Analytic Solution of Linear Difference Equations with Applications
تحسين المغربي, جميلة حمدي زامل أبو مارية
جامعة القدس · فلسطين
The purpose of this work is to study and investigate topics related to the theory and applications of linear difference equations. After introducing the basic concepts related to the existence and uniqueness of the solution for the non-homogeneous equation [ 2 ], we present the notion of equilibrium points, stability, asymptotic stability and the phenomena of chaotic behavior of a solution [ 7 ], where we discuss the differences that arise about a strict definition of chaos which was initiated by Devany in 1986 and it is now the widely known and accepted definition. Moreover, we investigate the unpredictable behavior of a solution in the interval where the solution jumps randomly, this phenomena it explained by considering the logistic map and using simulation. We further extend our study to higher order linear difference equations where the general solution of homogeneous equationsis presented. For the non-homogeneous case the method of undetermined coefficients is employed.Real –life problems whose behavior for large time can be described by difference equations can be solved using the theory and methods contained in this thesis. This is illustrated with specific models from different fields: From communication we consider the transmission of information model, our next model comes from economics where we represent the analysis of national income of a certain country. Another important model comes from biological sciences where the concern is on the description of the number of plants in any desired generation during growth seasons. vThese models which represent real-life phenomena in diverse fields where difference equations are the natural setting for such problems