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INTRODUCTION TO

MATHEMATICAL MODELING

Chapter – 1

INTRODUCTION TO MATHEMATICAL MODELING

1.1. INTRODUCTION

A model is an abstraction of reality or a representation of a real object or situation. In

other words, a model presents a simplified version of something. It may be as simple as a

drawing of house plans, or as complicated as a miniature. But it is functional representation

of a complex piece of machinery. A model of aero plane may be assembled and glued

together from a kit by a child, or it actually may contain an engine and a rotating propeller

that allows it to fly like a real aero plane.

The modeling means study of processes and objects in one physical environment by

using processes and objects in other physical environment as models that duplicate the

behavior of the systems under study.

Mathematical Modeling is an experimental approach where a problem is solved and

continually refined over time in order to be more efficient, faster, or more accurate. It is “the

process of scientific inquiry” and formal part of the curriculum for mathematics.

Mathematical modeling is a branch of mathematical logic or a discipline, which helps

us in shaping the real life problems into mathematical models and then solving them

accordingly. As matter of fact mathematical modeling is not a new subject. It has been there

since ancient ages. Scientists, engineers, statisticians, astronomers have been studying a good

number of variety of problems through mathematical models. In the generalized sense for

every process, the problem is modeled into mathematical equations be called mathematical

modeling. Thus there is hardly any area of study and research, which escapes from this

definition. However, the importance of mathematical modeling as a separate subject has

been realized only in recent times. This is evident from the fact that large number of books

and journals have appeared in this area and also a large number of conferences which have

been organized all over the world.

Ever since Issac Newton published his fundamental work Mathematical Principles

of Natural Philosophy in 1687 where the fundamental laws of force and motion were

formulated and the conclusion within the scientific community had been drawn that the

Nature has laws, and we can find them. The importance of this statement cannot be

overestimated. It implies that every system-mechanical, electrical, biological or whatever –

can be accurately described by a mathematical model. In combination with the rapid

development of computers during the last fifty years, the number of available models with

every scientific area has been exploded. The models can today also be applied in practice as

the computation allows us to numerically solve process of such complexity that could hardly

be imagined a couple of decades ago.

In an ideal world, process modeling would be a trivial task. Models would be

constructed in a simple manner yet in every way reproduced the true process behavior. Not

only would the models be accurate, but they would be concise, easy to use and reveal

everything about the internal cause and effect relationships within the process. Each model

would be built for a specific task to a prescribed accuracy. Unfortunately, our world is not

ideal although the above modeling perspective may serve as an excellent long term goal for

everyone dealing with modeling. In the real world it must be realized that a model is always

a simplification of reality. This is especially true when trying to model natural systems

containing living organisms.

Roughly defined, mathematical modeling is the process of constructing mathematical

objects whose behaviors or properties correspond in some way to a particular real- world

system. In this description, a mathematical object could be a system of equations, a stochastic

process, a geometric or algebraic structure, an algorithm, or even just a set of numbers. The

term real- world system could refer to a physical system, a financial system, a social system,

or essentially any other system whose behaviors can be observed.

To predict or simulate very often we wish to know what a real-world system will do

in the future, but it is expensive, impractical, or impossible to experiment directly with the

system. Examples include nuclear reactor design, space flight, extinction of species, weather

prediction, drug efficacy in humans, and so on.

From commercial point of view, it is clear that an improved ability to simulate,

predict, or understand certain real-world systems through mathematical modeling provides a

distinct competitive advantage. Examples: the stock market, aircraft design, oil production

and semiconductor manufacturing. Furthermore, just as in pure science, as computing power

becomes cheaper, modeling becomes an increasingly cost-effective to direct experimentation.

Therefore it is unbeatable that modeling is crucial in this regard.

But unfortunately, there is no definite “algorithm” to construct a mathematical model

that will work in all situations. Modeling is sometimes viewed as an art. It involves taking

whatever knowledge one may possess in mathematics and of the system of interest and using

that knowledge to create something. Since everyone has a different knowledge base, a

preferred bag of tricks, and a unique way of looking at problems, different people may come

up with different models for the same system. There are usually several arguments about the

suitability of the model at the best.

It is very important to understand that for any real system, there is no “perfect” model.

