Homework Title / No. : ___TERM-PAPER________________Course Code : MTH 204

Course Instructor : __ Ms. Ritu Sharma _ Tutor (if applicable) : ____________

Date of Allotment : _____________________ Date of submission : __03/12/2009_____

Student’s Roll No._______21______________ Section No. : D2702________ Declaration: I declare that this assignment is my individual work. I have not copied from any other student’s work or from any other source except where due acknowledgment is made explicitly in the text, nor has any part been written for me by another person.

Student’s Signature : _____ishan arora__

Evaluator’s comments: ____________________________________________________ Marks obtained : ___________ out of ______________________ Content of Homework should start from this page only:

TERM PAPER

OF

NUMERICAL ANALYSIS

(MTH 204)

TOPIC : APPLICATIONS OF DIFFERENTIAL EQUATIONS IN ENGINEERING

Submitted To:Ms. Ritu Sharma Submitted By: Ishan Arora B.Tech-M.B.A (IT) Section: D2702 Roll No. : 21

CONTENTS

1. Introduction to Differential Equations2. Nomenclature3. Applications of Differential Equations3.1. Radioactive Decay3.2. Newton’s Law of Cooling3.3. Orthogonal Trajectories3.4. Population Dynamics4. Some other Applications to Engineering and Sciences

ACKNOWLEDGEMENT

I Ishan Arora, student of B.Tech – M.B.A. (IT) (Section - D2702 ) express my deep gratitude to my teacher Ms. Ritu Sharma . I m very thankful to her for support that led me to the completion of this term paper. I m thankful to my parents who encouraged me and provided me all the necessary resources that had made possible for me to be able to accomplish this task. I also thank all my friends who assisted me in completing this work.

DIFFERENTIAL EQUATIONS

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics and other disciplines.

Differential equations arise in many areas of science and technology: whenever a deterministic relationship involving some continuously varying quantities (modelled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time varies. Newton's Laws allow one to relate the position, velocity, acceleration and various forces acting on the body and state this relation as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly.

An example of modelling a real world problem using differential equations is determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is constant but air resistance may be modelled as proportional to the ball's velocity. This means the ball's acceleration, which is the derivative of its velocity, depends on the velocity. Finding the velocity as a function of time requires solving a differential equation.

Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions, the set of functions that satisfy the equation. Only the simplest differential equations admit solutions given by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

Nomenclature

The theory of differential equations is quite developed and the methods used to study them vary significantly with the type of the equation.

An ordinary differential equation (ODE) is a differential equation in which the unknown function (also known as the dependent variable) is a function of a single independent variable. In the simplest form, the unknown function is a real or complex valued function, but more generally, it may be vector-valued or matrix-valued: this corresponds to considering a system of ordinary differential equations for a single function. Ordinary differential equations are further classified according to the order of the highest derivative with respect to the dependent variable appearing in the equation. The most important cases for applications are first order and second order differential equations. In the classical literature also distinction is made between differential equations explicitly solved with respect to the highest derivative and differential equations in an implicit form.

A partial differential equation (PDE) is a differential equation in which the unknown function is a function of multiple independent variables and the equation involves its partial derivatives. The order is defined similarly to the case of ordinary differential equations, but further classification into elliptic, hyperbolic, and parabolic equations, especially for second order linear equations, is of utmost importance. Some partial differential equations do not fall into any of these categories over the whole domain of the independent variables and they are said to be of mixed type.

Both ordinary and partial differential equations are broadly classified as linear and nonlinear. A differential equation is linear if the unknown function and its derivatives appear to the power 1 (products are not allowed) and nonlinear otherwise. The characteristic property of linear equations is that their solutions form an affine subspace of an appropriate function space, which results in much more developed theory of linear differential equations. Homogeneous linear differential equations are a further subclass for which the space of solutions is a linear subspace i.e. the sum of any set of solutions or multiples of solutions is also a solution. The coefficients of the unknown function and its derivatives in a linear differential equation are allowed to be (known) functions of the independent variable or variables; if these coefficients are constants then one speaks of a constant coefficient linear differential equation.

There are very few methods of explicitly solving nonlinear differential equations; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf Navier–Stokes existence and smoothness).

Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the

harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).

Examples

In the first group of examples, let u be an unknown function of x, and c and ω are known constants.

