Introduction to Computation and Modeling for Differential Equations

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 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 Schrödinger’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.

References.

Index.