Lattice, quantum analysis and applications to discrete D'Alembert problems

In this thesis, we study on lattice and quantum numbers. First, similar to the ordinary calculus, we build h-nabla, h-delta derivatives, h-binomial, h-exponential, h-trigonometric, h-hyperbolic functions and we introduce h-nabla antiderivative. Since h-analysis recovers real numbers as h tends to zero, the study on h-analysis can be called as "discretization" (h-discretization). As an application, we construct h-analogue of problems of wave equations in other words, h-D'Alembert problems and we find h-D'Alembert solutions in terms of h-binomial functions. Also we define h-analyticity and using h-analogue of Cauchy-Kovalevskaya Theorem we present some examples which have unique and analytic solutions. Next, we construct quantum analysis with related q-derivative, 1/q -derivative and q-integral, 1/q -integral properties. Finally we construct q and 1/q-D'Alembert problems and their solutions. By using q-analogue of Cauchy Kovalevskaya Theorem, we examine the uniqueness and q-analyticity of these solutions.