Lecture 2 (Cellular Network Lecture 2 (Cellular Network

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1. INTRODUCTION TO MATHEMATICAL MODELLING

2. OUTLINES OF THE PRESENTATION DIMENSIONAL ANALYSIS WHAT IS MATHEMATICAL MODELLING? WHY MATHEMATICAL MODEL IS NECESSARY? USE OF MATHEMATICAL MODEL TYPES OF MATHEMATICAL MODEL MATHEMATICAL MODELLING PROCESS

3. WHAT IS MATHEMATICAL MODELLING? Representation of real world problem in mathematical form with some simplified assumptions which helps to understand in fundamental and quantitative way. It is complement to theory and experiments and often to integrate them. Having widespread applications in all branches of Science and Engineering & Technology, Biology, Medicine and several other interdisciplinary areas. 2 3 1

4. WHY MATHEMATICAL MODEL IS NECESSARY? To perform experiments and to solve real world problems which may be risky and expensive or time consuming or impossible like astrophysics. Emerged as a powerful, indispensable tool for studying a variety of problems in scientific research, product and process development and manufacturing. Improves the quality of work and reduced changes, errors and rework However, mathematical model is only a complement but does not replace theory and experimentation in scientific research. 1 2 3

5. USE OF MATHEMATICAL MODEL Solves the real world problems and has become wide spread due to increasing computation power and computing methods. Facilitated to handle large scale and complicated problems. Some areas where mathematical models are highly used are : Climate modeling, Aerospace Science, Space Technology, Manufacturing and Design, Seismology, Environment, Economics, Material Research, Water Resource, Drug D esign , Populations Dynamics, Combat and War related problems, Medicine, Biology etc. 1 2 3

6. TYPES OF MATHEMATICAL MODEL EMPIRICAL MODELS THEORETICAL MODELS EXPERIMENTS OBSERVATIONS STATISTICAL MATHEMATICAL COMPUTATIONAL

7. TYPES OF MATHEMATICAL PROCESS REAL WORLD PROBLEM WORKING MODEL MATHEMATICAL MODEL RESULT / CONCLUSIONS COMPUTATIONAL MODEL SIMPLIFY REPRESENT TRANSLATE SIMULATE INTERPRET

8. FORMULATION PROBLEM Modelling PRARAMETERS START SOLUTION /Simulation EVALUATION SATISFIED STOP NO YES

9. TYPES OF MODELS QUALITATIVE AND QUANTITATIVE STATIC OR DYNAMIC DISCRETE OR CONTINUOUS DETERMINISTIC OR PROBABILISTIC LINEAR OR NONLINEAR EXPLICIT OR IMPLICIT 1 2 3 4 5 6

10. STATIC OR DYNAMIC MODEL STATIC MODEL A static (or steady-state) model calculates the system in equilibrium, and thus is time-invariant. A static model cannot be changed, and one cannot enter edit mode when static model is open for detail view. DYNAMIC MODEL A dynamic model accounts for time-dependent changes in the state of the system. Dynamic models are typically represented by differential equations.

11. DISCRETE OR CONTINUOUS MODEL DISCRETE MODEL A discrete model treats objects as discrete, such as the particles in a molecular model. A clock is an example of discrete model because the clock skips to the next event start time as the simulation proceeds. CONTINUOUS MODEL A continuous model represents the objects in a continuous manner, such as the velocity field of fluid in pipe or channels, temperatures and electric field.

12. DETERMINISTIC OR PROBABILISTIC (STOCHASTIC) MODEL DETERMINISTIC MODEL A deterministic model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables. Deterministic models describe behaviour on the basis of some physical law. PROBABILISTIC (STOCHASTIC) MODEL A probabilistic / stochastic model is one where exact prediction is not possible and randomness is present, and variable states are not described by unique values, but rather by probability distributions.

13. LINEAR OR NONLINEAR MODEL LINEAR MODEL If all the operators in a mathematical model exhibit linearity, the resulting mathematical model is defined as linear. A linear model uses parameters that are constant and do not vary throughout a simulation. NONLINEAR MODEL A nonlinear model introduces dependent parameters that are allowed to vary throughout the course of a simulation run, and its use becomes necessary where interdependencies between parameters cannot be considered.

