6-мавзу. Математик моделлаштириш элемент­лари

6-мавзу. Математик моделлаштириш элемент­лари
1. Гидрометеорологик жараёнлар математик моделларининг тузилиши.
2. Моделлар таснифи.
3. Имитацион моделлар. Моделларга қўйиладиган асосий талаблар.
4. Моделлар ёрдамида гидрометеорологик объектларни тавсифлаш намуналари.

Nazariy jihatdan tizimlarni tahlil qilishning ikkita yondashuvi mavjud:

• Analitik usul tizimni uning tarkibiy qismlariga bo'lishdan iborat bo'lib, keyinchalik ular birma-bir tahlil qilinadi.
• Tizimning tarkibiy qismlariga emas, balki ularning o‘zaro ta’siriga tegishli bo‘lgan xulq-atvor va xususiyatlarga ega bo‘lgan murakkab hodisa va jarayonlarni yaxlit holda o‘rganuvchi tizimli usul.
Tizimli yondashuv tizimning holatini, uning tashkil etilishini va dinamikligini o'rganishni talab qiladi. Ushbu tadqiqot uchun tahlil qilinadigan tizimning tuzilishi, funksionalligi va xatti-harakati haqida yangi ma'lumotlarni olish imkonini beruvchi turli xil modellar qo'llaniladi.
Har qanday model, umuman olganda, voqelikning soddalashtirilgan tasvirini tashkil etadi, uning asosiy xususiyatlariga qisqartiriladi. Modellashtirishning asosiy bosqichlari quyidagilardan iborat:
a) Birinchi bosqichda dastlabki S tizimni tahlil qilish va shu asosda uning muhim xususiyatlarini o'rnatish sodir bo'ladi.
Masalan, quyidagilar o'rnatilmoqda: tizimning boshlang'ich holatlari, bu evolyutsiya sodir bo'lgan qonunlar, quyi tizimlar o'rtasidagi aloqalar va ularning intensivligi va boshqalar. Aslida, bu elementlar S tizimning M modelini va boshlang'ich holatlarini tashkil qiladi. (boshlang'ich shartlar).
Model haqiqatni qabul qilinadigan aniqlik bilan aks ettirishi kerakligi sababli, modelni shakllantirishdan keyin uni tasdiqlash majburiydir; validatsiya nazariy echimlarni yo'q qiladigan yoki holat o'zgaruvchilari yoki chiqish o'zgaruvchilarining o'lchangan qiymatlariga ega bo'lgan ba'zi vaziyatlarni takrorlashdan iborat.
b) Ikkinchi bosqichda ilmiy izlanishlar orqali M modelni o'rganish amalga oshiriladi; shunday qilib, M modeliga oid yangi ma'lumotlar olinadi, uni dastlabki S sistemasini tekshirish orqali idrok etib bo'lmaydi;
c) Uchinchi bosqichda modellashtirish orqali olingan yangi ma'lumotlarning dastlabki tizimga o'tishi sodir bo'ladi, natijada modellashtirilgan tizim bo'yicha bilim darajasi oshadi.
1. Гидрометеорологик жараёнлар математик моделларининг тузилиши.
1. Theoretically, there are two approaches for systems' analysis:

• 
  The analytic method, consisting in dividing a system in its components, which are afterwards analysed one by one.
  
• 
  The systemic method, examining complex phenomena and processes as a whole, having behaviour and properties, which do not belong to the system's components, but to their interaction.
  

Systemic approach requires the study of the system's states, its organization and its dynamic. For this study various models are being used which make it possible to obtain new information concerning the structure, functionality and behaviour of the analysed system.
Any model constitutes in general a simplified representation of reality, reduced to its essential characteristics. The main stages of the modelling are the following:
a) In the first stage the analysis of the original system S occurs and the establishment on this basis of its essential properties.
For example, the following are being established: the system's initial states, the laws according to which this evolution occurs, the links between the subsystems and their intensity etc. In fact, these elements form the model M of the system S and the initial states (the initial conditions).
As the model must reflect reality with an acceptable accuracy, the model's formulation is followed compulsory by its validation; the validation consists in the reproduction of some situations for which one disposes of theoretical solutions or for which one has measured values of the state variables or of the output variables.
b) In the second stage, through scientific investigations the study of model M takes place; thus, new information concerning the model M is obtained that cannot be perceived by examining the original system S;
c) In the third stage, the transfer over the original system of the new information obtained through modelling takes place, thus resulting an increase of the degree of knowledge concerning the modelled system.
By synthesizing the information gathered until a certain moment, the modelling leads at the same time to the improving of the knowledge process. As a consequence, modelling implies an iterative process of elaboration of new models, which are more and more perfected, and the model (considered to be a satisfying representation of reality) is the result of some iteration covering each and every time the cycle reality-model-validation-reality.
The models thus represent instruments for a better and better approximation of reality, to which they tend to approach in an asymptotic way. The development of computers has led to increased modelling performances, due to the fact that in a short time the examination of tens of variants is possible, without affecting in any way the structure of the studied system. Direct experimentation on the reality of some decision variants, except the fact that it implies high costs and lots of time, may lead in certain cases to unwanted results, and the return to the situation previous to the experiment is either dear, either impossible, due to some transformations which may take place.
7.2. Physical, analogical and mathematical models
The model may be defined as a conventional image of a phenomenon or process, built in such a way as to reflect essential characteristics for the research's purpose. This specific feature results from the modelling limits themselves; the models cannot reproduce the reality in all its complexity, for in this case the model would be a substitute of reality itself (which study is, as it has been shown, difficult, dear, and sometimes even impossible).
The concept of model was used for the first time in 1868 by Italian mathematician Bertrami, who built an Euclidian model for non-Euclidian geometry. Previously, Descartes had put the bases of the analytical geometry, conceived as a model for the Euclidian geometry. The new geometry could be validated by the agreement of the results, which were obtained through the Euclidian, and the new geometry. Subsequently, analytical geometry developed in a special way, outrunning the Euclidian geometry's possibilities and creating the premises of the apparition of non-Euclidian geometries.
The model, as schematic representation of the reality, is largely used in current practice, as well as in the scientific activity. The concept of system itself was introduced through some elementary concepts which are in fact models; one reminds thus of the input-output models (Figure 1.1) or of the input-state-output models (Figure 1.4) for the schematic representation of the systems.
Any system may be modelled; moreover, a system's model is a system itself, being characterized by input, states as well as by output. Once the model is established, it represents the original system, any succession of the model's states being interpreted as a succession of the system's states.
For practical study, physical and mathematical models are being used.