One is always faced with tradeoffs between accuracy, flexibility and cost. Increasing the

accuracy of a model generally increasing cost and decreases flexibility. The object in creating

a model is usually to obtain a “sufficiently accurate” and flexible model at a lower cost.

Mathematical model is one of the main instruments of man‟s knowledge of

phenomena of surrounding world. Under the mathematical models to understand the basic

laws and communication inherent in the phenomenon.

1.2. MATHEMATICAL MODELING IN BIOLOGICAL SCIENCES

The mathematical modeling in Biological systems is relatively recent development

field. The 20th

century has been a period of dramatic technological and scientific

breakthroughs and discoveries. A general feeling among scientists in all areas is that these

developments will continue, perhaps most notably in the life and computer sciences. It is so

based on the testimony by the current rapid progress in biotechnology and information

technology. New sub disciplines have been created such as computational biology and

bioinformatics, which are at the crossroads of biology, physics, chemistry, informatics and

mathematics.

During the past fifty years major discoveries in biology have changed the direction of

science and today it has the Queen of sciences. All hardcore fields, such as physics,

mathematics, chemistry, and computer science are now necessary for the big adventure of

unraveling the secrets of life. Contrastingly, the mathematical sciences are all now

enthusiastically inspired by biological concepts, to the extent that more and more

theoreticians are interacting with biologists. An important part of biology, besides amassing

new experimental information, is the explanation of new phenomena. In order to explain

how a pure theoretician can contribute to the analysis of biological systems, the need of the

hour for mathematics in biology has just arrived.

During the last decade, life science research has become more quantitative and its

reliance on computers seems to increase steadily. The development and (computational)

analysis of the partial differential equation models, finite element analysis models from

mathematical biology is not only of increasing importance for the understanding of biological

processes and for the verification of hypotheses about the underlying biology, but also for the

development of new medicines. It requires a close cooperation with both theoretical and

experimental biologists.

New mathematical models are needed, as well as a new conceptual framework for

biological questions. A new generation of theoretical biologists is needed, trained in

statistical physics, mathematical analysis, stochastic processes, differential geometry, partial

differential equations, with a good overview of electrical engineering, chemical, physics, and

who had spent time in a real laboratory. Biology cannot be learnt from textbooks and

certainly not from mathematicians, but only from biologists. New methods and concepts

have to be created to solve the new problems. Sticking to old concepts is a waste of time and

money. Historically, mathematics has been used extensively in the other sciences and social

sciences to describe, explain, and ultimately predict the behavior of complex systems.

It is planned to discuss and implement the major facts of mathematical modeling

using examples from the biological sciences. These include: a) examining underlying

assumptions, b) translating the “real world” into mathematics, c) generating testable

predications, d) generalizing models to new or different situations, and e) examining the fit

between the mathematics employed and the underlying system being modeled.

There is increasing use of mathematics thought the biological sciences, yet the

training of most biologists still woefully lacks crucial mathematical tools.

1.3. NEED OF THE MATHEMATICAL MODEL

It should be apparent that much of modern science involves mathematical modeling.

The old age “mathematics is the language of science” is true. Scientists use mathematics to

describe real phenomena, and in fact much of this activity constitutes mathematical modeling.

As computers become cheaper and powerful and their usage becomes more widespread,

mathematical models play increasingly important role in science.

The word „model‟ has a wide spectrum of interpretations, e.g., mental model,

linguistic model, visual model, physical model and mathematical model. This work is aimed

and confined to expound to mathematical models, that is, models within a mathematical

framework where equations of various types are defined to relate inputs, outputs and

characteristics of a system.

Primarily, mathematical models are an excellent method of conceptualizing

knowledge about a process and to convey it to other people. Models are also useful for

formulating hypotheses and for incorporating new ideas that can later be verified (or

discarded) in reality. An accurate model of a process allows us to predict the process

behavior for different conditions and thereby we can optimize and control a process for a

specific purpose of our choice. Finally, models serve as an excellent tool for any purpose.

1.4. GENERAL MODELING STRATEGY

The reasons why needed mathematical models suggested in the previous section are

by no means exhaustive. However, once it is concluded that models are useful, it is needed a

general strategy for model building. Such a strategy is discussed in this section.

