基本説明
Features: Clear introduction to Laplace- A Mathematica package for computing Laplace- and z-transforms, including numerical inversion of Laplace transforms.
Full Description
The theory of Laplace transformation is an important part of the mathematical background required for engineers, physicists and mathematicians. Laplace transformation methods provide easy and effective techniques for solving many problems arising in various fields of science and engineering, especially for solving differential equations. What the Laplace transformation does in the field of differential equations, the z-transformation achieves for difference equations. The two theories are parallel and have many analogies. Laplace and z transformations are also referred to as operational calculus, but this notion is also used in a more restricted sense to denote the operational calculus of Mikusinski. This book does not use the operational calculus of Mikusinski, whose approach is based on abstract algebra and is not readily accessible to engineers and scientists. The symbolic computation capability of Mathematica can now be used in favor of the Laplace and z-transformations. The firstversion of the Mathematica Package LaplaceAndzTransforrns developed by the author appeared ten years ago. The Package computes not only Laplace and z-transforms but also includes many routines from various domains of applications. Upon loading the Package, about one hundred and fifty new commands are added to the built-in commands of Mathematica. The code is placed in front of the already built-in code of Laplace and z-transformations of Mathematica so that built-in functions not covered by the Package remain available. The Package substantially enhances the Laplace and z-transformation facilities of Mathematica. The book is mainly designed for readers working in the field of applications.
Contents
1 Laplace Transformation.- 1.1 The One-Sided Laplace Transform.- 1.2 The Two-Sided Laplace Transform.- 1.3 Ordinary Linear Differential Equations.- 2 z-Transformation.- 2.1 z-Transforms and Inverse z-Transforms.- 2.2 Difference Equations.- 3 Laplace Transforms with the Package.- 3.1 Basics.- 3.2 The Use of Transformation Rules.- 3.3 The Finite Laplace Transform.- 3.4 Special Functions.- 3.5 Inverse Laplace Transformation.- 3.6 Differential Equations.- 4 z-Transformation with the Package.- 4.1 Basics.- 4.2 Use of Transformation Rules.- 4.3 Difference Equations.- 5 Applications To Automatic Control.- 5.1 Controller Configurations.- 5.2 State-Variable Analysis.- 5.3 Second Order Differential Systems.- 5.4 Stability.- 5.5 Frequency Analysis.- 5.6 Sampled-Data Control Systems.- 6 Laplace Transformation: Further Topics.- 6.1 The Complex Inversion Formula.- 6.2 Laplace Transforms and Asymptotics.- 6.3 Differential Equations.- 7 z-Transformation: Further Topics.- 7.1 The Advanced z-Transformation.- 7.2 Applications.- 7.3 Use of the Package.- 8 Examples from Electricity.- 8.1 Transmission Lines.- 8.2 Electrical Networks.- 9 Examples from Control Engineering.- 9.1 Control of an Inverted Pendulum.- 9.2 Controling a Seesaw-Pendulum.- 9.3 Control of a DC Motor.- 9.4 A Magnetic-Ball-Suspension-System.- 9.5 A Sampled-Data State-Variable Control System.- 10 Heat Conduction and Vibration Problems.- 10.1 Flow of Heat.- 10.2 Waves and Vibrations in Elastic Solids.- 11 Further Techniques.- 11.1 Duhamel's Formulas.- 11.2 Green's Functions.- 11.3 Fundamental Solutions.- 11.4 Finite Fourier Transforms.- 12 Numerical Inversion of Laplace Transforms.- 12.1 Inversion by the Use of Laguerre Functions.- 12.2 Inversion by Use of Fourier Analysis.- 12.3 The Use of Gaussian Quadrature Formulas.- 12.4 The Method of Gaver and Stehfest.- 12.5 Example.- Appendix: Package Commands.
