Read an Excerpt
An Introduction to Differential Equations and their Applications
By Stanley J. Farlow
Dover Publications, Inc.Copyright © 1994 Stanley J. Farlow
All rights reserved.
Introduction to Differential Equations
1.1 BASIC DEFINITIONS AND CONCEPTS
1.2 SOME BASIC THEORY
There was a lot of excitement in the air when on July 1, 1940, local dignitaries cut the ribbon that opened the Tacoma Narrows Bridge over Puget Sound in the state of Washington, but the excitement didn't stop there. Because it tended to experience undulating vibrations in the slightest breeze, the bridge gained a great deal of attention and was nicknamed "Galloping Gertie." Although one might have thought that people would have been afraid to cross the bridge, this was not so. People came from hundreds of miles just for the thrill of crossing "Gertie." Although a few engineers expressed concern, authorities told the public that there was "absolutely nothing to worry about." They were so sure of this that they even planned to drop the insurance on the bridge.
However, at about 7:00 A.M. on November 7, 1940, Gertie's undulations became more violent, and entire portions of the bridge began to heave wildly. At one time, one side of the roadway was almost 30 feet higher than the other. Then, at 10:30 A.M. the bridge began to crack up. Shortly thereafter it made a final lurching and twisting motion and then crashed into Puget Sound. The only casualty was a pet dog owned by a reporter who was crossing the bridge in his car. Although the reporter managed to reach safety by crawling on his hands and knees, clinging to the edge of the roadway, the dog lost its life.
Later, when local authorities tried to collect the insurance on the bridge, they discovered that the agent who had sold them the policy hadn't told the insurance company and had pocketed the $800,000 premium. The agent, referring to the fact that authorities had planned on canceling all policies within a week, wryly observed that if the "damn thing had held out just a little longer no one would have been the wiser." The man was sent to prison for embezzlement. The collapse also caused embarrassment to a local bank, whose slogan was "As safe as the Tacoma Bridge." After the bridge collapsed into Puget Sound, bank executives quickly sent out workers to remove the billboard.
Of course, after the collapse the government appointed all sorts of commissions of inquiry. The governor of the State of Washington made an emotional speech to the people of Washington proclaiming that "we are going to build the exact same bridge, exactly as before." Upon hearing this, the famous engineer Theodor von Karman rushed off a telegram stating, "If you build the exact same bridge, exactly as before, it will fall into the same river, exactly as before."
After the politicians finished their analysis of the bridge's failure, several teams of engineers from major universities began a technical analysis of the failure. It was thegeneral consensus that the collapse was due to resonance caused by an aerodynamical phenomenon known as "stall flutter."
Roughly, this phenomenon has to do with frequencies of wind currents agreeing with natural frequencies of vibration of the bridge. The phenomenon can be analyzed by comparing the driving frequencies of a differential equation with the natural frequencies of the equation.
FISHES, FOXES, AND THE NORWAY RAT
Although at one time Charlie Elton suspected that sunspot activity might be the cause of the periodic fluctuation in the rodent population in Norway, he later realized that this fluctuation probably had more to do with the ecological balance between the rats and their biological competitors.
At about the same time, in the 1920s an Italian marine biologist, Umberto D'Ancona, observed that certain populations of fish in the northern Adriatic varied periodically over time. More specifically, he noted that when the population of certain predator fish (such as sharks, skates, and rays) was up, the population of their prey (herbivorous fish) was down, and vice versa. To better understand these "boom and bust" cycles, D' Ancona turned to the famous Italian mathematician and differential equations expert Vito Volterra. What Volterra did was to repeat for biology what had been done in the physical sciences by Newton 300 years earlier. In general, he developed a mathematical theory for a certain area of biology; in particular, he developed a mathematical framework for the cohabitation of organisms. One might say that he developed the mathematical theory for the "struggle for existence" and that current research in ecological systems had its beginnings in the differential equations of Volterra.
WHERE WERE YOU WHEN THE LIGHTS WENT OUT?
Most readers of this book were probably pretty young during the New York City power failure of 1977 that plunged the entire northeastern section of the United States and a large portion of Canada into total darkness. Although the lessons learned from that disaster have led to more reliable power grids across the country, there is always the (remote) possibility that another failure will occur at some future time.
