Euler method
1
Numerical Methods for Differential
Equations
1
2
NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS
Introduction
Differential equations can describe nearly all systems undergoing change. They are ubiquitous is science and engineering as well as economics, social science, biology, business, health care, etc. Many mathematicians have studied the nature of these equations for hundreds of years and there are many well-developed solution techniques.
Often, systems described by differential equations are so complex, or the systems that they describe are so large, that a purely analytical solution to the equations is not tractable. It is in these complex systems where computer simulations and numerical methods are useful.
The techniques for solving differential equations based on numerical approximations were developed before programmable computers existed. During World War II, it was common to find rooms of people (usually women) working on mechanical calculators to numerically solve systems of differential equations for military calculations.
Before programmable computers, it was also common to exploit analogies to electrical systems to design analog computers to study mechanical, thermal, or chemical systems. As programmable computers have increased in speed and decreased in cost, increasingly complex systems of differential equations can be solved with simple programs written to run on a common PC. Currently, the computer on your desk can tackle problems that were inaccessible to the fastest supercomputers just 5 or 10 years ago.
This chapter will describe some basic methods and techniques for programming simulations of differential equations. First, we will review some basic concepts of numerical approximations and then introduce Euler’s method, the simplest method. We will provide details on algorithm development using the Euler method as an example. Next we will discuss error approximation and discuss some better techniques. Finally we
Numerical Methods for Differential
Equations
1
2
NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS
Introduction
Differential equations can describe nearly all systems undergoing change. They are ubiquitous is science and engineering as well as economics, social science, biology, business, health care, etc. Many mathematicians have studied the nature of these equations for hundreds of years and there are many well-developed solution techniques.
Often, systems described by differential equations are so complex, or the systems that they describe are so large, that a purely analytical solution to the equations is not tractable. It is in these complex systems where computer simulations and numerical methods are useful.
The techniques for solving differential equations based on numerical approximations were developed before programmable computers existed. During World War II, it was common to find rooms of people (usually women) working on mechanical calculators to numerically solve systems of differential equations for military calculations.
Before programmable computers, it was also common to exploit analogies to electrical systems to design analog computers to study mechanical, thermal, or chemical systems. As programmable computers have increased in speed and decreased in cost, increasingly complex systems of differential equations can be solved with simple programs written to run on a common PC. Currently, the computer on your desk can tackle problems that were inaccessible to the fastest supercomputers just 5 or 10 years ago.
This chapter will describe some basic methods and techniques for programming simulations of differential equations. First, we will review some basic concepts of numerical approximations and then introduce Euler’s method, the simplest method. We will provide details on algorithm development using the Euler method as an example. Next we will discuss error approximation and discuss some better techniques. Finally we