Abstract: In the late 1800s, Oliver Heaviside popularized a technique for solving differential equations by treating derivatives and integrals as variables. Heaviside was able to derive correct results, but did not rigorously justify his methods. In the early 1900s, many mathematicians attempted to formalize Heaviside’s work by use of integral transforms. These attempts were successful enough to make their way into many undergraduate differential equations curricula. Other mathematicians, however, seeked justification in Heaviside’s work through algebraic means. In the 1950s, Jan Mikusiński developed a formalization of Heaviside’s methods by use of a convolution ring. In this talk, we will explore Mikusiński’s operational calculus and give meaning to the expressions \(\frac{1}{\frac{d}{dt}-1}\), \(\sqrt{\frac{d}{dt}}\), and \(e^{-\frac{d}{dt}}\).
Thackeray Hall 703