{"product_id":"introduction-to-computation-and-modeling-for-differential-equations-isbn-9781119018445","title":"Introduction to Computation and Modeling for Differential Equations","description":"\u003cp\u003e\u003cb\u003eUses mathematical, numerical, and programming tools to solve differential equations for physical phenomena and engineering problems\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003ci\u003eIntroduction to Computation and Modeling for Differential Equations, Second Edition \u003c\/i\u003efeatures the essential principles and applications of problem solving across disciplines such as engineering, physics, and chemistry. The \u003ci\u003eSecond Edition \u003c\/i\u003eintegrates 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.\u003c\/p\u003e \u003cp\u003eThe 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, \u003ci\u003eIntroduction to Computation and Modeling for Differential Equations, Second Edition \u003c\/i\u003eincludes:\u003c\/p\u003e \u003cul\u003e \u003cli\u003eNew 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\u003c\/li\u003e \u003cli\u003eNumerous 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®\u003c\/li\u003e \u003cli\u003eAdditional exercises that introduce new methods, projects, and problems to further illustrate possible applications\u003c\/li\u003e \u003cli\u003eA related website with select solutions to the exercises, as well as the MATLAB data sets for ordinary differential equations (ODEs) and PDEs\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003e\u003ci\u003eIntroduction to Computation and Modeling for Differential Equations, Second Edition \u003c\/i\u003eis 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.\u003c\/p\u003e \u003cp\u003e \u003c\/p\u003e \u003cp\u003ePreface xi\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Introduction 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 What is a Differential Equation? 1\u003c\/p\u003e \u003cp\u003e1.2 Examples of an Ordinary and a Partial Differential Equation, 2\u003c\/p\u003e \u003cp\u003e1.3 Numerical Analysis, a Necessity for Scientific Computing, 5\u003c\/p\u003e \u003cp\u003e1.4 Outline of the Contents of this Book, 8\u003c\/p\u003e \u003cp\u003eBibliography, 10\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Ordinary Differential Equations 11\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Problem Classification, 11\u003c\/p\u003e \u003cp\u003e2.2 Linear Systems of ODEs with Constant Coefficients, 16\u003c\/p\u003e \u003cp\u003e2.3 Some Stability Concepts for ODEs, 19\u003c\/p\u003e \u003cp\u003e2.3.1 Stability for a Solution Trajectory of an ODE System, 20\u003c\/p\u003e \u003cp\u003e2.3.2 Stability for Critical Points of ODE Systems, 23\u003c\/p\u003e \u003cp\u003e2.4 Some ODE models in Science and Engineering, 26\u003c\/p\u003e \u003cp\u003e2.4.1 Newton’s Second Law, 26\u003c\/p\u003e \u003cp\u003e2.4.2 Hamilton’s Equations, 27\u003c\/p\u003e \u003cp\u003e2.4.3 Electrical Networks, 27\u003c\/p\u003e \u003cp\u003e2.4.4 Chemical Kinetics, 28\u003c\/p\u003e \u003cp\u003e2.4.5 Control Theory, 29\u003c\/p\u003e \u003cp\u003e2.4.6 Compartment Models, 29\u003c\/p\u003e \u003cp\u003e2.5 Some Examples from Applications, 30\u003c\/p\u003e \u003cp\u003eBibliography, 36\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Numerical