14. EXPLICIT OR IMPLICIT MODEL EXPLICIT MODEL An explicit model calculates the next state of a system directly from the current state without solving complex equations . IMPLICIT MODEL An explicit model calculates the next state of a system directly from the current state without solving complex equations.

15. QUALITATIVE OR QUANTITATIVE MODEL QUALITATIVE MODEL It is basically a conceptual model that display visually of the important components of an ecosystem and linkages between them. It is a simplification of a complex system. The humans are good at common sense with qualitative reasoning. QUANTITATIVE MODEL Models are mathematically focused and many times are based on complex formulas. In addition quantitative models generally through an input-output matrix. Quantitative modelling and simulation give precise numerical answers.

16. DEDUCTIVE MODEL A deductive model is a logical structure based on theory. A single conditional statement is made and a hypothesis ( P ) is stated. The conclusion ( Q ) is then deduced from the statement and hypothesis. (What this model represents ?) P Q (Conditional statement) P (Hypothesis stated) | Q (Conclusion deducted) Example #1 All men are mortal, Ahmed is man, Therefore, Ahmed is mortal. 1 2 3 Example #2 If an angle satisfies 90 <A<180 , then A is an obtuse angle, A=120 , Therefore, A is an obtuse angle. 1 2 3

17. DEDUCTIVE MODEL An inductive model arises from empirical findings and generalizations from them. This is known as “Bottom-up” approach (Qualitative). Focus on generating new theory which is used to form hypothesis. THEORY HYPOTHESIS OBSERVATION CONFIRMATION Deductive model is more narrow in nature and is concerned with confirmation of hypothesis.

18. DEDUCTIVE MODEL Deductive model is a “Top-down” approach (Quantitative). It focus on existing theory and usually begins with hypothesis. OBSERVATION PATTERN TENTATIVE HYPOTHESIS Inductive model is open ended and explanatory, specially at the beginning. THEORY

19. REAL WORLD PROBLEM FALLS IN WHICH CATEGORY? This is based on how much priori information is available on the system. There are two type of models : BLACK BOX MODEL and WHITE BOX MODEL. BLACK BOX MODEL is a system of which there is no priori information available. WHITE BOX MODEL is a system where all necessary information is available.

20. DIMENSIONAL ANALYSIS A method with which non-dimensional can be formed from the physical quantities occurring in any physical problem is known as dimensional analysis. This is a practice of checking relations among physical quantities by identifying their dimensions. The dimension analysis is based on the fact that a physical law must be independent of units used to measure the physical variables. 2 3 1

21. DIMENSIONAL ANALYSIS The practical consequence is that any model equations must have same dimensions on the left and right sides. One must check before developing any mathematical model. 4

22. DIMENSIONAL ANALYSIS EXAMPLE Let us take an example of heat transfer problem. We start with the Fourier’s law of heat transfer. Rate of heat transfer Temperature gradient Let us consider a uniform rod of length l with non-uniform temp. Lying on the x-axis form x=0 to x=l. The density of the rod ( ), specific heat (c), thermal conductivity (K) and cross-sectional area (A) are all constant. (1)

23. DIMENSIONAL ANALYSIS EXAMPLE Change of heat energy of the segment in time ( ) = Heat in from the left side – Heat out from the right side After rearranging (2) (3) After taking the limit where (4)

24. DIMENSIONAL ANALYSIS EXAMPLE (5) (6) (7) , , , , , , , ,

25. DIMENSIONAL ANALYSIS ASSIGNMENT #1 , where D and are diffusion coefficient and coefficient of kinematic viscosity respectively. CALCULATE FOR 1-D DIFFUSION EQUATION AND 1-D FLUID EQUATION IN DIMENSIONLESS FORM AS : THERE ARE GENERALLY THREE ACCEPTED METHODS OF DIMENSIONAL ANALYSIS : RAYLEIGH METHOD (1904): Conceptual method expressed as a functional relationship of some variable | BUCKINGHAM METHOD (1914): The use of Buckingham Pi ( ) theorem as the dimensional parameters was introduced by the Physicist Edger Buckingham in his classical paper | P. W. BRIDGMAN METHOD (1946): Developed on pressure physics)