In overview, the modeling of any system occurs in five rather distinct steps. Step one

is to delineate the system being modeled as functional specification. A quantitative

understanding of the structure and parameters describing the process is required. Typically

for waste-water applications this functional specification may include such information as

equipment type and size flow-sheet layout, environment variables, nominal operating

conditions.

The modeling objectives are then decided and then the desired model type selected.

A model building strategy is then followed to arrive at the appropriate model for the desired

application. In the following subsections it is assumed that the first step, i.e., the functional

process specification, has been successfully accomplished and lights closer look at the

following four steps.

1.4.1. Modeling Objectives

Any given process may have different „appropriate‟ models. The chosen appropriate

model depends on its objectives. So it is the first step about the model and must be prepare

before beginning of model construction. Some of the more relevant objectives concern model

purpose, system boundaries, time constraints and accuracy.

1.4.2. Purpose of the Model

A wide variety of models are possible, each of which may be suitable for a different

applications. For example, simple models suitable for model-based control algorithms may

be totally inadequate for simulating and predating the entire process behavior for safety and

operational analysis. A clear statement of the model intention is needed as first step in setting

the model objectives. This gives entire relevant process variables and the accuracy to which

they must be modeled.

For example, within the field of wastewater treatment number of general purposes can

be defined for mathematical models. These are listed below.

Design – models allow the exploration of the impact of changing system parameters and

development of plants designed to meet the desired process objectives at minimal cost.

Research – models serve as a tool develop and test hypotheses and thereby gaining new

knowledge about the processes.

Process control – models allow for the development of new control strategies by investigating

the system response to a wide range of inputs without endangering the actual plant.

Forecasting – models are used to predict future plant performance when exposed to foreseen

input changes and provide a framework for testing appropriate counteractions.

Performance analysis – models allow for analyzing of total plant performance over time

when compared with laws and regulations and what the impact of new effluent requirements

on plant design and operational costs.

Education – models provide students with a tool to actively explore new ideas and improve

the learning process as well as allowing plant operators training facilities and thereby

increasing their ability to handle unforeseen situations.

1.4.3. System Boundaries

The system boundaries define the scope of model. A correct choice of the system

boundaries is necessary so that all the important dynamics in the process are modeled.

Choice of boundaries which include too many insignificant details lead to large model. This

may cloud an understanding of how and why the system dynamics are occurring as well as

being computationally more expensive. Conversely, the definition of boundaries which fail

to include significant features of the real process could lead to inaccurate dynamic responses

and a loss of confidence in the final model.

In uncertainty exists about the correct choice of boundary, a criterion for boundary

selection is to cheek whether the streams crossing the proposed boundary are easy to

characterize (e.g., constant, step impulses). If the streams are well characterized, then the

correct boundary has been chosen.

1.4.4. Time Constraints

Time constraints are important model restrictions to be chosen before construction of

dynamic models. Frequently, the process under investigation contains a wide range of

dynamic activity with widely varying speeds of response. Characteristic time constants in the

process may range over many orders of magnitude. Invariably the modeler is interested in a

simulation over a defined period of time. For example, in an activated sludge process the

dynamics of the dissolved oxygen concentration have a time constant in the range of minutes

whereas the dynamics of the biomass population are more in the range of days-weeks.

To produce an appropriate model, the modeler should therefore identify a „time-scale-

of-interest‟ and not model any latent dynamic effects outside this time-scale. This

identification should be in the form of maximum and minimum characteristic time constant.

Selection of an appropriate timescale will also have the added advantage of possibly avoiding

ultra-stable or stiff problems in the model numerical solution. These numerical problems

occur in systems with widely varying time stable or speed or response.

1.4.5. Accuracy

The appropriateness of the model depends on the ability to predict the system

performance within a prescribed accuracy. The accuracy sought will affect the degree of

simplification which can be achieved in building the model. It is important that the desired

accuracy of the model be specified before the model is constructed and that this accuracy

reflects the purpose of the model. A measure of accuracy must be created to confirm this, or

the accuracy must be confirmed during the model validation.

1.5. TYPES OF MODELS

A more usable concept of a model is that of an abstraction, from the real problem, of

key variables and relationships. These are abstracted in order to simplify the problem itself.

Modeling allows the user to better understanding of the problem and presents a means for

manipulating the situation in order to analyze the results of various inputs ("what if" analysis)

by subjecting it to a changing set of assumptions.