Methods for Initial Value Problems 37\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Graphical Representation of Solutions, 38\u003c\/p\u003e \u003cp\u003e3.2 Basic Principles of Numerical Approximation of ODEs, 40\u003c\/p\u003e \u003cp\u003e3.3 Numerical Solution of IVPs with Euler’s method, 41\u003c\/p\u003e \u003cp\u003e3.3.1 Euler’s Explicit Method: Accuracy, 43\u003c\/p\u003e \u003cp\u003e3.3.2 Euler’s Explicit Method: Improving the Accuracy, 46\u003c\/p\u003e \u003cp\u003e3.3.3 Euler’s Explicit Method: Stability, 48\u003c\/p\u003e \u003cp\u003e3.3.4 Euler’s Implicit Method, 53\u003c\/p\u003e \u003cp\u003e3.3.5 The Trapezoidal Method, 55\u003c\/p\u003e \u003cp\u003e3.4 Higher Order Methods for the IVP, 56\u003c\/p\u003e \u003cp\u003e3.4.1 Runge–Kutta Methods, 56\u003c\/p\u003e \u003cp\u003e3.4.2 Linear Multistep Methods, 60\u003c\/p\u003e \u003cp\u003e3.5 Special Methods for Special Problems, 62\u003c\/p\u003e \u003cp\u003e3.5.1 Preserving Linear and Quadratic Invariants, 62\u003c\/p\u003e \u003cp\u003e3.5.2 Preserving Positivity of the Numerical Solution, 64\u003c\/p\u003e \u003cp\u003e3.5.3 Methods for Newton’s Equations of Motion, 64\u003c\/p\u003e \u003cp\u003e3.6 The Variational Equation and Parameter Fitting in IVPs, 66\u003c\/p\u003e \u003cp\u003eBibliography, 69\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Numerical Methods for Boundary Value Problems 71\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Applications, 73\u003c\/p\u003e \u003cp\u003e4.2 Difference Methods for BVPs, 78\u003c\/p\u003e \u003cp\u003e4.2.1 A Model Problem for BVPs, Dirichlet’s BCs, 79\u003c\/p\u003e \u003cp\u003e4.2.2 A Model Problem for BVPs, Mixed BCs, 83\u003c\/p\u003e \u003cp\u003e4.2.3 Accuracy, 86\u003c\/p\u003e \u003cp\u003e4.2.4 Spurious Solutions, 87\u003c\/p\u003e \u003cp\u003e4.2.5 Linear Two-Point BVPs, 89\u003c\/p\u003e \u003cp\u003e4.2.6 Nonlinear Two-Point BVPs, 91\u003c\/p\u003e \u003cp\u003e4.2.7 The Shooting Method, 92\u003c\/p\u003e \u003cp\u003e4.3 Ansatz Methods for BVPs, 94\u003c\/p\u003e \u003cp\u003e4.3.1 Starting with the ODE Formulation, 95\u003c\/p\u003e \u003cp\u003e4.3.2 Starting with the Weak Formulation, 96\u003c\/p\u003e \u003cp\u003e4.3.3 The Finite Element Method, 100\u003c\/p\u003e \u003cp\u003eBibliography, 103\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Partial Differential Equations 105\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Classical PDE Problems, 106\u003c\/p\u003e \u003cp\u003e5.2 Differential Operators Used for PDEs, 110\u003c\/p\u003e \u003cp\u003e5.3 Some PDEs in Science and Engineering, 114\u003c\/p\u003e \u003cp\u003e5.3.1 Navier–Stokes Equations for Incompressible Flow, 114\u003c\/p\u003e \u003cp\u003e5.3.2 Euler’s Equations for Compressible Flow, 115\u003c\/p\u003e \u003cp\u003e5.3.3 The Convection–Diffusion–Reaction Equations, 116\u003c\/p\u003e \u003cp\u003e5.3.4 The Heat Equation, 117\u003c\/p\u003e \u003cp\u003e5.3.5 The Diffusion Equation, 117\u003c\/p\u003e \u003cp\u003e5.3.6 Maxwell’s Equations for the Electromagnetic Field, 117\u003c\/p\u003e \u003cp\u003e5.3.7 Acoustic Waves, 118\u003c\/p\u003e \u003cp\u003e5.3.8 Schrödinger’s Equation in Quantum Mechanics, 119\u003c\/p\u003e \u003cp\u003e5.3.9 Navier’s Equations in Structural Mechanics, 119\u003c\/p\u003e \u003cp\u003e5.3.10 Black–Scholes Equation in Financial Mathematics, 120\u003c\/p\u003e \u003cp\u003e5.4 Initial and Boundary Conditions for PDEs, 121\u003c\/p\u003e \u003cp\u003e5.5 Numerical Solution of PDEs, Some General Comments, 121\u003c\/p\u003e \u003cp\u003eBibliography, 122\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Numerical Methods for Parabolic Partial Differential Equations 123\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Applications, 125\u003c\/p\u003e \u003cp\u003e6.2 An Introductory Example of Discretization, 127\u003c\/p\u003e \u003cp\u003e6.3 The Method of Lines for Parabolic PDEs, 130\u003c\/p\u003e \u003cp\u003e6.3.1 Solving the Test Problem with MoL, 130\u003c\/p\u003e \u003cp\u003e6.3.2 Various Types of Boundary Conditions, 134\u003c\/p\u003e \u003cp\u003e6.3.3 An Example of the Use of MoL for a Mixed Boundary Condition, 135\u003c\/p\u003e \u003cp\u003e6.4 Generalizations of the Heat Equation, 136\u003c\/p\u003e \u003cp\u003e6.4.1 The Heat Equation with Variable Conductivity, 136\u003c\/p\u003e \u003cp\u003e6.4.2 The Convection – Diffusion – Reaction PDE, 138\u003c\/p\u003e \u003cp\u003e6.4.3 The General Nonlinear Parabolic PDE, 138\u003c\/p\u003e \u003cp\u003e6.5 Ansatz Methods for the Model Equation, 139\u003c\/p\u003e \u003cp\u003eBibliography, 140\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Numerical Methods for Elliptic Partial Differential Equations 143\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Applications, 145\u003c\/p\u003e \u003cp\u003e7.2 The Finite Difference Method, 150\u003c\/p\u003e \u003cp\u003e7.3 Discretization of a Problem with Different BCs, 154\u003c\/p\u003e \u003cp\u003e7.4 Ansatz Methods for Elliptic PDEs, 156\u003c\/p\u003e \u003cp\u003e7.4.1 Starting with the PDE Formulation, 156\u003c\/p\u003e \u003cp\u003e7.4.2 Starting with the Weak Formulation, 158\u003c\/p\u003e \u003cp\u003e7.4.3 The Finite Element Method, 159\u003c\/p\u003e \u003cp\u003eBibliography, 164\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Numerical Methods for Hyperbolic PDEs 165\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Applications, 171\u003c\/p\u003e \u003cp\u003e8.2 Numerical Solution of Hyperbolic PDEs, 174\u003c\/p\u003e \u003cp\u003e8.2.1 The Upwind Method (FTBS), 175\u003c\/p\u003e \u003cp\u003e8.2.2 The FTFS Method, 177\u003c\/p\u003e \u003cp\u003e8.2.3 The FTCS Method, 178\u003c\/p\u003e \u003cp\u003e8.2.4 The Lax–Friedrichs Method, 178\u003c\/p\u003e \u003cp\u003e8.2.5 The Leap-Frog Method, 179\u003c\/p\u003e \u003cp\u003e8.2.6 The Lax–Wendroff Method, 179\u003c\/p\u003e \u003cp\u003e8.2.7 Numerical Method for the Wave Equation, 181\u003c\/p\u003e \u003cp\u003e8.3 The Finite Volume Method, 183\u003c\/p\u003e \u003cp\u003e8.4 Some Examples of Stability Analysis for Hyperbolic PDEs, 185\u003c\/p\u003e \u003cp\u003eBibliography, 187\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Mathematical Modeling with Differential Equations 189\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Nature Laws, 190\u003c\/p\u003e \u003cp\u003e9.2 Constitutive Equations, 192\u003c\/p\u003e \u003cp\u003e9.2.1 Equations in Heat Transfer Problems, 192\u003c\/p\u003e \u003cp\u003e9.2.2 Equations in Mass Diffusion Problems, 193\u003c\/p\u003e \u003cp\u003e9.2.3 Equations in Mechanical