Introduction to Computation and Modeling for Differential Equations (eBook)

Lennart Edsberg (Autor)

eBook Download: EPUB

2015 | 2. Auflage
John Wiley & Sons (Verlag)
978-1-119-01846-9 (ISBN)

Infos

Uses mathematical, numerical, and programming tools to solve differential equations for physical phenomena and engineering problems

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.

 

LENNART EDSBERG, PhD, 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.

Textauszug

Chapter 1
Introduction

It is probably no exaggeration to say that differential equations are the most common and important mathematical model in science and engineering. Whenever we want to model a system where the state variables vary with time and/or space, differential equations are the natural tool for describing its behavior. The construction of a differential equation model demands a thorough understanding of what takes place in the process we want to describe.

However, setting up a differential equation model is not enough, we must also solve the equations. The process of finding useful solutions of a differential equation is much a symbiosis of modeling, mathematics and choosing a method, analytical or numerical. Therefore, when you are requested to solve a differential equation problem from some application, it is useful to know facts about its modeling background, its mathematical properties, and its numerical treatment. The last part involves choosing appropriate numerical methods, adequate Software, and appealing ways of visualizing the result.

The interaction among modeling, mathematics, numerical methods, and programming is nowadays referred to as scientific computing and its purpose is to perform simulations of processes in science and engineering.

1.1 What is a Differential Equation?

A differential equation is a relation between a function and its derivatives. If the function depends on only one variable , i.e., , the differential equation is called ordinary. If depends on at least two variables and , i.e., , the differential equation is called partial.

1.2 Examples of an ordinary and a partial differential equation

An example of an elementary ordinary differential equation (ODE) is

1.1

where is a parameter, in this case a real constant. It is frequently used to model, e.g., the growth of a population or the decay of a radioactive substance . The ODE (1.1) is a special case of differential equations called linear with constant coefficients (see Chapter 2).

The differential equation (1.1) can be solved analytically, i.e., the solution can be written explicitly as an algebraic formula. Any function of the form

1.2

where is an arbitrary constant satisfies (1.1) and is a solution. The expression (1.2) is called the general solution. If is known to have a certain value, however, we get a unique solution, which, when plotted in the -plane, gives a trajectory (solution curve). This solution is called a particular solution.

The constant can be determined, e.g., by selecting a point in the -plane through which the solution curve shall pass. Such a point is called an initial point and the demand that the solution shall go through this point is called the initial condition. A differential equation together with an initial condition is called an initial value problem (IVP) (Figure 1.1).

Figure 1.1 General and particular solution

Observe that the differential equation alone does not define a unique solution, we also need an initial condition or other conditions. A plot of all trajectories, i.e., all solutions of the ODE (1.1) in the -plane will result in a graph that is totally black as there are infinitely many solution curves filling up the whole plane.

In general, it is not possible to find analytical solutions of a differential equation. The “simple” differential equation

1.3

cannot be solved analytically. If we want to plot some of its trajectories, we have to use numerical methods.

An example of an elementary partial differential equation (PDE) is

1.4

where is a parameter, in this case a real constant. The solution of (1.4) is a function of two variables . This differential equation is called the 1D (one space dimension, ) advection equation. Physically it describes the evolution of a scalar quantity, e.g., temperature carried along the -axis by a flow with constant velocity . It is also known as the linear convection equation and is an example of a hyperbolic PDE (see Chapter 5).

The general solution of this differential equation is (see Exercise 1.2.4)

1.5

where is any arbitrary differentiable function of one variable. This is indeed a large family of solutions! The three functions

are just three solutions out of the infinitely many solutions of this PDE.

To obtain a unique solution for we need an initial condition. If the differential equation is valid for all , i.e., and is known for , i.e., where is a given function, the initial value function, we get the particular solution (Figure 1.2)

1.6

Figure 1.2 Propagation of a solution of the advection equation

Physically, (1.6) corresponds to the propagation of the initial function along the -axis with velocity . The propagation is to the right if and to the left if .

The graphical representation can alternatively be done in 3D (Figure 1.3).

