Partial dynamic equations on time scales

摘要

In this work, we generalize existing ideas of the univariate case of the time scales calculus to the bivariate case. Formal definitions of partial derivatives and iterated integrals are offered, and bivariate partial differential operators are examined. In particular, solutions of the homogeneous and nonhomogeneous heat and wave operators are found when initial distributions given are in terms of elementary functions by means of the generalized Laplace Transform for the time scale setting. Finally, the so-termed mixed time scale setting is discussed. Examples are given and solutions are provided in tabular form.