Partial Dynamic Equations: Wave, Parabolic and Elliptic Equations on Time Scales

This book is devoted to the qualitative theory of partial dynamic equations on arbitrary time scales. The results in the book generalize the classical results, and they unify the discrete and continuous cases. The book starts with classification and canonical forms for second-order PDEs. Next, the Laplace transform method and the Fourier transform method are introduced. The Fourier transform is applied to solving second-order PDEs. The method of separation of variables is considered later in the book. The following few chapters are devoted to factoring second-order PDEs, including the wave equation, the heat equation, and the Laplace equation. It proves the weak maximum principle and as its application is investigated the stability of the solutions of the Poisson equation. Finally, the reduction of some nonlinear PDEs to the wave equation, the heat equation, and the Laplace equation are discussed. Тhe main advantage of the book is that it offers a variety of analytical techniques for the study of partial dynamical equations and that the results obtained over arbitrary time scales can be used to derive results in the classical case and in the discrete case.