1.6. MODEL CLASSIFICATIONS

Some models are replicas of the physical properties (relative shape, form, and

weight) of the object they represent. Others are physical models but do not have the same

physical appearance as the object of their representation. A third type of model deals with

symbols and numerical relationships and expressions. Each of these fits within an overall

classification of four main categories: physical models, schematic models, verbal models, and

mathematical models.

1.6.1. Physical Models

Physical models are the ones that look like the finished object they represent. Iconic

models are exact or extremely similar replicas of the object being modeled. Model aero

planes, cars, ships, and even models of comic book super-heroes look exactly like their

counterpart but in a much smaller scale. Scale models of municipal buildings, shopping

centers, and property developments such as subdivisions, homes, and office complexes all

hopefully look exactly as the "real thing" will look when it is built. The advantage here is the

model‟s correspondence with the reality of appearance. In other words, the model user can

tell exactly what the proposed object will look like, in three dimensions, before making a

major investment.

In addition to looking like the object they represent, some models perform as their

counterparts would. This allows experiments to be conducted on the model to see how it

might perform under actual operating conditions. Scale models of aeroplanes can be tested in

wind tunnels to determine aerodynamic properties and the effects of air turbulence on their

outer surfaces. Model automobiles can be exposed to similar tests to evaluate how wind

resistance affects such variables as handling and gas mileage. Models of bridges and dams

can be subjected to multiple levels of stress from wind, heat, cold, and other sources in order

to test such variables as endurance and safety. A scale model that behaves in a manner that is

similar to the "real thing" is far less expensive to create and test than its actual counterpart.

These types of models often are referred to prototypes.

Additionally, some physical models may not look exactly like their object of

representation but are close enough to provide some utility. Many modern art statues

represent some object of reality, but are so different that many people cannot clearly

distinguish the object they represent. These are known as analog models. An example is the

use of cardboard cutouts to represent the machinery being utilized within a manufacturing

facility. This allows planners to move the shapes around enough to determine an optimal

plant layout.

1.6.2. Schematic Models

Schematic models are more abstract than physical models. While they do have some

visual correspondence with reality, they look much less like the physical reality they

represent. Graphs and charts are schematic models that provide pictorial representations of

mathematical relationships. Plotting a line on a graph indicates a mathematical linear

relationship between two variables. Two such lines can meet at one exact location on a graph

to indicate the break-even point, for instance. Pie charts, bar charts, and histograms can all

model some real situation, but really bear no physical resemblance to anything.

Diagrams, drawings, and blueprints also are versions of schematic models. These are

pictorial representations of conceptual relationships. This means that the model depicts a

concept such as chronology or sequence. A flow chart describing a computer program is a

good example. The precedence diagrams used in project management or in assembly-line

balancing show the sequence of activities that must be maintained in order to achieve a

desired result.

1.6.3. Verbal Models

Verbal models use words to represent some object or situation that exists, or could

exist, in reality. Verbal models may range from a simple word presentation of scenery

described in a book to a complex business decision problem (described in words and

numbers). A firm's mission statement is a model of its beliefs about what business it is in and

sets the stage for the firm's determination of goals and objectives.

Verbal models frequently provide the scenario necessary to indicate that a problem is

present and provide all the relevant and necessary information to solve the problem, make

recommendations, or at least determine feasible alternatives. Even the cases presented in

management textbooks are really verbal models that represent the workings of a business

without having to take the student to the firm's actual premises. Oftentimes, these verbal

models provide enough information to later depict this problem in mathematical form. In

other words, verbal models are frequently converted into mathematical models so that an

optimal, or at least functional, solution may be found utilizing some mathematical technique.

A look into in any mathematics book, operations management book, or management science

text generally provides some problems that appear in word form. The job of the student is to

convert the word problem into a mathematical problem and seek a solution.

1.6.4. Mathematical Models

Mathematical models are perhaps the most abstract of the four classifications. These

models do not look like their real-life counterparts at all. Mathematical models are built using

numbers and symbols that can be transformed into functions, equations, and formulas. They

also can be used to build much more complex models such as matrices or linear programming

models. The user can then solve the mathematical model (seek an optimal solution) by

utilizing simple techniques such as multiplication and addition or more complex techniques

such as matrix algebra or Gaussian elimination. Since mathematical models frequently are

easy to manipulate, they are appropriate for use with calculators and computer programs.