Inhomogeneous first order linear constant coefficient ordinary differential equation:

Homogeneous second order linear ordinary differential equation:

Homogeneous second order constant coefficient linear ordinary differential equation describing the harmonic oscillator:

First order nonlinear ordinary differential equation:

Second order nonlinear ordinary differential equation describing the motion of a pendulum of length L:

In the next group of examples, the unknown function u depends on two variables x and t or x and y.

Homogeneous first order linear partial differential equation:

Homogeneous second order linear constant coefficient partial differential equation of elliptic type, the Laplace equation:

Third order nonlinear partial differential equation, the Korteweg–de Vries equation:

Applications of Differential Equations

We present examples where differential equations are widely applied to model natural phenomena, engineering systems and many other situations.

Radioactive Decay

Many radioactive materials disintegrate at a rate proportional to the amount present. For example, if X is the radioactive material and Q(t) is the amount present at time t, then the rate of change of Q(t) with respect to time t is given by

where r is a positive constant (r>0). Let us call the initial quantity of the material X, then we have

Clearly, in order to determine Q(t) we need to find the constant r. This can be done using what is called the half-life T of the material X. The half-life is the time span needed to

disintegrate half of the material. So, we have . An easy calculation gives

. Therefore, if we know T, we can get r and vice-versa. Many chemistry text-books contain the half-life of some important radioactive materials. For example, the

half-life of Carbon-14 is . Therefore, the constant r associated with

Carbon-14 is . As a side note, Carbon-14 is an important tool in the archeological research known as radiocarbon dating.

Example: A radioactive isotope has a half-life of 16 days. You wish to have 30 g at the end of 30 days. How much radioisotope should you start with?

Solution: Since the half-life is given in days we will measure time in days. Let Q(t) be

the amount present at time t and the amount we are looking for (the initial amount). We know that

,

where r is a constant. We use the half-life T to determine r. Indeed, we have

Hence, since

,

we get

Newton's Law of Cooling

From experimental observations it is known that (up to a ``satisfactory'' approximation) the surface temperature of an object changes at a rate proportional to its relative temperature. That is, the difference between its temperature and the temperature of the surrounding environment. This is what is known as Newton's law of cooling. Thus, if

is the temperature of the object at time t, then we have

where S is the temperature of the surrounding environment. A qualitative study of this phenomena will show that k >0. This is a first order linear differential equation. The

solution, under the initial condition , is given by

Hence,

,

which implies

This equation makes it possible to find k if the interval of time is known and vice-versa.

Example: Time of Death Suppose that a corpse was discovered in a motel room at midnight and its temperature was . The temperature of the room is kept constant at

. Two hours later the temperature of the corpse dropped to . Find the time of death.

Solution: First we use the observed temperatures of the corpse to find the constant k. We have

.

In order to find the time of death we need to remember that the temperature of a corpse at time of death is (assuming the dead person was not sick!). Then we have

which means that the death happened around 7:26 P.M.

Orthogonal Trajectories

We have seen before that the solutions of a differential equation may be given by an implicit equation with a parameter something like

This is an equation describing a family of curves. Whenever we fix the parameter C we get one curve and vice-versa. For example, consider the families of curves

where m and C are parameters. Clearly, we may change the names of the variables and still have the same geometric curves. For example, the above families define the same geometric object as

Note that the first family describes all the lines passing by the origin (0,0) while the second the family describes all the circles centered at the origin (including the limit case when the radius 0 which reduces to the single point (0,0)) (see the pictures below).

and

In this page, we will only use the variables x and y. Any family of curves will be written as

One may ask whether any family of curves may be generated from a differential equation? In general, the answer is no. Let us see how to proceed if the answer were to be yes. First differentiate with respect to x, and get a new equation involving in general x, y,

, and C. Using the original equation, we may able to eliminate the parameter C from the new equation.

Example. Find the differential equation satisfied by the family

Answer. We differentiate with respect to x, to get

Since we have

then we get

You may want to do some algebra to make the new equation easy to read. The next step is to rewrite this equation in the explicit form

this is the desired differential equation.

