Partial dynamic equations on time scales
Jackson, Billy Joe
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In this work, we introduce the basic terminology of the time scales calculus. In particular, the notions of derivatives and integrals on arbitrary closed sets are presented and examples are given. From there, ordinary dynamic equations and operators are presented and examples are given. Solutions are formed in terms of the exponential function of the particular time scale involved. Within this context, we present the definition of the exponential function on an arbitrary time scale and again give examples. In the final chapter, we present several new ideas based on the generalization of the ideas of the univariate case from the previous chapters to the bivariate case. In particular, three partial dynamic operators are introduced and discussed, and some solutions to these equations are provided in the instances where the functions involved are elementary by means of the Laplace Transform.