Mathematical models can be classified according to use (description or optimization), degree

of randomness (deterministic and stochastic), and degree of specificity (specific or general).

Following is a more detailed discussion of different types of mathematical models.

1.7. TYPES OF MATHEMATICAL MODELS

According to usage, mathematical models are classified into descriptive models and

optimization models.

1.7.1. Descriptive Models

Descriptive models are used merely to describe something mathematically. Common

statistical models in this category include the mean, median, mode, range, and standard

deviation. Consequently, these phrases are called "descriptive statistics." Balance sheets,

income statements, and financial ratios are also descriptive in nature.

1.7.2. Optimization Models

Optimization models are used to find an optimal solution. The linear programming

models are mathematical representations of constrained optimization problems. These models

share certain common characteristics. Knowledge of these characteristics enables us to

recognize problems that can be solved using linear programming. For example, suppose that

a firm that assembles computers and computer equipment is about to start production of two

new types of computers. Each type will require assembly time, inspection time, and storage

space. The amounts of each of these resources that can be devoted to the production of the

computers is limited. The manager of the firm would like to determine the quantity of each

computer to produce in order to maximize the profit generated by their sale.

Many different classifications have been produced for the different model types which

are available. It is possible to separate mathematical models based on the philosophy of the

approach, with regard to the mathematical form of the model, in sometimes also depending

on the application area of the model. Philosophies of different models will enlighten

discussion as follows.

1.7.3. Reductionist Vs Holistic Models

Reductionist models are based on the attempt to include as many details as possible

into the model and to describe the behavior of a system as the net effect of all processes. In

contrast to this approach, holistic models are based on a few important global parameters and

on general principles.

1.7.4. Internal Vs External Models

Internal (or mechanistic) models describe system response as a consequence of input

using the mechanistic structure of the system, whereas external (or input/output, black-box,

empirical) models are based on empirical relationships between the input and the output.

Typical external models are time series models (e.g., “ARMAX” models) and neural

networks. A mechanistic model is a model based on fundamental engineering and scientific

knowledge stand the physical, chemical and biological mechanisms that affect a system. A

model based on elementary principles tends to produce more reliable results when used for

extrapolation. In complex systems it can be very difficult to obtain the necessary

fundamental relationships of the process and consequently, a model must be based on

empirical relationships. In practice, models are often a mixture of mechanistic and empirical

models, using different examples at different levels of resolution. As an example, microbial

growth rates are in most cases parameterized empirically at the cell level, but macroscopic

water flow and substance mass balances are treated in a mechanistic way. External models

may even be used to obtain simplified descriptions of situations in which the validity of an

internal model is widely accepted. As an example, empirical parameterizations of turbulent

correlations are used in equations describing mean values of turbulent flow, because the

solution of the underlying Navier-Stokes equations is too difficult.

1.7.5. Dynamic Vs Static Models

This classification arises between models that do or do not vary with time.

Static models are often referred to as steady-state models. They model the equilibrium

behavior of the system. Conversely dynamic models account for the time varying responses

of a system. Both these types are used extensively in engineering applications. This is

evidenced by the large number of commercially available „simulators‟ for both types. While

it may appear that the dynamic simulators are dominating, they have received a more limited

acceptance outside of an academic environment.

1.7.6. Deterministic Vs Stochastic (Probabilistic) Models

Another classification arises between models that contain uncertainty or

insolences in their final results and those that do not. Stochastic models are models in which

the final outcome is not known with certainty but can be expressed as a distribution of all

possible outcomes. In deterministic models all future outcomes are known with precision by

the present state and the future values of external variables (inputs) of the model. Stochastic

models also take into account the random influences of the temporal evolution of the system

itself. Although the stochastic description of systems may be more realistic, the large

majority of models formulated so far are deterministic. The main reasons for this fact may be

the lack of date for the characterization of random variables, high requirements of

computational resources for solving stochastic differential equations and the success of

deterministic models in describing average future behavior.

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.

Therefore, deterministic models perform the same way for a given set of initial conditions.

