice applications of distributions in various fields. This includes generalized Fourier series, fundamental solutions for linear differential equations with constant coefficients in a myriad of applications. Linear circuits and input-output relations by convolution of input signals with the impulse or step response form general examples. For 3D problems, the fundamental solution for the potential equation is calculated and applied in examples. Furthermore, as one of the most important practice applications, the method of finite elements (FEM) is treated, and its solution in a suitable Sobolev space by the Lax-Milgram theorem is shown and applied in examples, e.g., the Poisson boundary value problem. The problem of the vibrating string from the beginning in Chap. 1 is now solved in the sense of distributions, i.e., weak solutions are obtained as in the FEM problems.