Approximation Methods in Science and Engineering

__Approximation Methods in Engineering and Science__ covers fundamental and advanced topics in three areas: Dimensional Analysis, Continued Fractions, and Stability Analysis of the Mathieu Differential Equation. Throughout the book, a strong emphasis is given to concepts and methods used in everyday calculations. Dimensional analysis is a crucial need for every engineer and scientist to be able to do experiments on scaled models and use the results in real world applications. Knowing that most nonlinear equations have no analytic solution, the power series solution is assumed to be the first approach to derive an approximate solution. However, this book will show the advantages of continued fractions and provides a systematic method to develop better approximate solutions in continued fractions. It also shows the importance of determining stability chart of the Mathieu equation and reviews and compares several approximate methods for that. The book provides the energy-rate method to study the stability of parametric differential equations that generates much better approximate solutions. Preface Level of the Book Organization of the Book Method of Presentation Prerequisites Unit System Symbols Contents Part I Dimensional Analysis 1 Static Dimensional Analysis 1.1 Base Quantities and Units 1.2 Dimensional Homogeneity 1.3 Conversion of Units 1.4 Chapter Summary 1.5 Key Symbols Exercises References 2 Dynamic Dimensional Analysis 2.1 Buckingham pi-Theorem 2.2 Nondimensionalization 2.3 Model and Prototype Similarity Analysis 2.4 Size Effects 2.5 Chapter Summary 2.6 Key Symbols Exercises References Part II Continued Fractions 3 Numerical Continued Fractions 3.1 Rational and Irrational Numbers 3.2 Convergents of Continued Fractions 3.3 Convergence of Continued Fractions 3.4 Algebraic Equations 3.5 Chapter Summary 3.6 Key Symbols Exercises References 4 Functional Continued Fractions 4.1 Power Series Expansion of Functions 4.2 Continued Fractions of Functions 4.3 Series Solution of Differential Equations 4.3.1 Substituting Method 4.3.2 Derivative Method 4.4 Continued Fractions Solution of Differential Equations 4.4.1 Second-Order Linear Differential Equations 4.4.2 Series Solution Transformation 4.5 Chapter Summary 4.6 Key Symbols Exercises References Part III Approximation Tools 5 Mathieu Equation 5.1 Periodic Solutions of Order n=1 5.2 Periodic Solutions of Order nN 5.3 Recursive Method 5.4 Determinant Method 5.5 Continued Fractions of Characteristic Numbers 5.6 Chapter Summary 5.7 Key Symbols Exercises References 6 Energy-Rate Method 6.1 Differential Equations 6.2 Mathieu Stability Chart 6.3 Initial Conditions 6.4 Chapter Summary 6.5 Key Symbols Exercises References A Trigonometric Formulas B Unit Conversions General Conversion Formulas Conversion Factors Index "Approximation Methods in Engineering and Science covers fundamental and advanced topics in three areas: Dimensional Analysis, Continued Fractions, and Stability Analysis of the Mathieu Differential Equation. Throughout the book, a strong emphasis is given to concepts and methods used in everyday calculations. Dimensional analysis is a crucial need for every engineer and scientist to be able to do experiments on scaled models and use the results in real world applications. Knowing that most nonlinear equations have no analytic solution, the power series solution is assumed to be the first approach to derive an approximate solution. However, this book will show the advantages of continued fractions and provides a systematic method to develop better approximate solutions in continued fractions. It also shows the importance of determining stability chart of the Mathieu equation and reviews and compares several approximate methods for that. The book provides the energy-rate method to study the stability of parametric differential equations that generates much better approximate solutions. Covers practical model-prototype analysis and nondimensionalization of differential equations; Coverage includes approximate methods of responses of nonlinear differential equations; Discusses how to apply approximation methods to analysis, design, optimization, and control problems; Discusses how to implement approximation methods to new aspects of engineering and physics including nonlinear vibration and vehicle dynamics." -- Prové de l'editor