GVIC

BHARATH EDUCATIONAL SOCIETYS

GROUP OF INSTITUTIONS:

MADANAPALLI

GOLDEN

VALLEY COLLEGE OF

ENGINEERING [GVIC]

Branch: ECE-A

ROLL NO-109M1A0416 ROLL NO-109M1A0435

Laplace Transform & Fourier transforms & Its Applications in Engineering ABSTRACT:

Laplace transform is useful in all branches of Engineering, in probability theory ,in

electrical and mechanical systems. These transform we are using often.

Inverse Laplace transform is also useful in many domains

In periodic function problems and also to deal with differential equations, they

are all very useful.

Fourier transforms are useful for electronics and communications students

and to solve many problems in their subjects. Particularly Z-transforms have

properties similar to that of Laplace .The main difference is Laplace transforms is

defined for the functions of continuous variables and Z-transform defined for

sequences.

The partial differential equation in a boundary value problem gets transform to

an ordinary differential equation by the application of Fourier transform.

Introduction To Laplace Transform:-

Laplace transform is useful since

1. Particular solution is obtained without first determining the general solution.

2. Non homogenous equations are solved without obtaining the complementary integral.

3. Solution of mechanical or electrical problems involving discontinuous force function (RHS function

F(x)) (or) periodic functions other than cosine and sine are obtained easily.

4. System of DE, PDE and integral functions

Before the advent of calculators and computers logarithms were extensively used to replace

multiplication (or division) of two large numbers. The crucial idea which made the Laplace transform

(L.T.). A very powerful technique is that it replaces operations of calculus by operations of algebra. For

example, with application of Laplace transform to an initial value problem, consisting of an ordinary (or

partial) differential equation (O.D.E) together with initial conditions (I.C.) is reduced to problem of

solving an algebraic equation (with any given initial conditions automatically taken care) as shown in

fig. below.

The z-transform plays an important role in the field of communication engineering and control

engineering at the stage of

analy

sis

and

repre

senta

tion of discrete time linear

shift invariance system.

Original space

t-shape

O.D.E

+ (particular) solution of D.E

IC

:

:

The z-transform plays an important role in the field of communication engineering and control

engineering at the stage of analysis and representation of discrete time linear shift invariance system.

In mathematics ,the Laplace transform is widely used integral transform denoted L{f(t)}, it

is a linear operator of a function f(t) with a real argument t(t0) that transforms to a function F(s) with

a complex argument S. this transformation is essentially bijective for the majority of practical uses ; the

respective pairs of f(t) and f(s) are matched in the tables. The Laplace transform has the useful property

that many relationships and operations over the originals of f(t) correspond to similar relationships and

operations over the images f(s)[1] The Laplace transform has many important applications in his work on

probability theory.

The Laplace transform is related to the Fourier transform, but whereas the Fourier

transform resolves a function or signal into its modes of vibrations, the Laplace resolves a function into

its moments like the Fourier transform, the Laplace transform is used for solving differential and integral

equations. In Physics and engineering, it is used for analysis of linear time-in variant systems such as

electrical circuits, harmonic oscillations, optical devices, and mechanical systems.

Given a simple mathematical or functional description of an input or output of a system, the Laplace

transform provides an alternative functional description that often simplifies the process of analyzing

the behavior of the system, or in synthesizing a new system based on a set of qualification

INTRODUCTION OF FOURIER TRANSFORMS :-

It may seem unusual that we begin a course on Geodynamics by reviewing Fourier transforms and

Fourier series. However you will that Fourier analysis is used in almost every aspect of the subject

ranging from solving linear differential equation to depending computer modules, to the processing and

analysis of data .we wont be using Fourier analysis for first few lectures, but Ill introduce the concepts

today so that people who are use familiar with the topic can have time for review. In the first few

lectures, Ill also discuss plate tectonics. I imagine that students with physics and math backgrounds may

Laplace

Transform

Image space

s-space

Algebraic solution of AE

(or) subsidiary (by pure algebraic manipulations)

Equation (AE)

have to spend some time reviewing plate tectonics. Hopefully everyone will busy the first two weeks

next class Ill give homework assignment involving Fourier transforms. Later well have a short quiz on

plate tectonics

FOURIER TRANSFORMS:-

Definitions of Fourier transforms:-

The 1-dimensional Fourier transform is defined as:

f(k) = ()2

Forward Transform

f(k) = ()2

Inverse Transform

Where x is distance &

K is wave number

Where k=1

and is wavelength.

These equations are commonly written in terms of time t and frequency v where = 1/ and T is the

period.

The 2-dimensional Fourier transform is defined as:-

F(k) = ()2( .)2

F(x) = ()2( .)2

Where X=(x,y) is the position vector

K = (Kx,Ky) is the wave number vector

& (k.x) = kxx+kyy

Why use Fourier transforms on a nearly special earth?