Figure 1.3 3D graph of a propagating solution

When a PDE is formulated on a semi-infinite or finite -interval, boundary conditions are needed in addition to initial conditions to specify a unique solution.

Most PDEs can only be solved with numerical methods. Only for very special classes of PDE problems it is possible to find an analytic solution, often in the form of an infinite series.

Exercise 1.2.1

If is a complex constant what is the real and imaginary part of ?

Exercise 1.2.2

What conditions are necessary to impose on and if for is to be

1. a. exponentially decreasing,
2. b. exponentially increasing,
3. c. oscillating with constant amplitude,
4. d. oscillating with increasing amplitude,
5. e. oscillating with decreasing amplitude?

Exercise 1.2.3

If is a complex constant what condition on is needed if is to be bounded for ?

Exercise 1.2.4

Show that the general solution of is by introducing the transformation

Transform the original problem to a PDE in the variables and , and solve this PDE. Sketch the two coordinate systems in the same graph.

Exercise 1.2.5

Show that a solution of (1.4) starting at , is constant along the straight line . This means that the initial value is transported unchanged along this line, which is called a characteristic of the hyperbolic PDE (1.4).

1.3 Numerical analysis, a necessity for scientific computing

In scientific computing, the numerical methods used to solve mathematical models should be robust, i.e., they should be reliable and give accurate values for a large range of parameter values. Sometimes, however, a method may fail and give unexpected results. Then, it is important to know how to investigate why an erroneous result has occurred and how it can be remedied.

Two basic concepts in numerical analysis are stability and accuracy. When choosing a method for solving a differential equation problem, it is necessary to have some knowledge about how to analyze the result of the method with respect to these concepts. This necessity has been well expressed by the late Prof. Germund Dahlquist, famous for his fundamental research in the theory of numerical treatment of differential equations: “There is nothing as practical as a little good theory.”

As an example of what may be unexpected results, choose the well-known vibration equation, occurring in, e.g., mechanical vibrations, electrical vibrations, and sound vibrations. The form of this equation with initial conditions is

1.7

In mechanical vibrations, is the mass of the vibrating particle, the damping coefficient, the spring constant, an external force acting on the particle, the initial position, and the initial velocity of the particle. The five quantities are referred to as the parameters of the problem.

Solving (1.7) numerically for a set of values of the parameters is an example of simulation of a mechanical process and it is desirable to choose a robust method, i.e., a method for which the results are reliable for a large range of values of the parameters. The following two examples based on the vibration equation show that unexpected results depending on instability and/or bad accuracy mayoccur.

Example 1.1

Assume that (free vibrations) and the following values of the parameters: , , , , . Without too much knowledge about mechanics, we would expect the solution to be oscillatory and damped, i.e., the amplitude of the vibrations is decreasing. If we use the simple Euler method with constant stepsize (see Chapter 3), we obtain the following numerical solution, visualized together with the exact solution (Figure 1.4).

Figure 1.4 An example of numerical instability

The graph shows a numerical solution that is oscillatory but unstable with increasing amplitude. Why? The answer is given in Chapter 3. For the moment just accept that insight...

Produkt-Details

• Erscheint lt. Verlag: 16.9.2015
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• Schlagworte: boundary value problems • BVPs • computer programming • Computer Science • Comsol Multiphysics • Differential Equations • Differentialgleichung • Differentialgleichungen • Electrical & Electronics Engineering • Elektrotechnik u. Elektronik • Finite Element Method • Finite Volume Methods • initial value problems • IVPs • Mathematical Applications • Mathematical Modeling • Mathematics • Mathematik • Mathematische Modellierung • MATLAB • Modell (Math.) • multiphysics • Numerical Methods • Numerical Methods & Algorithms • numerical solutions • Numerische Methoden u. Algorithmen • odes • Ordinary differential equations • Partial differential equations • PDEs • Scientific Computing
• ISBN-10: 1-119-01846-3 / 1119018463
• ISBN-13: 978-1-119-01846-9 / 9781119018469