Conversely, in a stochastic model, randomness is present, and variable states are not

described by unique values, but rather by probability distributions. A static model does not

account for the element of time, while a dynamic model does. Dynamic models are typically

represented with difference equations or differential equations.

1.7.7. Continuous Time Vs Discrete – Time Models

Many courses of events which are of interest to modeling are distributed for

only in time but also in space. Mathematically, variables distributed in space can be

described by partial differential equations and the resulting models are called distributed

models. Application of such equations will, however, result in a complex simulation

problem. A common way of overcoming this difficulty is to use the lumped – parameter

approximation of these distributed equations. To use this approach, isotropic regions in the

process are identified. These are regions in which composition, specific energy and

momentum are approximately invariant with spatial dimension. The time-varying properties

of this „lump‟ are then calculated from the transfer of mass, energy and momentum over the

boundary of the region.

1.7.8. Linear Vs Nonlinear

Mathematical models are usually composed by variables, which are abstractions of

quantities of interest in the described systems, and operators that act on these variables, which

can be algebraic operators, functions, differential operators, etc. If all the operators in a

mathematical model present linearity, the resulting mathematical model is defined as linear.

A model is considered to be nonlinear otherwise.

The question of linearity and nonlinearity is dependent on context, and linear models

may have nonlinear expressions in them. For example, in a statically linear model, it is

assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor

variables. Similarly, a differential equation is said to be linear if it can be written with linear

differential operators, but it can have nonlinear expressions in it. In a mathematical

programming model, if the objective functions and constraints are represented entirely by

linear equations, then the model is regarded as a linear model. If one or more of the objective

functions or constraints are represented with a nonlinear equation, then the model is known as

nonlinear model. Nonlinearity, even in fairly simple systems, is often associated with

phenomena such as chaos and irreversibility. Although there are exceptions, nonlinear

systems and models tend to be more difficult to study than linear ones. A common approach

to nonlinear problems is linearization, but this can be problematic if one is trying to study

aspects such as irreversibility, which are strongly tied to nonlinearity.

1.7.9. Lumped Vs Distributed Parameters

If the model is homogeneous (Consistent state throughout the entire system) the

parameters are distributed. If the model is heterogeneous (varying state within the system),

then the parameters are lumped. Distributed parameters are typically represented with partial

differential equations.

A crucial part of the modeling process is the evaluation of whether or not a given

mathematical model describes a system accurately. This question can be difficult to answer

as it involves several different types of evaluation.

1.8. IMPORTANCE OF MATHEMATICAL MODELING

Mathematical models essentially find the relationship between certain variables.

Basically any physical quantity in a system can be governed by a mathematical model, simple

or complicated. The better the model is, the more accurate it will reflect the real system.

1.9. ADVANTAGES OF MATHEMATICAL MODELING

The mathematical model is a tool that allows us to investigate the static and dynamic

behavior of a system without doing – or at least reducing the number of practical

experiments. The use of mathematical models in the experimental analysis of behavior has

increased over the years, and they offer several advantages. Mathematical models require

theorists to be precise and unambiguous, often allowing comparisons of competing theories

that sound similar when stated in words. Sometimes different mathematical models may

make equally accurate predictions for a large body of data. In such cases, it is important to

find and investigate situations for which the competing models make different predictions

because, unless two models are actually mathematically equivalent, they are based on

different assumptions about the psychological processes that underlie an observed behavior.

Mathematical models developed in basic behavioral research have been used to predict and

control behavior in applied settings, and they have guided research in other areas of

psychology. A good mathematical model can provide a common framework for

understanding what might otherwise appear to be diverse and unrelated behavioral

phenomena. Because psychologists vary in their quantitative skills and in their tolerance for

mathematical equations, it is important for those who develop mathematical models of

behavior to find ways (such as verbal analogies, pictorial representations, or concrete

examples) to communicate the key premises of their models to non specialists.

Models provide the most effective means developed for predicting performance. It is

hard to conceive a prediction system that is not finally a model. To construct a model of a

real process or system, careful consideration of the system elements that must be abstracted is

required. This in itself usually is a profitable activity, for it develops insights into the

problem.

It is important for the model user to realize that model development and model

solution are not completely separable. While the most accurate representation possible may

seem desirable, the user still must be able to find a solution to the modeled problem. Model

users need to remember that they are attempting to simplify complex problems so that they

may be analyzed easily, quickly, and inexpensively without actually having to perform the

task. It is also desirable for a model that allows the user to manipulate the variables so that

"what if" questions can be answered.