It you have taken Geomagnetism or global seismology, we were taught to expand a function of

latitude and longitude in spherical harmonics. Later in the course we will also use spherical harmonics to

represent large scale variations in the gravity field and to represent viscous mantle flow. However,

throughout the course we will dealing with problems related to the crust and lithosphere. In these cases

a flat earth approximation is both adequate and practical for the following reasons:

Cartesian geometry is good approximation consider a small patch of crust or litho sphere on the surface

of a sphere. If the area of patch is A is much less than the area of the earth and thickness L of the patch

is much less than the radius of the earth Re, then the Cartesian geometry will be adequate

A

Fourier sine and cosine transforms:- Any function f(x) can be decomposed into add O(x) and even E(x) components

f(x)=E(x)+O(x)

E(x =1

2[f(x)+f(-x)]

O(x) =1

2[f(x)-f(-x)]

f(k)= ()2

=

f(k) = cos 2

s in 2

odd part cancels even part cancels

Fourier Transform :- Fourier series can be generalized to complex numbers, and further transform

Forward Fourier transform

f(k) = ()2

*inverse Fourier transform

f(x) = ()2

Note = cos(x)+isin(x)

Fourier Transform:- Fourier transform maps a time series.( e.g. : Audio samples) into the series of frequencies(their

amplitudes and phases) that composed the time series.

Inverse Fourier transform maps the series of frequencies (their amplitudes and phases) back

into the corresponding time series.

The two functions and inverses of each other.

Discrete Fourier transform If we wish to find the frequency spectrum of a function that we have sampled, the

continuous Fourier transform is not so useful.

We need a discreet version.

Discreet Fourier transforms.

Forward DFT:

=

The complex numbers f0.fn are transformed into complex numbers f0.fn

Inverse DFT :

=

The complex numbers f0..fn are transformed into complex numbers f0fn.

DFT Example Interpreting a DFT can be slightly difficult, because the DFT for really data includes complex

numbers.

Basically :The magnitude of the complex number for a DFT component is the power at that

frequency.

The phase of the wave form can be determined from the relatives values of the real

and imaginary coefficients

Also both POSITIVE and NEGATIVE frequencies show up.

Fourier Transform :- The continuous Fourier transform is evaluating the bilateral Laplace transform with

complex argument.

S = iw (or) S = 2fi:

f(w) =F{f(t)}

= =

= f(t)dt

This expression excludes the scaling factor , which is often included in definitions of the Fourier

transform. This relationship between the Laplace and Fourier transform is often used to determine the

frequency spectrum of a single or dynamical system.

Applications:-

The solution of a IBVP consisting of a partial differential equation together with if the

boundary & initial conditions can be solved by the Fourier transform method. If the boundary

conditions are of the Dirichlet type where the function value is prescribed on the boundary,

then the Fourier sine transform is used .If the boundary conditions are of the Neumann type

where the derivative of function is prescribed on the boundary, then Fourier cosine transform is

applied . In either case, the P.D.E reduces to an O.D.E in Fourier transform which is solved then

the inverse Fourier sine (or cosine) transforms will give the solution to the problem.

Fourier transforms are widely used to solve various boundary values problems

of engineering such as

1. Vibrations of a string,

2. Conductions of heat ,

3. Oscillations of electric beam ,

4. Transmission lines etc.,

conclusion:- The partial differential equation in a boundary value problem gets transformed to an ordinary differential equation by the application of Fourier sine and cosine transform are useful when

f(x) is defined ina finite interval for say x in o

transformation or Fourier transform is an integral transform which is used to solve the partial

differential equations. The advantages of applying Fourier transform or any integral transform is to

reduce the number of independent variables by one. Fourier transforms are widely used to solve various

boundary value problems of engineeringsuch as vibrations of a string, conductions of heat, oscillations

of an electric beam, transmission lines.

Laplace Transform:-

Definition:- Let f(t) be the given function defined for all t 0. Laplace transforms of f(t) denoted by L{f(t)}

or simply L{f} is defined as

L{f(t)} =

0 = ()

L is known as Laplace transform operator. The original given function f(t) known as determining function depends on t, while the new function to be determined f(s), called as generating function to be depends only on S (because the improper integral on the R.H.S of (1) is integrated with respect to T ). f(s) in equation (1) is known as the Laplace transform of f(t). Eq. (1) is known as direct transform or simply transform, in which f(t) is given and f(s) is to be determined. Thus, Laplace transform transforms one class of complicated function f(t) to produce another class of simpler functions F(s).

APPLICATIONS:- Laplace transforms is very useful in obtaining solutions of linear differential equations , both ordinary and partial , solutions of system of simultaneous differential equations , solution of integral equations , solutions of linear difference equations and in the evaluation of definite integrals. In the PHYSICS and ENGINEERINGLaplace transforms are widely used for analysis of linear time variantsystems such as

1. Electric circuits, 2. Harmonic oscillators, 3. Optical devices,& 4. Mechanical systems. 5. In the analysis, the Laplace transform is often interpreted as a transformation from the

TIME-DOMAIN , in which inputs and outputs are functions of time , to the frequency domain , where the same inputs and outputs are functions of complex angular frequency , in radians per unit time

6. Laplace transforms are used in the Area of digital signal processing and digital filters

APPLICATION OF THE LAPLACE TRANSFORM IN CIRCUIT ANALYSIS:- The Laplace transform is an attractive tool in circuit analysis . it transforms a set of linear constant - coefficient differential equations into a set of linear polynomial equations . It automatically

introduces into the polynomial equations the initial values of the current and voltage variables. In the circuit analysis, we can develop the s-domain circuit models for various elements and s-domain equations can be written directly. CIRCUIT ELEMENTS IN THE S-DOMAIN :-

We take the Laplace transform of the time domain equation . This gives an algebraic relation between s-domain current and voltage . The dimensions of a transformed voltage is volt-seconds and the dimensions of a transformed current is ampere-seconds .

