Introduction to Computation and Modeling for Differential Equations 2nd Edition by Lennart Edsberg 9781119018445 1119018447

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ISBN 10: 1119018447

ISBN 13: 9781119018445

Author: Lennart Edsberg

Introduction to Computation and Modeling for Differential Equations, Second Edition features the essential principles and applications of problem solving across disciplines such as engineering, physics, and chemistry. The Second Edition integrates the science of solving differential equations with mathematical, numerical, and programming tools, specifically with methods involving ordinary differential equations; numerical methods for initial value problems (IVPs); numerical methods for boundary value problems (BVPs); partial differential equations (PDEs); numerical methods for parabolic, elliptic, and hyperbolic PDEs; mathematical modeling with differential equations; numerical solutions; and finite difference and finite element methods.

The author features a unique “Five-M” approach: Modeling, Mathematics, Methods, MATLAB®, and Multiphysics, which facilitates a thorough understanding of how models are created and preprocessed mathematically with scaling, classification, and approximation and also demonstrates how a problem is solved numerically using the appropriate mathematical methods. With numerous real-world examples to aid in the visualization of the solutions, Introduction to Computation and Modeling for Differential Equations, Second Edition includes:

• New sections on topics including variational formulation, the finite element method, examples of discretization, ansatz methods such as Galerkin’s method for BVPs, parabolic and elliptic PDEs, and finite volume methods
• Numerous practical examples with applications in mechanics, fluid dynamics, solid mechanics, chemical engineering, heat conduction, electromagnetic field theory, and control theory, some of which are solved with computer programs MATLAB and COMSOL Multiphysics®
• Additional exercises that introduce new methods, projects, and problems to further illustrate possible applications
• A related website with select solutions to the exercises, as well as the MATLAB data sets for ordinary differential equations (ODEs) and PDEs

Introduction to Computation and Modeling for Differential Equations, Second Edition is a useful textbook for upper-undergraduate and graduate-level courses in scientific computing, differential equations, ordinary differential equations, partial differential equations, and numerical methods. The book is also an excellent self-study guide for mathematics, science, computer science, physics, and engineering students, as well as an excellent reference for practitioners and consultants who use differential equations and numerical methods in everyday situations.

Table of contents:

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

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 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

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

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

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
 5.3.7 Acoustic Waves
 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

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 Test 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
6.5 Ansatz Methods for the Model Equation

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

8 Numerical Methods for Hyperbolic PDEs
8.1 Applications
8.2 Numerical Solution of Hyperbolic PDEs
 8.2.1 The 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

9 Mathematical Modeling with Differential Equations
9.1 Nature Laws
9.2 Constitutive Equations
 9.2.1 Equations in 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 Chemical Reaction Engineering Problems
 9.2.6 Equations in Electrical Engineering Problems
9.3 Conservative Equations
 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

10 Applied Projects on Differential Equations

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Tags: Lennart Edsberg, Introduction, Computation, Modeling