The problem is incredibly complicated. How to match the energy needs of the millions of customers with the energy output from the hundreds of generating stations? And this must be done so that the entire network remains synchronized at 60 cycles per second and the customer's voltage levels stay at acceptable levels! Everything would not be quite so difficult if demand remained constant and if there were never any breakdowns. As one system engineer stated, "It's easy to operate a power grid if nothing breaks down. The trick is to keep it working when you have failures." However, there will always be the possibility of a generator breaking down or lightning hitting a transformer. And when this happens, there is always the possibility that the entire network may go down with it.
To help design large-scale power grids to be more reliable (stable), engineers have constructed mathematical models based on systems of differential equations that describe the dynamics of the system (voltages and currents through power lines). By simulating random failures the engineers are able to determine how to design reliable systems. They also use mathematical models to determine after the fact how a given failure can be prevented in the future. For example, after a 1985 blackout in Colombia, South America, mathematical models showed that the system would have remained stable if switching equipment had been installed to trip the transmission lines more quickly.
DIFFERENTIAL EQUATIONS IN WEATHER PATTERNS
Meteorologist Edward Lorenz was not interested in the cloudy weather outside his M.I.T. office. He was more interested in the weather patterns he was generating on his new Royal McBee computer. It was the winter of 1961, and Lorenz had just constructed a mathematical model of convection patterns in the upper atmosphere based on a system of three nonlinear differential equations. In the early 1960s there was a lot of optimism in the scientific world about weather forecasting, and the general consensus was that it might even be possible in a few years to modify and control the weather. Not only was weather forecasting generating a great deal of excitement, but the techniques used in meteorology were also being used by physical and social scientists hoping to make predictions about everything from fluid flow to the flow of the economy.
Anyway, on that winter day in 1961 when Edward Lorenz came to his office, he decided to make a mathematical shortcut, and instead of running his program from the beginning, he simply typed into the computer the numbers computed from the previous day's run. He then turned on the computer and left the room to get a cup of coffee. When he returned an hour later, he saw something unexpected—something that would change the course of science.
The new run, which should have been the same as the previous day's run, was completely different. The weather patterns generated on this day were completely different from the patterns generated on the previous day, although their initial conditions were the same.
Initially, Lorenz thought he had made a mistake and keyed in the wrong numbers, or maybe his new computer had a malfunction. How else could he explain how two weather patterns had diverged if they had the same initial conditions? Then it came to him. He realized that the computer was using six-place accuracy, such as 0.209254, but only three places were displayed on the screen, such as 0.209. So when he typed in the new numbers, he had entered only three decimal places, assuming that one part in a thousand was not important. As it turned out insofar as the differential equations were concerned, it was very important.
The "chaotic" or "randomlike behavior" of those differential equations was so sensitive to their initial conditions that no amount of error was tolerable. Little did Lorenz know it at the time, but these were the differential equations that opened up the new subject of chaos. From this point on scientists realized that the prediction of such complicated physical phenomena as the weather was impossible using the classical methods of differential equations and that newer theories and ideas would be required. Paradoxically, chaos theory provides a way to see the order in a chaotic system.
ENGINEERS TEACH SMART BUILDING TO FOIL QUAKES
Engineers and applied mathematicians are now designing self-stabilizing buildings that, instead of swaying in response to an earthquake, actively suppress their own vibrations with computer-controlled weights. (See Figure 1.1.) In one experimental building, the sway was said to be reduced by 80 percent.
During an earthquake, many buildings collapse when they oscillate naturally with the same frequency as seismic waves traveling through the earth, thus amplifying their effect, said Dr. Thomas Heaton, a seismologist at the U.S. Geological Survey in Pasadena, California. Active control systems might prevent that from happening, he added.
Figure 1.1 How a self-stabilizing building works. Instead of swaying in response to an earthquake, some new buildings are designed as machines that actively suppress their own vibrations by moving a weight that is about 1 percent of the building's weight.
