Introduction to computation and modeling for differential equations. (2015)

• Record Type: Book
• Further Information: Note: Lennart Edsberg.
• Authors: Edsberg, Lennart, 1946-
• Contents: Chapter 1, Introduction 1.1 What is a Differential Equation? 1.2 Examples of an ordinary and a partial differential equation 1.3 Numerical analysis, a necessity for scientific computing 1.4 Outline of the contents of this book Bibliography Chapter 2, Ordinary differential equations 2.1 Problem classification 2.2 Linear systems of ODEs with constant coefficients 2.3 Some stability concepts for ODEs 2.3.1 Stability for a Solution Trajectory of an ODE system 2.3.2 Stability for critical points of ODE-systems 2.4 Some often used ODE-models in science and engineering 2.4.1 Newton’s second law 2.4.2 Hamilton’s equations 2.4.3 Electrical networks 2.4.4 Chemical kinetics 2.4.5 Control theory 2.4.6 Compartment models 2.5 Some examples from applications Bibliography Chapter 3, Numerical methods for initial value problems 3.1 Graphical representation of solutions 3.2 Basic principles of numerical approximation of ODEs 3.3 Numerical solution of IVPs with Euler’s method 3.3.1 Euler’s explicit method: Accuracy 3.3.2 Euler’s explicit method: Improving the Accuracy 3.3.3 Euler’s explicit method: Stability 3.3.4 Euler’s implicit method 3.3.5 The trapezoidal method 3.4 Higher order methods for the IVP 3.4.1 Runge-Kutta methods 3.4.2 Linear Multistep methods 3.5 Special methods for special problems 3.5.1 Preserving linear and quadratic invariants 3.5.2 Preserving positivity of the numerical solution 3.5.3 Methods for Newton’s equations of motion 3.6 The variational equation and parameter fitting in IVPs Bibliography Chapter 4. Numerical methods for boundary value problems 4.1 Applications 4.2 Difference Methods for BVPs 4.2.1 A model problem for BVPs, Dirichlet’s BCs 4.2.2 A model problem for BVPs, mixed BCs 4.2.3 Accuracy 4.2.4 Spurious solutions 4.2.5 Linear Two-Point BVPs 4.2.6 Nonlinear Two-Point BVPs 4.2.7 The shooting method 4.3 Ansatz methods for BVPs 4.3.1 Starting with the ODE formulation 4.3.2 Starting with the weak formulation 4.3.3 The Finite Element Method Bibliography Chapter 5, Partial differential equations 5.1 Classical PDE-problems 5.2 Differential operators used for PDEs 5.3 Some PDEs in science and engineering 5.3.1 Navier-Stokes equations for incompressible flow 5.3.2 Euler’s equations for compressible flow 5.3.3 The Convection-Diffusion-Reaction Equations 5.3.4 The heat equation 5.3.5 The diffusion equation 5.3.6 Maxwell’s equations for the electromagnetic field 5.3.8 Schrödinger’s equation in quantum mechanics 5.3.9 Navier’s equations in structural mechanics 5.3.10 Black-Scholes equation in financial mathematics 5.4 Initial and boundary conditions for PDEs 5.5 Numerical solution of PDEs, some general comments Bibliography Chapter 6. Numerical Methods for Parabolic Partial Differential Equations 6.1 Applications 6.2 An Introductory Example of Discretization 6.3 The Method of Lines for Parabolic PDEs 6.3.1 Solving the Model Problem with MoL 6.3.2 Various Types of Boundary Conditions 6.3.3 An Example of the Use of MoL for a Mixed Boundary Condition 6.4 Generalizations of the Heat Equation 6.4.1 The Heat Equation with Variable Conductivity 6.4.2 The Convection-Diffusion-Reaction PDE 6.4.3 The General Nonlinear Parabolic PDE Ansatz Methods for the Model Problem Bibliography Chapter 7. Numerical methods for elliptic partial differential equations 7.1 Applications 7.2 The Finite Difference Method 7.3 Discretization of a Problem with Different BCs 7.4 Ansatz methods for elliptic PDEs 7.4.1 Starting with the PDE formulation 7.4.2 Starting with the weak formulation 7.4.3 The Finite Element Method Bibliography Chapter 8. Numerical methods for hyperbolic PDEs 8.1 Applications 8.2 Numerical solution of hyperbolic PDEs 8.2.1The Upwind Method (FTBS) 8.2.2 The FTFS method 8.2.3 The FTCS method 8.2.4 The Lax-Friedrichs method 8.2.5 The leap-frog method 8.2.6 The Lax-Wendroff Method 8.2.7 Numerical method for the wave equation 8.3 The Finite Volume Method 8.4 Some examples of stability analysis for hyperbolic PDEs Bibliography Chapter 9, Mathematical Modeling with Differential Equations 9.1 Laws of Nature 9.2 Constitutive Equations 9.2.1 Equations of Heat Transfer Problems 9.2.2 Equations in Mass Diffusion Problems 9.2.3 Equations in Mechanical Moment Diffusion Problems 9.2.4 Equations in Elastic Solid Mechanics Problems 9.2.5 Equations in Chamical Reaction Engineering Problems 9.2.6 Equations in Electrical Engineering Problems 9.3 Conservation Laws 9.3.1 Some Examples of Lumped Models 9.3.2 Some Examples of Distributed Models 9.4 Scaling of Differential Equations to Dimensionless Form Bibliography Chapter 10. Applied projects on Differential Equations Project 1. Signal Propagation in a long electrical conductor Project 2. Flow in a cylindrical pipe Project 3. Soliton waves Project 4. Wave scattering in a wave guide Project 5. Metal block with heat sourse and thermometer Project 6. Deformation of a circular metal plate Project 7. Cooling of a chrystal glass Project 8. Rotating fluid in a cylinder A. Appendix: Some Numrical and Mathematical Tools A.1 Newton’s Method for Systems of Nonlinear Algebraic Equations A.1.1 Square Systems A.1.2 Overdetermined Systems A.2 Some Facts about Linear Difference Equations A.3 Derivation of Difference Approximations A.4 The Interpretations of Grad, Div and Curl A.5 Numerical Solution of Algebraic Systems of Equations A.5.1 Direct Methods A.5.2 Iterative Methods for Linear Systems of Equations A.6 Some results for Fourier Transforms B. Appendix: Software for Scientific Computing B.1 MATLAB (R) B.2 COMSOL Multiphysics (R) Bibliography C. Appendix: Computer Exercises to Support the Chapters
• Edition: Second edition
• Publisher Details: Hoboken, New Jersey : John Wiley & Sons
• Publication Date: 2015
• Extent: 1 online resource
• Subjects: 515.350285; Differential equations -- Data processing; Mathematical models
• Languages: English
• ISBNs: 9781119018469
• Related ISBNs: 9781119018452
• Notes: Note: Includes bibliographical references and index.; Note: Description based on CIP data; resource not viewed.
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