ADVANTAGES:- 1. With the application of L.T. particular solution of differential equation(D.E.) is obtained directly

without the necessity of first determining general solution and then obtaining the particular solution (by substitution of initial condition ).

2. L.T. solves non-homogenous D.E. without the necessity of first solving the corresponding homogenous D.E.

3. L.T. is applicable not only to continuous functions but also to piecewise continuous functions, complicated periodic functions, step functions and impulse functions.

SUFFICIENT CONDITIONS FOR THE EXISTENCE OF LAPLACE

TRANSFORMS OF f(t):- The L.T. of f(t) exists i.e., the improper integral in the R.H.S. of (1) converges (has a finite valve ) when the following sufficient conditions are satisfied :- (a). f(t) is piecewise (or section ally ) continuous

i.e., f(t) is continuous in every sub interval and has finite limits at end points of each of these sub intervals and

(b). f(t) is of exponential order of i.e., there exists M, such that |f(t)| 2

3 = finite,

f(t) = t2 is of exponential order say 3.

ex:- since lim>

2

= not finite,

f(t) = 2 is not of exponential order.

NOTE:- Above conditions (a) and (b) are not necessary conditions.

Conclusion:- The partial differential equation in a boundary value problem gets transformed to an ordinary differential equation by the application. Laplace Transform can be used to solve linear differential equations with constant coefficients. The advantages by using Laplace transform is that the particular solution can be obtained for given initial condition with obtaining the general solution. The formulae to be used in applying Laplace transform to solve differential equations are L [ (t)] =s (s) f(0)

L [ (t)] = 2 (s) s f(0) - (0)

L [ (t)] = 3 (s) 2f (0) s (0) - (0)

While apply the above formulae, replace , replace f(t) by Y(x), (s) by (s).

2. Laplace Transforms and their Applications

To

Differential Equations

Authors:-

K.Ravi kiran, E.E.E, MITS

K.B.Venu gopal, E.E.E, MITS

Email id [email protected]

[email protected]

Contents:-

1. Definition of Laplace Transforms

2.Conditions for Existance

3. Applications of Differential Equations

4. Problem solving

Abstract:-

The main idea behind Laplace Transformation is ordinary differential equations with constant coefficients can be easily solved by the laplace transform method, with out

the necessity of first finding the general solution and then evaluating the arbitrary

constants. This method is, in general, shorter than our earlier method and is especially

suitable to obtain the solution of linear non-homogeneous ordinary differential equations

with constant coefficients.

INTRODUCTION TO THE LAPLACE TRANSFORM METHOD

The Laplace Transform method is a technique for solving linear differential

equations with initial conditions. It is commonly used to solve electrical circuit

and systems problems.

What is a Transform Method?

The simplest way to describe a transform method is to consider an example.

Suppose we wish to compute the product of VI and XIV, both Roman numerals,

and express the answer as a Roman numeral. Unless you are a Roman(!), the

first thing to do is transform the Roman numerals to Arabic numerals. VI is 6

and XIV is 14. The transformed problem is: compute the product of 6 and 14. We

can all do this! The solution to the transformed problem is 84. We then convert

the solution of the transformed problem to the solution to original problem. 84 in

Roman numerals is LXXXIV. This last step is called the inverse transformation.

The following diagram summarizes what we have done.

Why use a transform method? Some problems are difficult to solve directly.

With a transform method, the hope is that the transformed problem is easy

to solve. That is certainly the case for the simple example above. One must

also take into account the difficulty of transforming the original problem

and inverse transforming the solution to the transformed problem.

DEFINITION OF LAPLACE TRANSFORM

CONDITIONS FOR THE EXISTENCE OF LT

While finding the laplace transforms of elementary functions, it can be

noticed that the integral exists under certain conditions, such as s > 0 or s > a etc.

in general, the functions f(t) must satisfy the following conditions for the

existence of the laplace transform.

(i) The function f(t) must be piece- wise continuous or sectionally

continuous in any limited interval 0 < a t b.

(ii) The function f(t) is of exponential order.

APPLICATION TO DIFFERENTIAL EQUATIONS

Consider the linear differential equation with constant coefficients

under the initial conditions

The Laplace transform directly gives the solution without going through the general solution.

The steps to follow are:

(1)

Evaluate the Laplace transform of the two sides of the equation (C);

(2)

;

(3)

After algebraic manipulation, write down

;

(4)

Make use of the properties of the inverse Laplace transform , to find the solution y(t).

Example: Find the solution of the IVP

,

where

.