Models come in many varieties and forms, ranging from the simple and crude to the

elegant and exotic. Whatever category they are in, all models share the distinction of being

simplifications of more complex realities that should, with proper use, result in a useful

decision-making aid.

Translating a verbal hypothesis into mathematical model forces a theorist to be

precise and unambiguous and this can point to ways of testing competing theories that

sound as if they make similar predictions when they are stated in words.

Even when the shapes of two mathematical functions are quite similar, these functions

may make distinctly different predictions about behavior with profound theoretical

and applied implications.

In some cases, competing mathematical models may account for large data sets about

equally well. But unless they are actually mathematically equivalent, different

mathematical models are based on different assumptions about the psychological

processes underlying an observed behavior.

Mathematical models of behavior that are developed through basic behavioral

research can be used to predict or control behavior in applied settings. The models

have been used in neuroscience and psychopharmacology to help researchers identify

the functions of different brain structures and to assess the behavioral effects of

different drugs.

A mathematical model can provide a common framework that unites diverse

behavioral phenomena.

The goal of modeling use is to adequately portray realistic phenomenon. Once

developed properly, a great deal can be learned about the real-life counterpart by

manipulating a model's variables and observing the results.

Real-world decisions involve an overwhelming amount of detail, much of which may

be irrelevant for a particular problem or decision. Models allow the user to eliminate

the unimportant details so that the user can concentrate on the relevant decision

variables that are present in a situation. This increases the opportunity to fully

understand the problem and its solution.

Models generally are easy to use and less expensive than dealing with the actual

situation.

Models require users to organize and sometimes quantify information and, in the

process; often indicate areas where additional information is needed.

Models provide a systematic approach to problem solving.

Models develop understanding of the problem.

Models enable managers to analyze "what if" questions.

Models require users to be very specific about objectives.

Models serve as a consistent tool for evaluation.

Models enable users to bring the power of mathematics to bear on a problem.

Models provide a standardized format for analyzing a problem.

1.10. LIMITATIONS OF MATHEMATICAL MODELING

Although some of these models are relatively straightforward (e.g., those representing

the negative law of effect and the avoidance theory of punishment), for others the equations

are quite complex, as are the derivations that allow them to be applied to specific examples,

and not everyone is able to follow them. This is one of the drawbacks of mathematical

modeling, but it is a cost that is more than offset by the advantages. The mathematical

precision of these theories allows them to be tested rigorously, and in testing these theories

their strengths can be demonstrated and their weaknesses can be exposed. For instance, the

Rescorla -Wagner model is a landmark in the field of classical conditioning, and it has

stimulated a great deal of research. However, the Rescorla-Wagner model does have some

well-documented limitations, and these have prompted the development of alternative

models. Because of the empirical and theoretical work that was stimulated by the Rescorla-

Wagner model, we now have a much better understanding of the richness and complexity of

classical conditioning than we did before this model was introduced.

In practice an experimental approach often has serious limitations that make it

necessary to work with mathematical models instead. Some other extreme examples of such

limitations are given below.

Too expensive: It is somewhat expensive to launch rockets to the moon until one successfully

hits the surface, then rebuild this type of rocket in order to use it for the intended purpose.

Too dangerous: Starting to train nuclear power plant operators at full-scale running plants is

not advisable.

Too time-consuming: It would take far too much time to investigate all variations of

combinations mixtures, temperature and pressure in a complex chemical process to identify

the optimum combination. With a few experiments, the rest of the experimental domain cab

be stimulated by a model.

Non-existing system: While designing a suspension bridge it is necessary to simulate how

different designs will be affected by, for example, high winds.

1.11. FLUIDS

A fluid is a substance which is capable of flowing or a fluid is a substance which

deforms continuously when subjected external shearing force. According to viscosity fluids

are mainly divided in to two types they are ideal fluids and real fluids.

1.11.1. Ideal Fluids

An ideal fluid is one which has no viscosity and surface tension and is

incompressible. In true sense no such fluid exists in nature. However, fluid which has low

viscosities such as water and air can be treated as ideal fluids under certain conditions. The

assumption of ideal fluids helps in simplifying the mathematical analysis.

