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## Über dieses Buch

Science used to be experiments and theory, now it is experiments, theory and computations. The computational approach to understanding nature and technology is currently flowering in many fields such as physics, geophysics, astrophysics, chemistry, biology, and most engineering disciplines. This book is a gentle introduction to such computational methods where the techniques are explained through examples. It is our goal to teach principles and ideas that carry over from field to field. You will learn basic methods and how to implement them. In order to gain the most from this text, you will need prior knowledge of calculus, basic linear algebra and elementary programming.

## Inhaltsverzeichnis

### Chapter 1. Computing Integrals

In Oslo, there is a chain of small cafés called Bagel and Juice that serve fresh bagels and tasty juice. We know of such a café on Hegdehaugsveien, fairly close to the University of Oslo. The owner of this café, as well as all the other owners, faces one particular problem each night: She has to determine how many bagels to order for the next day. Obviously, on the one hand, she wants to have a sufficient supply for the customers. However, on the other hand, she does not want to order more than she will be able to sell, because the surplus has to be discarded or sold elsewhere at a loss.
Aslak Tveito, Hans Petter Langtangen, Bjørn Frederik Nielsen, Xing Cai

### Chapter 2. Differential Equations: The First Steps

If you know the characteristics of something today, and you know the laws of change, then you can figure out what the characteristicswill be tomorrow. This is the basic idea of modeling lots of natural processes. Since we know the weather today and we know the equations modeling the changes of the weather, we can predict it some days ahead. We know the heat of an object now and we know the equations describing how heat changes; thus we can predict how warm an object will be later on. This is really at the heart of science and has been so for quite a while. But how do we express change? How do we make these vague statements precise and suitable for computer simulations? We do so by expressing the laws of change in terms of differential equations.
Aslak Tveito, Hans Petter Langtangen, Bjørn Frederik Nielsen, Xing Cai

### Chapter 3. Systems of Ordinary Differential Equations

In Chap. 2, we saw that models of the form$$y{\prime}(t) = F(y),\,\,\,y(0) = y_{0,}$$can be used to model natural processes.
Aslak Tveito, Hans Petter Langtangen, Bjørn Frederik Nielsen, Xing Cai

### Chapter 4. Nonlinear Algebraic Equations

Suppose we want to solve the ordinary differential equation (ODE)
Aslak Tveito, Hans Petter Langtangen, Bjørn Frederik Nielsen, Xing Cai

### Chapter 5. The Method of Least Squares

Suppose you have a set of discrete data. (ti,yi), i D 1,...,n.
Aslak Tveito, Hans Petter Langtangen, Bjørn Frederik Nielsen, Xing Cai

### Chapter 6. About Scientific Software

When a science problem is solved with the aid of numerical computations, the solution procedure involves several steps:
1.
Understanding the problem and formulating a mathematical model

2.
Using numerical methods to solve the mathematical problems

3.
Implementing the numerical methods in a computer program

4.
Verifying that the results from the program are mathematically correct

5.
Applying the program to the scientific problem and interpreting the results

Aslak Tveito, Hans Petter Langtangen, Bjørn Frederik Nielsen, Xing Cai

### Chapter 7. The Diffusion Equation

This chapter treats the numerical simulation of diffusion processes. Diffusion takes place in space and time simultaneously and is an important phenomenon in nature and technology. Scientific computations of physical quantities such as temperature, pollution, and velocity are frequently based on models for diffusion. Diffusion is often coupled with other processes in physical problems, but in this chapter we will only consider diffusion as an isolated process.
Aslak Tveito, Hans Petter Langtangen, Bjørn Frederik Nielsen, Xing Cai

### 8. Analysis of the Diffusion Equation

In Chap. 7 we studied several aspects of the theory of diffusion processes. We saw how these equations arise in models of several physical phenomena and how they can be approximately solved by suitable numerical methods. The analysis of diffusion equations is a classic subject of applied mathematics and of scientific computing. Its impact on the field of partial differential equations (PDEs) has been very important, both from a theoretical and practical point of view. The purpose of this chapter is to dive somewhat deeper into this field and thereby increase our understanding of this important topic.
Aslak Tveito, Hans Petter Langtangen, Bjørn Frederik Nielsen, Xing Cai

### Chapter 9. Parameter Estimation and Inverse Problems

We have seen how mathematical models can be expressed in terms of differential equations.
Aslak Tveito, Hans Petter Langtangen, Bjørn Frederik Nielsen, Xing Cai

### Chapter 10. A Glimpse of Parallel Computing

It should not be too difficult to imagine that applications of scientific computing in the real world can require huge amounts of computation.
Aslak Tveito, Hans Petter Langtangen, Bjørn Frederik Nielsen, Xing Cai

### Backmatter

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