Solution: Let us follow these steps:

(1)

We have

;

(2)

Using properties of Laplace transform, we get

,

where . Since , we get

;

(3)

Inverse Laplace:

Using partial decomposition technique we get

,

which implies

Since

,

which gives

,

and

Hence,

PROBLEM SOLVING

Solve the following initial value problem by using laplace transform 4y"+2 y = 0, y (0) =2, y'(0) =0. Solution: Given equation is 4y"+2 y = 0 Taking laplace transform on both sides, we have 4L,y"-+2L{y}=0

i.e., 4[s2L{y}-sy(0)- y'(0)++2L[y]=0

=> 4[s2 L{y}-2s] + 2L{y}=0

i.e., L{y}[4s2+2] = 2s or L{y}= 2s

4s2+2

Y=(1/2) L-1 s =(1/2) cos t/2 S2+ 2/4

3. FOURIER TRANSFORMS AND ITS APPLICATIONS IN ENGINEERING

FIELDS

G.Sneha Geetha,

I B.Tech, EEE,

SDIT, Nandyal,

E-mail: [email protected]

S.Talat Misba,

I B.Tech, EEE,

SDIT, Nandyal,

E-mail: [email protected]

Abstract:

The Fourier transform is a mathematical operation that decomposes a signal into its

constituent frequencies. Thus the Fourier transform of a musical chord is a mathematical

representation of the amplitudes of the individual notes that make it up. The original signal

depends on time, and therefore is called the time domain representation of the signal, whereas

the Fourier transform depends on frequency and is called the frequency domain representation

of the signal. The term Fourier transform refers both to the frequency domain representation of

the signal and the process that transforms the signal to its frequency domain representation .In

this paper Fourier transform and its applications in a step by step procedure.

Introduction:

In mathematical terms, the Fourier transform transforms one complex-valued function of

a real variable into another. In effect, the Fourier transform decomposes a function into

oscillatory functions. The Fourier transform and its generalizations are the subject of Fourier

analysis. In this specific case, both the time and frequency domains are unbounded linear

continua. It is possible to define the Fourier transform of a function of several variables, which is

important for instance in the physical study of wave motion and optics. It is also possible to

generalize the Fourier transform on discrete structures such as finite groups. The efficient

computation of such structures, by fast Fourier transform, is essential for high-speed computing.

The motivation for the Fourier transform comes from the study of Fourier series. In the

study of Fourier series, complicated functions are written as the sum of simple waves

mathematically represented by sines and cosines. Due to the properties of sine and cosine it is

possible to recover the amount of each wave in the sum by an integral. In many cases it is

desirable to use Euler's formula, which states that e2i

= cos 2 + i sin 2, to write Fourier

series in terms of the basic waves e2i

. This has the advantage of simplifying many of the

formulas involved and providing a formulation for Fourier series that more closely resembles the

definition followed in this article. This passage from sines and cosines to complex exponentials

makes it necessary for the Fourier coefficients to be complex valued. The usual interpretation of

this complex number is that it gives both the amplitude (or size) of the wave present in the

function and the phase (or the initial angle) of the wave. This passage also introduces the need

for negative "frequencies". If were measured in seconds then the waves e2i and e2i would

both complete one cycle per second, but they represent different frequencies in the Fourier

transform. Hence, frequency no longer measures the number of cycles per unit time, but is

closely related.

There is a close connection between the definition of Fourier series and the Fourier transform for

functions which are zero outside of an interval. For such a function we can calculate its Fourier

series on any interval that includes the interval where is not identically zero. The Fourier

transform is also defined for such a function. As we increase the length of the interval on which

we calculate the Fourier series, then the Fourier series coefficients begin to look like the Fourier

transform and the sum of the Fourier series of begins to look like the inverse Fourier transform.

To explain this more precisely, suppose that T is large enough so that the interval [T/2,T/2]

contains the interval on which is not identically zero. Then the n-th series coefficient cn is

given by:

Comparing this to the definition of the Fourier transform it follows that since

(x) is zero outside [T/2,T/2]. Thus the Fourier coefficients are just the values of the Fourier

transform sampled on a grid of width 1/T. As T increases the Fourier coefficients more closely

represent the Fourier transform of the function.

Under appropriate conditions the sum of the Fourier series of will equal the function . In other

words can be written:

where the last sum is simply the first sum rewritten using the definitions n = n/T, and

= (n + 1)/T n/T = 1/T.

This second sum is a Riemann sum, and so by letting T it will converge to the integral for

the inverse Fourier transform given in the definition section. Under suitable conditions this

argument may be made precise (Stein & Shakarchi 2003).

In the study of Fourier series the numbers cn could be thought of as the "amount" of the wave in

the Fourier series of . Similarly, as seen above, the Fourier transform can be thought of as a

function that measures how much of each individual frequency is present in our function , and

we can recombine these waves by using an integral (or "continuous sum") to reproduce the

original function.

The following images provide a visual illustration of how the Fourier transform measures

whether a frequency is present in a particular function. The function depicted

oscillates at 3 hertz (if t measures seconds) and tends quickly to 0.

This function was specially chosen to have a real Fourier transform which can easily be plotted.