Moment Diffusion Problems, 193\u003c\/p\u003e \u003cp\u003e9.2.4 Equations in Elastic Solid Mechanics Problems, 194\u003c\/p\u003e \u003cp\u003e9.2.5 Equations in Chemical Reaction Engineering Problems, 194\u003c\/p\u003e \u003cp\u003e9.2.6 Equations in Electrical Engineering Problems, 195\u003c\/p\u003e \u003cp\u003e9.3 Conservative Equations, 195\u003c\/p\u003e \u003cp\u003e9.3.1 Some Examples of Lumped Models, 196\u003c\/p\u003e \u003cp\u003e9.3.2 Some Examples of Distributed Models, 197\u003c\/p\u003e \u003cp\u003e9.4 Scaling of Differential Equations to Dimensionless Form, 201\u003c\/p\u003e \u003cp\u003eBibliography, 204\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Applied Projects on Differential Equations 205\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eProject 1 Signal propagation in a long electrical conductor, 205\u003c\/p\u003e \u003cp\u003eProject 2 Flow in a cylindrical pipe, 206\u003c\/p\u003e \u003cp\u003eProject 3 Soliton waves, 208\u003c\/p\u003e \u003cp\u003eProject 4 Wave scattering in a waveguide, 209\u003c\/p\u003e \u003cp\u003eProject 5 Metal block with heat sourse and thermometer, 210\u003c\/p\u003e \u003cp\u003eProject 6 Deformation of a circular metal plate, 211\u003c\/p\u003e \u003cp\u003eProject 7 Cooling of a chrystal glass, 212\u003c\/p\u003e \u003cp\u003eProject 8 Rotating fluid in a cylinder, 212\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix A Some Numerical and Mathematical Tools 215\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eA.1 Newton’s Method for Systems of Nonlinear Algebraic Equations, 215\u003c\/p\u003e \u003cp\u003eA.1.1 Quadratic Systems, 215\u003c\/p\u003e \u003cp\u003eA.1.2 Overdetermined Systems, 218\u003c\/p\u003e \u003cp\u003eA.2 Some Facts about Linear Difference Equations, 219\u003c\/p\u003e \u003cp\u003eA.3 Derivation of Difference Approximations, 223\u003c\/p\u003e \u003cp\u003eBibliography, 225\u003c\/p\u003e \u003cp\u003eA.4 The Interpretations of Grad, Div, and Curl, 225\u003c\/p\u003e \u003cp\u003eA.5 Numerical Solution of Algebraic Systems of Equations, 229\u003c\/p\u003e \u003cp\u003eA.5.1 Direct Methods, 229\u003c\/p\u003e \u003cp\u003eA.5.2 Iterative Methods for Linear Systems of Equations, 233\u003c\/p\u003e \u003cp\u003eA.6 Some Results for Fourier Transforms, 237\u003c\/p\u003e \u003cp\u003eBibliography, 239\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix B Software for Scientific Computing 241\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eB.1 MATLAB, 242\u003c\/p\u003e \u003cp\u003eB.1.1 Chapter 3: IVPs, 242\u003c\/p\u003e \u003cp\u003eB.1.2 Chapter 4: BVPs, 244\u003c\/p\u003e \u003cp\u003eB.1.3 Chapter 6: Parabolic PDEs, 245\u003c\/p\u003e \u003cp\u003eB.1.4 Chapter 7: Elliptic PDEs, 246\u003c\/p\u003e \u003cp\u003eB.1.5 Chapter 8: Hyperbolic PDEs, 246\u003c\/p\u003e \u003cp\u003eB.2 COMSOL MULTIPHYSICS, 247\u003c\/p\u003e \u003cp\u003eBibliography and Resources, 249\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix C Computer Exercises to Support the Chapters 251\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eC.1 Computer Lab 1 Supporting Chapter 2, 251\u003c\/p\u003e \u003cp\u003eC.1.1 ODE Systems of LCC Type and Stability, 251\u003c\/p\u003e \u003cp\u003eC.2 Computer Lab 2 Supporting Chapter 3, 254\u003c\/p\u003e \u003cp\u003eC.2.1 