similar
The process of mathematical modeling is to express the actual problem into a mathematical problem with mathematical language, mathematical concepts and mathematical symbols after analysis, abstraction and generalization. What kind of problem is expressed depends on the thinker's intention to solve the problem. Generally speaking, analogy modeling is based on the concrete analysis of various factors of practical problems, through the analysis of association and induction, comparing with known models, transforming unknown relationships into known relationships, looking for the same or similar relationships among different objects or completely unrelated objects, comparing with some conclusions of known models, obtaining mathematical methods to solve "similar" problems, and finally establishing a model to solve problems.
Dimensional analysis method
Dimensional analysis is a method of establishing mathematical model in the field of physics, which was put forward in the early 20th century. Based on experience and experiments, it uses the dimensional homogeneity of physical laws to determine the relationship between physical quantities. It is a mathematical analysis method. Through dimensional analysis, we can correctly analyze the relationship between variables, simplify the experiment and facilitate the arrangement of results.
In the international system of units, there are seven basic quantities: mass, length, time, current, temperature, light intensity and material quantity. Their dimensions are M, L, T, I, H, J and N, which are called basic dimensions.
Dimensional analysis is often used to study some relations and properties qualitatively, and to seek the relationship between physical quantities by using the principle of dimensional homogeneity. In the process of mathematical modeling, dimensionless is often needed. Dimensionless means that according to the idea of dimensional analysis, the characteristic scale is selected appropriately, and the dimensional quantity is quantized into dimensionless quantity, thus reducing parameters and simplifying the model.
difference method
The mathematical idea of difference method is to discretize the derivative in the control equation by using the difference quotient of function values on grid nodes instead of Taylor series expansion, so as to establish an equation group with the values on grid nodes as unknowns, and transform the differential problem into an algebraic problem, which is an effective method to establish the mathematical model of discrete dynamic systems.
There are many ways to construct the difference, and Taylor series expansion is the main method at present. The basic difference expressions are mainly in the following forms: first-order forward difference, first-order backward difference, first-order central difference and second-order central difference, in which the first two formats are first-order calculation accuracy and the last two formats are second-order calculation accuracy. Through the combination of several different difference formats, time and space can be combined into different difference calculation formats.
The solution steps of difference method are: establishing differential equation; Constructing a difference scheme; Solve the difference equation; Accuracy analysis and testing.
calculus of variations
Variational method is a mathematical field dealing with functions of functions, that is, functional problems, as opposed to ordinary calculus dealing with functions of numbers. Such a functional can be constructed from the integrals of unknown functions and their derivatives, and finally the extremum function can be obtained. In reality, many phenomena can be expressed as functional minimization problems, that is, variational problems. There are usually two methods to solve variational problems: classical variational method and optimal cybernetics. Limited by basic knowledge, junior college students seldom use variational method in mathematical modeling competition.
Graph theory method
Graph theory method in mathematical modeling is a unique method. Graph theory modeling refers to the process of abstracting and simplifying some abstract things and describing their characteristics and internal relations with graphics. Graph theory is the theory of studying point sets connected by lines. Nodes in the graph represent objects, and the connection between two points indicates that there is a certain relationship between two objects (order relationship, winning relationship, transitive relationship, connection relationship, etc.). In fact, any system containing binary relations can be simulated by graphics. Therefore, graph theory is an important modern mathematical tool for studying natural science, engineering technology, economic problems, management and other social problems, and has become an essential tool for mathematical modeling.