The first image contains its graph. In order to calculate we must integrate e2i(3t)(t). The

second image shows the plot of the real and imaginary parts of this function. The real part of the

integrand is almost always positive, this is because when (t) is negative, then the real part of

e2i(3t)

is negative as well. Because they oscillate at the same rate, when (t) is positive, so is the

real part of e2i(3t)

. The result is that when you integrate the real part of the integrand you get a

relatively large number (in this case 0.5). On the other hand, when you try to measure a

frequency that is not present, as in the case when we look at , the integrand oscillates

enough so that the integral is very small. The general situation may be a bit more complicated

than this, but this in spirit is how the Fourier transform measures how much of an individual

frequency is present in a function (t).

Applications in Engineering Fields:

Analysis of differential equations

Fourier transforms and the closely related Laplace transforms are widely used in solving

differential equations. The Fourier transform is compatible with differentiation in the following

sense: if f(x) is a differentiable function with Fourier transform , then the Fourier transform

of its derivative is given by . This can be used to transform differential equations into

algebraic equations. Note that this technique only applies to problems whose domain is the

whole set of real numbers. By extending the Fourier transform to functions of several variables

partial differential equations with domain Rn can also be translated into algebraic equations.

Fourier transform spectroscopy

The Fourier transform is also used in nuclear magnetic resonance (NMR) and in other

kinds of spectroscopy, e.g. infrared (FTIR). In NMR an exponentially-shaped free induction

decay (FID) signal is acquired in the time domain and Fourier-transformed to a Lorentzian line-

shape in the frequency domain. The Fourier transform is also used in magnetic resonance

imaging (MRI) and mass spectrometry.

One of the most basic tasks in spectroscopy is to characterize the spectrum of a light

source: How much light is emitted at each different wavelength. The most straightforward way

to measure a spectrum is to pass the light through a monochromator, an instrument that blocks all

of the light except the light at a certain wavelength (the un-blocked wavelength is set by a knob

on the monochromator). Then the intensity of this remaining (single-wavelength) light is

measured. The measured intensity directly indicates how much light is emitted at that

wavelength. By varying the monochromator's wavelength setting, the full spectrum can be

measured. This simple scheme in fact describes how some spectrometers work.

Fourier transform spectroscopy is a less intuitive way to get the same information. Rather

than allowing only one wavelength at a time to pass through to the detector, this technique lets

through a beam containing many different wavelengths of light at once, and measures the total

beam intensity. Next, the beam is modified to contain a different combination of wavelengths,

giving a second data point. This process is repeated many times. Afterwards, a computer takes all

this data and works backwards to infer how much light there is at each wavelength.

To be more specific, between the light source and the detector, there is a certain

configuration of mirrors that allows some wavelengths to pass through but blocks others (due to

wave interference). The beam is modified for each new data point by moving one of the mirrors;

this changes the set of wavelengths that can pass through.

Conclusion:

This paper mainly deals with the introduction of Fourier Transforms and its applications

to Engineering Fields.

APPLICATIONS OF LAPLACE TRANSFORMS

1. V.SRAVANTHI 1stECE, AITS, [email protected]

2.D.VIJITHA 1stECE, AITS,Rajampet [email protected]

ABSTRACT

In mathematics the Laplace transformation is a widely used integral transform. Laplace

Transform has many important applications throughout the sciences. It is named for pierre-simon

Laplace who introduced the transform in his work on theory. The Laplace transform has the

useful property that many relationships and operations over the originals correspond to simpler

relationships and operations over the images. In this paper the initial value problem has been

arrived and solved by showing pictorial representations in a procedural way.

INTRODUCTION

The knowledge of Laplace transform is an essential part of mathematics required by engineers

and scientists. The Laplace Transforms is an excellent tool for solving linear differential

equations with given initial values of an unknown function and its derivatives without the

necessity of first finding the general solution (complementary function +particular integral) and

then evaluating from it the particular solution satisfying the given conditions. The technique is

useful to solve some partial differential equations as well. This is a powerful tool in diverse

fields of engineering.

What are Laplace Transforms?

A Laplace Transform is a type of integral transform.

Let f(t) be a function defined for all positive values of t. Then Laplace Transform of f(t),

denoted by L{f(t)} is defined by

0f(t)dt = F(s)

Plug one function in and get another function out.

The new function is in a different domain when

0f(t)dt = F(s)

F(s) is the Laplace Transform of f(t)

Write L {f (t)} =F(s),

L{y(t)}=Y(s),

L{x(t)}=X(s)etc

L{f(t)}=

0f(t)dt it is a Laplace Transform and it can be written as

f(t)=L-1

0f(t)dt So that the function f(t) is said to be Inverse Laplace Transform.

STANDARD FORMULAE:

1. L{1}=1/s

2. L{}=1/s-a

3. L{ }=1/s+a

4. L{coshat}=s/s2-a2

5. L{sinhat}=a/s2-a2

6. L{cosat}=s/ s2+a2

7. L{sinat}=a/ s2+a2

8. L{t}=1/s2

To what end does one use Laplace Transforms?

