Introduction to computation and modeling for differential equations /
Edsberg, Lennart,1946-
Hoboken, N.J. : Wiley,
c2008.
Includes bibliographical references and index.
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 IVPs. -- 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 The variational equation and parameter fitting in IVPs. -- 3.6 References. -- 4. Numerical methods for BVPs. -- 4.1 Applications. -- 4.2 Difference methods for BVPs. -- 4.2.1 A model problem for BVPs. -- 4.2.2 Accuracy. -- 4.2.3 Spurious oscillations. -- 4.2.4 Linear two-point boundary value problems. -- 4.2.5 Nonlinear two-point boundary value problems. -- 4.2.6 The shooting method. -- 4.3 Ansatz methods for BVPs. -- 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 in fluid dynamics. -- 5.3.2 The convection-diffusion-reaction equations. -- 5.3.3 The heat equation. -- 5.3.4 The diffusion equation. -- 5.3.5 Maxwell's equations for the electromagnetic field. -- 5.3.6 Acoustic waves. -- 5.3.7 Schrodinger's equation in quantum mechanics. -- 5.3.8 Navier's equations in structural mechanics. -- 5.3.9 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 PDEs. -- 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 a mixed BC. -- 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 PDEs. -- 7.1 Applications. -- 7.2 The Finite Difference Methods. -- 7.3 Discretization of a problem with different BCs. -- 7.4 The Finite Element Method. -- 8. Numerical methods for hyperbolic PDEs. -- 8.1 Applications. -- 8.2 Numerical solution of hyperbolic PDEs. -- 8.3 Introduction to numerical stability for hyperbolic PDEs. -- 9. Mathematical modeling with differential equations. -- 9.1 Nature laws. -- 9.2 Constitutive equations. -- 9.2.1 Equations in heat conduction 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. -- A. Appendix. -- A.1 Newton's method for systems of nonlinear algebraic equations. -- A.1.1 Quadratic 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 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. Software for scientific computing. -- B.1 Matlab. -- B.2 Comsol Multiphysics. -- C. Computer exercises to support the chapters.
Differential equations
Differential equations
Équations différentielles
Introduction to computation and modeling for differential equations.