Педагогика/ 7Современные методы

преподавапния

Ismagul R.

Kostanay State University named after A.Baitursynov, Kazakhstan

Differential equations in applications

In the mathematical study of any real-world problems can be divided into three main stages:

- Construction of a mathematical model of the phenomenon;

- study of this mathematical model and obtain the corresponding mathematical problem solving;

application of the results to the practical question, which arose from the permission given mathematical model, and the search for other issues to which it is applicable.

In constructing a mathematical model of the process or needed his idealization and formalization. When idealization phenomena separated conditions significantly affecting it, the conditions have no significant impact .

A classic example is an idealized model scheme for studying the motion of the pendulum - a mathematical pendulum. In this case, neglecting the dimensions and shape of cargo by air resistance, the friction at the point of suspension, and the flexibility of the thread , etc.

Investigation of this idealized scheme can already formalized, making differential equation. Then, you need to examine to what limits are acceptable approximations made as a condition will change in accounting dropped factors, etc. It should find out what other phenomena are described by the same formal mathematical model.

Application of differential equations in physics

In solving problems of a physical nature, leading to differential equations, the main difficulty is, as a rule, drawing themselves differential equations.

Solution of physical problem must consistently take place in three stages:

- Preparation of the differential equation;

- Solution of this equation;

- Study of the solution.

In this case we recommend the following steps:

1. Set the value in changing this phenomenon and to identify the physical laws that bind them.

2. Select the independent variable and function of this target variable.

3. Based on the conditions of the problem, define the initial or boundary conditions.

4. Express all appearing in the problem through the independent variable values , the desired function and derivatives of this function.

5. Based on the conditions of the problem and the physical laws that govern this phenomenon, to make the differential equation.

6. Find the general solution or general integral differential equation.

7. The initial or boundary conditions to find a particular solution.

8. Explore the solution obtained.

In many cases, drafting of the differential equation is based on the differentiability of the functions expressing the dependence of the quantities. Typically all participating in that particular process values within a short time interval are changed with a constant rate. This allows you to apply the known laws of physics, describing phenomena occurring uniformly, compiling relations between the values , i.e. between the magnitudes involved in the process, and their backups. The resulting equation is only approximate, since the values vary even within a short period of time, generally speaking, is uneven. But, if we divide both sides of the resulting equality in and go to the limit, when will the exact equality. It contains the time t, changing over time physical quantities and their derivatives, i.e. is a differential equation describing this phenomenon. The same equation in differential form can be obtained by replacing the increment on the differential dt, and increment functions - corresponding differentials.

Application of differential equations in biology

The models used in biology, divided into three categories:

1. Subject biological model, in which we study the general laws, pathological processes, the effects of various drugs, etc. This class of models include, for example, laboratory animals, isolated organs, cell culture suspension organelles, etc.

2. Physical (analog) model, i.e. physical models have the same behavior with the simulation object. For example, the deformations arising in the bone when different loads can be studied on a specially prepared bone layout. Movement of blood on large vessels modeled chain of resistors, capacitors and inductive coils.

3. Mathematical models are systems of mathematical expressions - formulas, functions, equations, etc., describing the various properties of the studied object, phenomenon or process. When you create a mathematical model using physical principles identified in the experimental study of object modeling. For example, the mathematical model of circulation based on the laws of hydrodynamics.

Mathematical modeling as a method of study has a number of undoubted advantages:

- The method of presentation of quantitative laws in mathematical language is precise and economical;

- Testing hypotheses formulated on the basis of experimental data can be performed by testing a mathematical model that is based on this hypothesis;

- A mathematical model to judge the behavior of such systems and in such conditions, which are difficult to create in the experiment or in the clinic, study the work of the studied system, as a whole, or the work of any individual part of it.

The general idea of replacing the functions on small intervals of linear functions of the argument underlying the solution of physical problems using differential equations, called linearization.

Literature:

1. Amyelkin V.V. “Differential equations in applications.” Science 1987-160p