We can use Laplace Transforms to turn an initial value problem.

solve for Y(t) into an algebraic equation

solve for Y(s)

Laplace Transforms are particularly effective on differential equation with forcing functions that

are piece-wise like the head wise function, and other functions that turn on and of

" 3 ' 4 ( 1)

(0) 1, '(0) 2

y y y t u t

y y

2

2 1( )*( 3 4) ( 1) ss

s eY s s s s

Then

If you solve the algebraic equation

and find the inverse Laplace Transform of the solution Y(s), you have the solution to the I.V.P

The Inverse Laplace Transform of

is

2

2 2

( 1) ( 1)( )

( 3 4)

s ss s e eY s

s s s

2

2 2

( 1) ( 1)( )

( 3 4)

s ss s e eY s

s s s

4 43 32 15 80 4 16

4325 5

( ) ( 1)( + ( ) )

( )( ( ) )

t tee

t t

y t u t e e t

u t e e

Thus

is the solution of Initial value problem

REFERENCES

1. Engineering Mathematics volume 1by Dr.T.K.Iyengar, Dr.B.KrishnaGandhi, S.Ranganatham,

M.V.S.S.N.Prasad.

2. Text Bookof Engineering Mathematics B.V.Ramana.

3. TextBookof Engineering Mathematics Thomson Book collection.

4 43 32 15 80 4 16

4325 5

( ) ( 1)( + ( ) )

( )( ( ) )

t tee

t t

y t u t e e t

u t e e

5. APPLICATIONS OF LAPLACE TRANSFORMS

Y. SILPA, I B.TECH (EEE)

Sri Sai Institute of Science & Technology, Rayachoti.

ABSTRACT Laplace transform or Laplace transformation is a method for solving linear differential

equation arising in physics and engineering. It reduces the problem of solving a differential

equation to an algebraic problem.

Let K(s,t) be a function of two variables s, and t, where s is a parameter (may be real or

complex) independent of t. The function F(s) defined by the integral (assumed to be

convergent).

sFdttftsK

),

is called the Integral transform of the function f(t) and denoted by T{f(t)}. The function

K(s, t) is called the kernal of the transformation.

If the kernal K(s, t) is defined as

0

00,

tfore

tfortsK

st

then sFdttfe st

0

(1)

The function F(s) defined by the integral (1) is called the Laplace transform of the

function f(t) and is also denoted by

L{f(t)} or F(s)

Thus Laplace transform is a function of a new variable s given by (1)

A semi-infinite insulated bar (x>0) which is initial at constant temperature (To > 0) and in

which the end is held at a temperature of 0oC is considered for presentation of the solution. The

solution for the problem deals with the determination of the temperature at any point of the semi-

infinite insulated bar.

INTRODUCTION

They are many partial differential equations problems in engineering cases, which their

quantities vary with the time. To solve these problems Laplace transforms, as a powerful

technique can be used to transform the original differential equation into integral algebraic

expression. For mechanical engineering problem, Laplace transform is addressed to analyze the

linear time invariant such as harmonic oscillators in vibration analysis and various problems in

mechanical systems. Basically Laplace transforms changes integral and differential equations

into polynomial equations.

FORMATION OF THE PROBLEM

A semi-infinite insulated bar (x > 0) which is initially at constant temperature (To> 0) and

in which the end is hold at a temperature at 0oC is considered.

We are to solve the diffusion equation

2

22

x

wc

t

w

(2)

Subject to the initial and boundary conditions

w(x, 0) = To

w(0, t) = 0 (3)

w(x, t) 0 as x

Applying the Laplace transform on both sides of equation 92)

2

22

x

wcL

t

wL

t

wL

x

wcL

2

22

t

wL

x

wLc

2

22

oTsxwssxwdx

dc ,,

2

22

Equation (4) in ordinary differential equation

O x

We have

x

c

sx

c

s

esBesAsxwFC

,:.

s

TsxwPI o,:

Solution is s

TesBesAsxw o

xc

sx

c

s

, (5)

Evidently B(s) = 0 from the third condition

s

TesAsxw o

xc

s

,

Since w(0, t) = 0, we have w(0, s) = 0

0s

TsA o

and so s

c

x

oo es

T

s

Tsxw

, (6)

Inverse Laplace Transforms:

111

sL

t

aerf

t

aerfc

s

eL

sa

21

2

1

where erf is the error function.

Applying the Inverse Laplace Transforms on both sides of equation (6)

s

c

x

oo es

T

s

TLsxwL 11 ,

s

eTL

s

TLtxw

sc

x

oo 11,

s

eLT

sLTtxw

sc

x

oo

11 1,

t

cxerfTTtxw oo

2

/11,

tc

xerfTTTtxw ooo

2,

tc

xerfTtxw o

2,

CONCLUSION:

The temperature of any the semi-infinite insulated at different points by calculated with the help

of Laplace transformations.

