Advanced Mathematical Modeling with Technology

William P. Fox, Robert E. Burks

• Publisher: Chapman and Hall/CRC
• Year: 2021
• ISBN:
  9781000388886
• Edition: 1
• Topic: Matematica
• Subtopic: Equazioni differenziali

Mathematical modeling is both a skill and an art and must be practiced in order to maintain and enhance the ability to use those skills. Though the topics covered in this book are the typical topics of most mathematical modeling courses, this book is best used for individuals or groups who have already taken an introductory mathematical modeling course. This book will be of interest to instructors and students offering courses focused on discrete modeling or modeling for decision making.

1

Perfect Partners: Mathematical Modeling and Technology

Objectives

1.Understand the mathematical modeling process.

2.Understand the process of decision modeling.

3.Understand that models have both strengths and limitations.

Consider the importance of decision-making in such areas as business (B), industry (I), and government (G). Decision-making under uncertainty is incredibly important in many areas of life but it is particularly important in business, industry, and government (BIG). BIG decision-making is essential to success at all levels and we do not encourage “shooting from the hip.” However, we do recommend good analysis for the decision-maker to examine and question in order to find the best alternative course of action. So, why mathematical modeling?

A mathematical model may be defined as a description of a real-world system using mathematical concepts to facilitate the explanation of change in the system or to study the effects of different elements of the system and to understand changes in the patterns of behavior.

Mathematical models are used not only in the natural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines (such as computer science, artificial intelligence), but also in the social sciences (such as business, economics, psychology, sociology and political science); physicists, engineers, statisticians, operations research analysts and economists use mathematical models most extensively. A model may help to explain a system and to study the effects of different components, and to make predictions about behavior.

Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include logical models, as far as logic is taken as a part of mathematics. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed.

1.1 Examples of Really Big Problems and the Process of Mathematical Modeling

Consider for a moment a basic real-world search situation. Two observation posts 5.43 miles apart pick up a brief radio signal. The sensing devices were oriented at 110∘ and 119∘ respectively when a signal was detected. The devices are accurate to within 2∘ (that is ±2∘ of their respective angle of orientation). According to intelligence, the reading of the signal came from a region of active terrorist exchange, and it is inferred that there is a boat waiting for someone to pick up the terrorists. It is dusk, the weather is calm, and there are no currents. A small helicopter leaves a pad from Post 1 and is able to fly accurately along the 110∘ angle direction. This helicopter has only one detection device, a searchlight. At 200 feet, it can just illuminate a circular region with a radius of 25 feet. The helicopter can fly 225 miles in support of this mission due to its fuel capacity. Some basic pre-launch decisions would include; where do we search for the boat? How many search helicopters should you use to have a “good” chance of finding the target?

Photochemical smog permeates the Los Angeles basin most days of the year. While this problem is not unique to the Los Angeles area, conditions in the basin are well suited to this phenomenon. The surrounding mountains and frequent inversion layers create the stagnant air that give rise to these conditions. Can we build a model to examine this? Can we use such a model to study or analyze the poor air quality in Los Angeles due to traffic pollution or measuring vehicle emissions?

In the sport of bridge jumping, a willing participant attaches one end of bungee cord to himself, attaches the other end to a bridge railing, and then drops off a bridge. Is this a safe sport? Can we describe the typical motion?

You are a new city manager in California. You are worried about earthquake survivability of your city’s water tower. You need to analyze the effects of an earthquake on your water tower and see if any design improvements are necessary. You want to prevent catastrophic failure.

If you have flown lately, you may have noticed that most airplanes are full. As a matter of fact, many times an announcement is made that the plane is overbooked, and the airlines are looking for volunteers to take a later flight. Why do airlines overbook? Should they overbook? What impact does this have on the passengers? What impact does it have on the airlines?

These events all share one common element – we can model them using mathematics to support making decisions. This textbook will help you understand what a mathematical modeler might do for you as a confident problem-solver using the techniques of mathematical modeling. As a decision-maker, understanding the possibilities and asking the key questions will enable better decision to be made.

1.2 The Modeling Proc...