Example. Find the differential equation (in the explicit form) satisfied by the family

Answer. We have already found the differential equation in the implicit form

Algebraic manipulations give

Let us reconsider the example of the two families

If we draw the two families together on the same graph we get

As we see here something amazing happened. Indeed, it is clear that whenever one line intersects one circle, the tangent line to the circle (at the point of intersection) and the line are perpendicular or orthogonal. We say the two curves are orthogonal at the point of intersection.

Definition. Consider two families of curves and . We say that and are

orthogonal whenever any curve from intersects any curve from , the two curves are orthogonal at the point of intersection.

For example, we have seen that the families y = m x and are orthogonal. One may then ask the following natural question:

Given a family of curves , is it possible to find a family of curves which is orthogonal to ?

The answer to this question has many implications in many areas such as physics, fluid-dynamics, etc... In general this question is very difficult. But in some cases, we may be able to carry on the calculations and find the orthogonal family. Let us show how.

Consider the family of curves . We assume that an associated differential equation may be found, say

We know that for any curve from the family passing by the point (x,y), the slope of the tangent at this point is f(x,y). Hence the slope of the line perpendicular (or orthogonal) to

this tangent is which happens to be the slope of the tangent line to the orthogonal curve passing by the point (x,y). In other words, the family of orthogonal curves are solutions to the differential equation

From this we see what we have to do. Indeed consider a family of curves . In order to find the orthogonal family, we use the following practical steps

Step 1. Find the associated differential equation. Step 2. Rewrite this differential equation in the explicit form

Step 3. Write down the differential equation associated to the orthogonal family

Step 4. Solve the new equation. The solutions are exactly the family of orthogonal curves. Step 5. You may be asked to give a geometric view of the two families. Also you may be asked to find a specific curve from the orthogonal family (something like an IVP).

Example. Find the orthogonal family to the family of circles

Answer. First, we look for the differential equation satisfied by the circles. We differentiate with respect to the variable x to get

We rewrite this equation in the explicit form

Next we write down the equation for the orthogonal family

This is a linear as well as a separable equation. If we use the technique of linear equations, we get the integrating factor

which gives

We recognize the family of lines and we confirm our earlier observation (that the two families are indeed orthogonal).

This example is somehow easy and was given here to illustrate the technique.

Example. Find the orthogonal family to the family of circles

Answer. We have seen before that the explicit differential equation associated to the family of circles is

Hence the differential equation for the orthogonal family is

We recognize an homogeneous equation. Let us use the technique developed to solve this

kind of equations. Consider the new variable (or equivalently y = x z). Then we have

and

Hence we have

Algebraic manipulations imply

This is a separable equation. The constant solutions are given by

which gives z=0. The non-constant solutions are found once we separate the variables

and then we integrate

Before we perform the integration for the left-hand side, we need to use partial decomposition technique. We have

We will leave the details to you to show that A = 1, B=-2, and C=0. Hence we have

Hence

which is equivalent to

where . Putting all the solutions together we get

Going back to the variable y, we get

which is equivalent to

We recognize a family of circles centered on the y-axis and the line y=0 (the x-axis which was easy to guess, isn't it?)

If we put both families together, we appreciate better the orthogonality of the curves (see the picture below).

Population Dynamics

Here are some natural questions related to population problems:

What will the population of a certain country be in ten years? How are we protecting the resources from extinction?

More can be said about the problem but, in this little review we will not discuss them in detail. In order to illustrate the use of differential equations with regard to this problem we consider the easiest mathematical model offered to govern the population dynamics of a certain species. It is commonly called the exponential model, that is, the rate of change of the population is proportional to the existing population. In other words, if P(t) measures the population, we have

,

where the rate k is constant. It is fairly easy to see that if k > 0, we have growth, and if k <0, we have decay. This is a linear equation which solves into

,

where is the initial population, i.e. . Therefore, we conclude the following:

if k>0, then the population grows and continues to expand to infinity, that is,

if k<0, then the population will shrink and tend to 0. In other words we are facing extinction.