One new idea for an active control system is being developed by the University of Southern California by Dr. Sami Masri and his colleagues in the civil engineering department. When wind or an earthquake imparts energy to the building, Dr. Masri said, it takes several seconds for the oscillation to build up to potentially damaging levels. Chaotic theory of differential equations, he said, suggests that a random source of energy should be injected into this rhythmic flow to disrupt the system.
At the present time, two new active stabilizing systems are to be added to existing buildings in the United States that sway excessively. Because the owners do not want their buildings identified, the names of the buildings are kept confidential.
Bridges and elevated highways are also vulnerable to earthquakes. During the 1989 San Francisco earthquake (the "World Series" earthquake) the double-decker Interstate 880 collapsed, killing several people, and the reader might remember the dramatic pictures of a car hanging precariously above San Francisco Bay where a section of the San Francisco-Oakland Bay Bridge had fallen away. Less reported was the fact that the Golden Gate Bridge might also have been close to going down. Witnesses who were on the bridge during the quake said that the roadbed underwent wavelike motions in which the stays connecting the roadbed to the overhead cables alternately loosened and tightened "like spaghetti." The bridge oscillated for about a minute, about four times as long as the actual earthquake. Inasmuch as an earthquake of up to ten times this magnitude (the "big one") is predicted for California sometime in the future, this experience reinforces our need for a deeper understanding of nonlinear oscillations in particular and nonlinear differential equations in general.
1.1 BASIC DEFINITIONS AND CONCEPTS
To introduce some of the basic terminology and ideas that are necessary for the study of differential equations. We introduce the concepts of
#x2022; ordinary and partial differential equations,
order of a differential equation,
linear and nonlinear differential equations.
THE ROLE OF DIFFERENTIAL EQUATIONS IN SCIENCE
Before saying what a differential equation is, let us first say what a differential equation does and how it is used. Differential equations can be used to describe the amount of money in a savings bank, the orbit of a spaceship, the amount of deformation of elastic structures, the description of radio waves, the size of a biological population, the current or voltage in an electrical circuit, and on and on. In fact, differential equations can be used to explain and predict new facts for about everything that changes continuously. In more complex systems we don't use a single differential equation, but a system of differential equations, as in the case of an electrical network of several circuits or in a chemical reaction with several interacting chemicals.
The process by which scientists and engineers use differential equations to understand physical phenomena can be broken down into three steps. First, a scientist or engineer defines a real problem. A typical example might be the study of shock waves along fault lines caused by an earthquake. To understand such a phenomenon, the scientist or engineer first collects data, maybe soil conditions, fault data, and so on. This first step is called data collection.
The second step, called the modeling process , generally requires the most skill and experience on the part of the scientist. In this step the scientist or engineer sets up an idealized problem, often involving a differential equation, which describes the real phenomenon as precisely as possible while at the same time being stated in such a way that mathematical methods can be applied. This idealized problem is called a mathematical model for the real phenomenon. In this book, mathematical models refer mainly to differential equations with initial and boundary conditions. There is generally a dilemma in constructing a good mathematical model. On one hand, a mathematical model may describe accurately the phenomenon being studied, but the model may be so complex that a mathematical analysis is extremely difficult. On the other hand, the model may be easy to analyze mathematically but may not reflect accurately the phenomenon being studied. The goal is to obtain a model that is sufficiently accurate to explain all the facts under consideration and to enable us to predict new facts but at the same time is mathematically tractable.
The third and last step is to solve mathematically the ideal problem (i.e., the differential equation) and compare the solution with the measurements of the real phenomenon. If the mathematical solution agrees with the observations, then the scientist or engineer is entitled to claim with some confidence that the physical problem has been "solved mathematically," or that the theory has been verified. If the solution does not agree with the observations, either the observations are in error or the model is inaccurate and should be changed. This entire process of how mathematics (differential equations in this book) is used in science is described in Figure 1.2.
WHAT IS A DIFFERENTIAL EQUATION?
Quite simply, a differential equation is an equation that relates the derivatives of an unknown function, the function itself, the variables by which the function is defined, and constants. If the unknown function depends on a single real variable, the differential equation is called an ordinary differential equation. The following equations illustrate four well-known ordinary differential equations.
Excerpted from An Introduction to Differential Equations and their Applications by Stanley J. Farlow. Copyright © 1994 Stanley J. Farlow. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.