Introduction to Mathematical Modeling
What is Mathematical Modeling?
Mathematical modeling is a mathematical thinking method developed by a group of mathematicians who want to use mathematical knowledge to solve practical problems. They once said that any practical problem is complicated, but we can use mathematical language and methods, through reasonable abstraction and simplification, use “quantitative” language or structure to describe the inherent laws of natural phenomena, approximate description and “solve” the actual problem.
Since the 1970s, with the rapid development of industry, manufacturing and other fields, the complexity of things has risen rapidly. Simple thinking and calculation of the human brain can no longer meet the needs of actual things, so mathematical modeling was born. Mathematical modeling can be said to be an interdisciplinary blend, turning pure mathematicians into physicists, biologists, economists, even psychologists and so on. Furthermore, mathematical modeling not only includes probability theory, statistics, operations research, and other disciplines, but also gradually comes into our life with the development of various majors (such as engineering, physics, finance).
Mathematical modeling is a bridge between the natural world and the rational world. Moreover, it has three fundamental ideas: transience, permanence, and optimality.
Simple Application of Mathematical Modeling
The problems solved by mathematical modeling can divide into five main categories. Evaluation, such as when interviewing new employees, to make an evaluation of their comprehensive ability. Operations, such as planning the day’s route before leaving. Equations of straight lines, like patterns of population growth. Prediction classes, such as predicting when the next rain will fall. Discriminant categories, such as determining whether a product is genuine or defective by data.
General Process of Mathematical Modeling
Before modeling, we should have as deep understanding of the actual background as possible. Moreover, clarify the purpose and requirements of the problem. And collect necessary data and information.
Based on fully digesting relevant information, the actual problem is idealized, simplified, and linearized, and the essence and main factors of the problem are tightly grasped, to make reasonable and mathematically processed assumptions.
Correctly describe the problem with mathematical language, simplify the model reasonably, and choose the appropriate method.
As for more complex practical problems, analytical solutions become preferable. But more often numerical solutions are required. The degree of mastery of calculation method and application software, as well as the level of programming ability, will determine the degree of optimization and accuracy of solution results.
(1) Result testing: translate the solution results into practical problems to test the rationality and applicability of the model.
(2) Stability analysis: analyze the “tolerance” of the model to parameter changes.
(3) Sensitivity analysis: analyze the sensitivity of the objective function to variable changes.