6. LAPLACE TRANSFORMS

NAME :D.SUSHMITHA

BRANCH:EEE

DEPARTMENT OF HUMANITIES

Email: [email protected]

Phone no: 9490059399

2) NAME :P. RAJYALAKSHMI

BRANCH: EEE

DEPARTMENT OF HUMANITIES

Email: [email protected]

ABSTRACT

In this paper we here analyzed different electrical circuits here we have used laplace transform

technique to save these electrical circuits. We have analyzed different parameters namely

capacitor,resistor,battery,transistors, in this paper the solved problems are represented graphically. In

this the applications of laplace transforms also discussed. The terms explained in this are inverse laplace

transformations, applications of laplace transforms, table of laplace transformations advantage of

laplace transformations,sufficient conditions for yjr exostamce pf ;a[;ace tramsfpr, pf f(t). general

properties of laplace transform. Linear properties, applications of simulataneous differential equations,

methods of solution to system of differential equations. In this the graphs involved the increasing curve

and alternativecurves. In this mainly discussed about of electrical circuits, with examples and

explanations. In this each expression is explained with an example circuit problems and with diagramic

representation, and explained graphically. These are the terms explained in this paper.

KEYWORDS: Capacitors, switch, registors, transistors, inductors etc.

Pierre Simon Laplace, after whom the Laplace Transform is named, lived from 1749 to 1827.

The Laplace Transform is a powerful tool that is very useful in Electrical Engineering. The

transform allows equations in the "time domain" to be transformed into an equivalent equation in

the Complex S Domain. The laplace transform is an integral transform, although the reader does

not need to have a knowledge of integral calculus because all results will be provided. This page

will discuss the Laplace transform as being simply a tool for solving and manipulating ordinary

differential equations.

Laplace transformations of circuit elements are similar to phasor representations, but they are not

the same. Laplace transformations are more general than phasors, and can be easier to use in

some instances. Also, do not confuse the term "Complex S Domain" with the complex power

ideas that we have been talking about earlier. Complex power uses the variable , while the

Laplace transform uses the variable s. The Laplace variable s has nothing to do with power.

The transform is named after the mathematician Pierre Simon Laplace, who lived in the 18th

century. The transform itself did not become popular until Oliver Heaviside, a famous electrical

engineer, began using a variation of it to solve electrical circuits.

The Transform

The mathematical definition of the Laplace transform is as follows:

[The Laplace Transform]

Note:

The letter s has no special significance, and is used with the Laplace Transform as a matter of common

convention.

The transform, by virtue of the definite integral, removes all t from the resulting equation,

leaving instead the new variable s, a complex number that is normally written as s = + j. In essence, this transform takes the function f(t), and "transforms it" into a function in terms of s,

F(s). As a general rule the transform of a function f(t) is written as F(s). Time-domain functions

are written in lower-case, and the resultant s-domain functions are written in upper-case.

There is a table of Laplace Transform pairs in

we will use the following notation to show the transform of a function:

We use this notation, because we can convert F(s) back into f(t) using the inverse Laplace

transform.

The Inverse Transform

The inverse laplace transform converts a function in the complex S-domain to its counterpart in

the time-domain. Its mathematical definition is as follows:

[Inverse Laplace Transform]

where c is a real constant such that all of the poles s1,s2,...,sn of F(s) fall in the region

. In other words, c is chosen so that all of the poles of F(s) are to the left of the

vertical line intersecting the real axis at s = c

Before the advent of calculators and computers, logarithms were extensively used to

replace multiplication (or) division of two large numbers by addition (or substraction) of

two numbers. The cucial idea which made the laplace transformation a very powerful

technique is that it replaces operations of calculus by operations of algebra for example

with the applications of laplace transform to an initial value problem, consisting of an

ordinary (or partial) differential equation. Together with initial conditions is reduced to a

problem of solving an algebraic equation (with any given initial conditions automatically

taken care).

APPLICATIONS OF A LAPLACETRANSFORMATIONS:

Laplace transform is very seful in obtaining solution of linear differential equations

both ordinary and partial, solution of system of simultaneous differential equations. And in

the evaluation of definite integrals.

ADVANTAGES:

1 . with the application of laplae transform, particular solution of differential equation is

obtained directly without the necessity of first determining general solution and then

obtaining the particular solution (by substitution of initial conditions).

2 . laplace transformations solves non-homogeneous differential equation without the

necessity of first solving the corresponding homogeneous differential equations.

3 . laplace transforms is applicable not only to continuous functions but also to piecewise

continuous functions, complicated periodic functions, step functions and impulse function.

4 . laplace transforms of various functions are readily available (in tabulated form). In

section used functions are tabulated.

SUFFICIENT CONDITIONS FOR THE EXISTENCE OF LAPLACE TRANSFORM OF

f(t):-

The laplacetransform of f(t) exists that is the improper integral in the R.H.S. of (1)

convages (has a finite value) when the following sufficient conditions are satisfied.

1 . f(t) is piecewise (or sectionally) continuous that is f(t) is continuous in every subinterval

and has finite limits at end points of each of these sub interval.

GENERAL PROPERTIES OF LAPLACE TRANSFORM:

Although theoretically F(s), the laplace transforms of f(t) is obtained from the

definition in practice most of the time laplace transforms are obtained by the judical

application of some of the following important properties. In a nutshell, they are:

1 . Linearity property states that laplace transform of a linear combination(sum) of laplace

transforms.

2 . in change of scale, where the argument t of f is multiplied by a constant a,s is replaced

by s/a in F(s) and then multiplied by 1/a.

3. First shift theorem proves that multiplication of f(t) by eat

amounts to replacement of s by

s-a in F(s).