1.11.2. Real Fluids

A real practical fluid is one which has viscosity, surface tension and compressibility in

addition to the density. The real fluids are actually available in nature.

1.12. TYPES OF FLUIDS

1.12.1. Newtonian Fluids

These fluids follow Newton‟s viscosity equation. For such fluids μ does not change

with rate of deformation.

Ex: Water, Kerosene, air etc.

1.12.2. Non Newtonian Fluids

Fluids which do not follow the linear relationship between shear stress and rate of

deformation are termed as Non Newtonian fluids. Such fluids are relatively uncommon.

Ex: solutions or suspensions (slurries), mud flows, polymer solutions, blood etc. These fluids

are generally complex mixtures and are studied under Rheology, a science of deformation

and flow.

1.12.3. Plastic Fluids

In the case of a plastic substance which is Non - Newtonian fluid an initial yield stress

is to be exceeded to cause a continuous deformation. These substances are represented by

straight line intersecting the vertical axis at the “yield stress”. An ideal plastic (or Binigham

plastic) has a definite yield stress and a constant linear relation between shear stress and the

rate of angular deformation.

Ex: Sewage sludge, drilling muds etc.

A thyxotropic substance, which is non - Newtonian fluid, has a non – linear

relationship between the shear stress and the rate of angular deformation, beyond an initial

yield stress.

Ex: The printer‟s ink.

1.13. BRIEF INTRODUCTION ABOUT NANO MATERIALS

Nano Science and Nanotechnology is considered to be one of the most promising fields

having huge potential to bring countless opportunities in many areas of research and

development. It is the study of tiny structures at nanometer scale which forms a basis for

number of core technologies.

“Men love to wonder and that is the seed of Science”. Undoubtedly a nanomaterial is

here to change the lives of people. It is already showing its impact on various fields, from

cosmetics to medicine to aerospace. It opens door for immense activities in various fields and

will expand with flourishing research activities and their products in the coming year.

Nanomaterials plays vital role in the areas of electronics, semiconductors, materials,

automobiles and aerospace industries, textiles, sports equipment, mechanics, Pharmaceuticals

including drug delivery, cosmetics, biotechnology, medical fields, optoelectronics,

environmental monitoring and control, food science including quality control and packaging,

forensics, university and lab research, military, etc.

The advent of nanotechnology has resulted in increased use of nanomaterial based

products in day to day life. A significant increase in surface area to volume ratio at the

nanoscale, giving rise to novel and enhanced Mechanical, Optical, Conducting, Electronic,

Catalytic, Magnetic, etc., properties to nanomaterials has made nanotechnology the most

promising field. Due to well defined geometry, Exceptional mechanical properties and

extraordinary electric characteristics, among other outstanding physical properties of

nanomaterials have potential applications.

Hence we prepare the mathematical models with nanomaterials in this work.

1.14. SCOPE OF THE PRESENT STUDY

“The sciences do not try to explain, they hardly even try to interpret, but they mainly

make models”. By a model is meant a mathematical construct which, with the addition of

certain verbal interpretations describes observed phenomena. The justification of such a

mathematical construct is solely and precisely that it is expected to work.

Mathematical aspects play an ever increasing role in biology. In recent years,

mathematical modeling of developmental processes has gained new respect and significance

task to combine diverse areas of knowledge. Not only have mathematical models been used

to validate hypotheses made from experimental data, but designing and testing these models

have led to testable experimental predictions. There are now impressive cases in which

mathematical models have provided fresh insight into biological systems.

In the last forty years or so mathematical biology has become very popular, and in the

last ten years or so the explosion in quantitative experimental data has meant that models are

more necessary and better validated or parameterized. Mathematical biology aims at the

mathematical representation, treatment and modeling of biological processes, using a variety

of applied mathematical techniques and tools. It has both theoretical and practical

applications in biological, biomedical and biotechnological research. It has a tremendous

economic potential for pharmaceutical, and biotech industry.

Besides being incorporating and recognizing the research developed so far, the

present research work is planned to establish to prepare systematic mathematical model for

biological systems especially for water treatment, bone grafting, heat transfer in biomagnetic

fluids, orientation of red blood cells in the magnetic fields with in the boundary conditions.