Clearly, the first case, k>0, is not adequate and the model can be dropped. The main argument for this has to do with environmental limitations. The complication is that population growth is eventually limited by some factor, usually one from among many essential resources. When a population is far from its limits of growth it can grow exponentially. However, when nearing its limits the population size can fluctuate, even chaotically. Another model was proposed to remedy this flaw in the exponential model. It is called the logistic model (also called Verhulst-Pearl model). The differential equation for this model is

,

where M is a limiting size for the population (also called the carrying capacity). Clearly, when P is small compared to M, the equation reduces to the exponential one. In order to solve this equation we recognize a nonlinear equation which is separable. The constant solutions are P=0 and P=M. The non-constant solutions may obtained by separating the variables

,

and integration

The partial fraction techniques gives

,

which gives

Easy algebraic manipulations give

where C is a constant. Solving for P, we get

If we consider the initial condition (assuming that is not equal to both 0 or M), we get

,

which, once substituted into the expression for P(t) and simplified, we find

It is easy to see that

However, this is still not satisfactory because this model does not tell us when a population is facing extinction since it never implies that. Even starting with a small population it will always tend to the carrying capacity M.

Some other Applications to Engineering and Sciences

Historically, it has been the needs of the physical sciences which have driven the development of many parts of mathematics, particularly analysis. The applications are sometimes difficult to classify mathematically, since tools from several areas of mathematics may be applied. We focus on these applications not by discussing the nature of their discipline but rather their interaction with mathematics.

Mechanics of particles and systems studies dynamics of sets of particles or solid bodies, including rotating and vibrating bodies. Uses variational principles (energy-minimization) as well as differential equations.

Mechanics of deformable solids considers questions of elasticity and plasticity, wave propagation, engineering, and topics in specific solids such as soils and crystals.

Fluid mechanics studies air, water, and other fluids in motion: compression, turbulence, diffusion, wave propagation, and so on. Mathematically this includes study of solutions of differential equations, including large-scale numerical methods (e.g the finite-element method).

Optics, electromagnetic theory is the study of the propagation and evolution of electromagnetic waves, including topics of interference and diffraction. Besides the usual branches of analysis, this area includes geometric topics such as the paths of light rays.

Classical thermodynamics, heat transfer is the study of the flow of heat through matter, including phase change and combustion. Historically, the source of Fourier series.

Quantum Theory studies the solutions of the Schrödinger (differential) equation. Also includes a good deal of Lie group theory and quantum group theory, theory of distributions and topics from Functional analysis, Yang-Mills problems, Feynman diagrams, and so on.

Statistical mechanics, structure of matter is the study of large-scale systems of particles, including stochastic systems and moving or

evolving systems. Specific types of matter studied include fluids, crystals, metals, and other solids.

Relativity and gravitational theory is differential geometry, analysis, and group theory applied to physics on a grand scale or in extreme situations (e.g. black holes and cosmology).

Astronomy and astrophysics : as celestial mechanics is, mathematically, part of Mechanics of Particles (!), the principal applications in this area appear to be concerning the structure, evolution, and interaction of stars and galaxies.

Geophysics applications typically involve material in Mechanics and Fluid mechanics, as above, but for large-scale problems (this subject deals with a very big solid and a large pool of fluid!)

Systems theory; control study the evolution over time of complex systems such as those in engineering. In particular, one may try to identify the system -- to determine the equations or parameters which govern its development -- or to control the system -- to select the parameters (e.g. via feedback loops) to achieve a desired state. Of particular interest are issues in stability (steady-state configurations) and the effects of random changes and noise (stochastic systems). While popularly the domain of "cybernetics" or "robotics", perhaps, this is in practice a field of application of differential (or difference) equations, functional analysis, numerical analysis, and global analysis (or differential geometry).

Biology and other natural sciences whose connections merit explicit connection in the MSC scheme include Chemistry, Biology, Genetics, and Medicine, In chemistry and biochemistry, it is clear that graph theory, differential geometry, and differential equations play a role. Medical technology uses techniques of information transfer and visualization. Biology (including taxonomy and archaeobiology) use statistical inference and other tools.

Game theory, economics, social and behavioral sciences including Psychology, Sociology, and other social sciences as a group. The more behavioural sciences (including Linguistics!) use a medley of statistical techniques, including experimental design and other rather combinatorial topics. Economics and finance also make use of statistical tools, especially time-series analysis; some topics, such as voting theory, are more combinatorial. This category also includes game theory, which is actually not about games at all but rather about optimization; which combination of strategies leads to an optimal outcome.

Observe that the branches of mathematics most closely allied with the fields of mathematical physics are the parts of analysis, particularly those parts related to differential equations. The other sciences draw on these as well as probability and statistics and, increasingly, numerical methods.