Numerical Solution of Initial Value Problems, 254\u003c\/p\u003e \u003cp\u003eC.3 Computer Lab 3 Supporting Chapter 4, 257\u003c\/p\u003e \u003cp\u003eC.3.1 Numerical Solution of a Boundary Value Problem, 257\u003c\/p\u003e \u003cp\u003eC.4 Computer Lab 4 Supporting Chapter 6, 258\u003c\/p\u003e \u003cp\u003eC.4.1 Partial Differential Equation of Parabolic Type, 258\u003c\/p\u003e \u003cp\u003eC.5 Computer Lab 5 Supporting Chapter 7, 261\u003c\/p\u003e \u003cp\u003eC.5.1 Numerical Solution of Elliptic PDE Problems, 261\u003c\/p\u003e \u003cp\u003eC.6 Computer Lab 6 Supporting Chapter 8, 263\u003c\/p\u003e \u003cp\u003eC.6.1 Numerical Experiments with the Hyperbolic Model PDE\u003c\/p\u003e \u003cp\u003eProblem, 263\u003c\/p\u003e \u003cp\u003eIndex 265\u003c\/p\u003e  \u003cp\u003e\u003cb\u003eLENNART EDSBERG, PhD,\u003c\/b\u003e is Associate Professor in the Numerical Analysis section within the Department of Mathematics at KTH-The Royal Institute of Technology in Stockholm, Sweden, where he has also been Director of the International Master Program in Scientific Computing since 1998-2008. Dr. Edsberg has over 30 years of academic experience and is the author of over 20 journal articles in the areas of numerical methods and differential equations.   \u003c\/p\u003e\u003cp\u003e\u003cb\u003eUses mathematical, numerical, and programming tools to solve differential equations for physical phenomena and engineering problems\u003c\/b\u003e \u003c\/p\u003e\u003cp\u003e\u003ci\u003eIntroduction to Computation and Modeling for Differential Equations, Second Edition\u003c\/i\u003e features the essential principles and applications of problem solving across disciplines such as engineering, physics, and chemistry. The \u003ci\u003eSecond Edition\u003c\/i\u003e 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. \u003c\/p\u003e\u003cp\u003eThe author features a unique \"Five-M\" approach: Modeling, Mathematics, Methods, MATLAB\u003csup\u003e®\u003c\/sup\u003e, 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, \u003ci\u003eIntroduction to Computation and Modeling for Differential Equations, Second Edition\u003c\/i\u003e includes: \u003c\/p\u003e\u003cul\u003e \u003cli\u003eNew 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\u003c\/li\u003e \u003cli\u003eNumerous 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\u003csup\u003e®\u003c\/sup\u003e\n\u003c\/li\u003e \u003cli\u003eAdditional exercises that introduce new methods, projects, and problems to further illustrate possible applications\u003c\/li\u003e \u003cli\u003eA related website with select solutions to the exercises, as well as the MATLAB data sets for ordinary differential equations (ODEs) and PDEs\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003e\u003ci\u003eIntroduction to Computation and Modeling for Differential Equations, Second Edition\u003c\/i\u003e 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.\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47989457453285,"sku":"NP9781119018445","price":86.5,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9781119018445.jpg?v=1761784177","url":"https:\/\/k12savings.com\/products\/introduction-to-computation-and-modeling-for-differential-equations-isbn-9781119018445","provider":"K12savings","version":"1.0","type":"link"}