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INTRODUCTION TO MATHEMATICAL MODELING · Mathematical model is one of the main instruments of...

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  • Chapter – 1



    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.


    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


    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.


  • 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.


    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


    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.


    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.


    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.


    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.


    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.


    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


    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


    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.


    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.


    1.12.1. 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.


    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.


  • “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.