Ebook: Difference Equations and Their Applications
- Tags: Difference and Functional Equations, Partial Differential Equations, Mathematical Modeling and Industrial Mathematics
- Series: Mathematics and Its Applications 250
- Year: 1993
- Publisher: Springer Netherlands
- Edition: 1
- Language: English
- pdf
The theory of difference equations is now enjoying a period of Renaissance. Witness the large number of papers in which problems, having at first sight no common features, are reduced to the investigation of subsequent iterations of the maps f· IR. m ~ IR. m, m > 0, or (which is, in fact, the same) to difference equations The world of difference equations, which has been almost hidden up to now, begins to open in all its richness. Those experts, who usually use differential equations and, in fact, believe in their universality, are now discovering a completely new approach which re sembles the theory of ordinary differential equations only slightly. Difference equations, which reflect one of the essential properties of the real world-its discreteness-rightful ly occupy a worthy place in mathematics and its applications. The aim of the present book is to acquaint the reader with some recently discovered and (at first sight) unusual properties of solutions for nonlinear difference equations. These properties enable us to use difference equations in order to model complicated os cillating processes (this can often be done in those cases when it is difficult to apply ordinary differential equations). Difference equations are also a useful tool of syn ergetics- an emerging science concerned with the study of ordered structures. The application of these equations opens up new approaches in solving one of the central problems of modern science-the problem of turbulence.
Content:
Front Matter....Pages i-xii
Introduction....Pages 1-13
Introduction to the Theory of Dynamical Systems....Pages 15-43
Periodic Trajectories....Pages 45-70
Behavior of Trajectories....Pages 71-94
Dynamical Systems for U-Maps....Pages 95-124
Nonlinear Difference Equations....Pages 125-158
Difference Equations with U-Nonlinearity....Pages 159-185
Front Matter....Pages 187-187
Completely Integrable Differential-Difference Equations....Pages 188-222
Differential-Difference Equations Close to Difference Ones....Pages 223-237
Singularly Perturbed Differential-Difference Equations....Pages 239-272
Front Matter....Pages 273-273
Reduction of Boundary Value Problems to Difference and Differential-Difference Equations....Pages 274-284
Boundary-Value Problem for a System with Small Parameter....Pages 285-304
Boundary-Value Problem for Systems with Two Spatial Variables....Pages 305-334
Back Matter....Pages 335-358
Content:
Front Matter....Pages i-xii
Introduction....Pages 1-13
Introduction to the Theory of Dynamical Systems....Pages 15-43
Periodic Trajectories....Pages 45-70
Behavior of Trajectories....Pages 71-94
Dynamical Systems for U-Maps....Pages 95-124
Nonlinear Difference Equations....Pages 125-158
Difference Equations with U-Nonlinearity....Pages 159-185
Front Matter....Pages 187-187
Completely Integrable Differential-Difference Equations....Pages 188-222
Differential-Difference Equations Close to Difference Ones....Pages 223-237
Singularly Perturbed Differential-Difference Equations....Pages 239-272
Front Matter....Pages 273-273
Reduction of Boundary Value Problems to Difference and Differential-Difference Equations....Pages 274-284
Boundary-Value Problem for a System with Small Parameter....Pages 285-304
Boundary-Value Problem for Systems with Two Spatial Variables....Pages 305-334
Back Matter....Pages 335-358
....