4 . Laplace transfomation of a first derivative amounts to multiplication of F(s) by s

(approximately but for the constant f(o).

5 . Laplace transform of an integral of f amount to division of F(s) bt s.

6 . Multiplication of f(t) by t power n amounts to differentiations of F(s) n times with respect to s

(with(-1)power n as sign).

APPLICATIONS OF LAPLAE TRANSFORMS TO SYSTEM OF SIMULTANEOUS

DIFFERENTIAL EQUATIONS:

Lplace transform can also be used to solve a system (or) family of m simultaneous

ordinary differential equations in m dependent variables which ar functions of independent

variable t.

EXAMPLE 1

In the circuit shown below, the capacitor is uncharged at time t = 0. If the switch is then closed, find the currents i1 and i2, and the charge on

C at time t greater than zero.

Answer

It is easier in this example to do the second method. In many examples, it is easier to do the first method.

For the first loop, we have:

For the second loop, we have:

Substituting (2) into (1) gives:

Next we take the Laplace Transform of both sides.

Note:

In this example, . So

Now taking Inverse Laplace:

And using result (2) from above, we have:

For charge on the capacitor, we first need voltage across the capacitor:

So, since , we have:

Graph of q(t):

EXAMPLE 2

In the circuit shown, the capacitor has an initial charge of 1 mC and the switch is in position 1 long enough to establish the steady state.

The switch is moved from position 1 to 2 at t = 0. Obtain the transient

current i(t) for t > 0.

Answer

Quiescent implies i1, i2 and their derivatives are zero for t = 0, ie

i1(0) = i2(0) = i1'(0) = i2'(0) = 0.

For loop 1:

For loop 2:

Substituting our result from (1) gives:

Taking Laplace transform:

Let

So

So

Taking Inverse Laplace:

So

Alternative answer using Scientific Notebook. (.tex file)

EXAMPLE 6

Consider a series RLC circuit where R = 20 W, L = 0.05 H and C = 10-

4 F and is driven by an alternating emf given by E = 100 cos 200t.

Given that both the circuit current i and the capacitor charge q are

zero at time t = 0, find an expression for i(t) in the region t > 0.

Answer

EXAMPLE 6

Consider a series RLC circuit where R = 20 W, L = 0.05 H and C = 10-

4 F and is driven by an alternating emf given by E = 100 cos 200t.

Given that both the circuit current i and the capacitor charge q are

zero at time t = 0, find an expression for i(t) in the region t > 0.

Answer

We use the following:

and obtain:

After multiplying throughout by 20, we have:

Taking Laplace transform and using the fact that i(0) = 0:

Using Scientific Notebook to find the partial fractions:

So

So

+ cos200t 2 sin 200t

NOTE: Scientific Notebook can do all this for us very easily. In one step, we have:

+ cos200t 2 sin 200t

Transient part:

Steady state part:

RESULTS AND DISCUSSION:

The solutions of problems are finally represented graphically. From example 1 problem the graph shows the graph of the q(t) . q(t) is a function of t. the graph is taken between q and t. there are directly independt on each other. There fpre ot the va;ie pf q os omcreased tjem the t os a;sp omcreased. This is about example 1. From exapmple2 we observed that the gra[h os taklenm between i and t if I increases t also increases.

CONCLUSION:

Finally we got the solution for the ordinary differential equations . and in the example 2 the value of current is also got by graph the value of current is also finding in the example by using laplace transforms solutions are easily got this is the conclusion of laplace transformations.

REFERENCES:

E.J. Berg, Hevisides operational calculus, end ed.

R.V. Chur chill,modern operational mathematics in engineering.

D.V. Widder, advances calculus, end ed.

Engineering mathematics B.V. Ramana.

7. LAPLACE TRANSFORMS

VAAGDEVI INSTITUTE OF TECHNOLOGY

AND SCIENCE

PRODDATUR

KADAPA(DIST)

PRESENTED BY:

P.KAVYA, N.VARA LAKSHMI,

I B-Tech, I B-Tech,

E-mail ID:[email protected] , E-Mail ID:[email protected]

Abstract:

In mathematics, the Laplace transform is a widely used integral transform. Denoted

, it is a linear operator of a function f(t) with a real argument t (t 0) that transforms it to a

function F(s) with a complex argument s. This transformation is essentially bijective for the

majority of practical uses; the respective pairs of f(t) and F(s) are matched in tables. The Laplace

transform has the useful property that many relationships and operations over the originals f(t)

correspond to simpler relationships and operations over the images F(s).[1]

The Laplace

transform has many important applications throughout the sciences. It is named for Pierre-Simon

Laplace who introduced the transform in his work on probability theory.

The Laplace transform is related to the Fourier transform, but whereas the Fourier transform

resolves a function or signal into its modes of vibration, the Laplace transform resolves a

function into its moments. Like the Fourier transform, the Laplace transform is used for solving

differential and integral equations. In physics and engineering, it is used for analysis oflinear

time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and

mechanical systems. In this analysis, the Laplace transform is often interpreted as a

transformation from the time-domain, in which inputs and outputs are functions of time, to

the frequency-domain, where the same inputs and outputs are functions of complex angular

frequency, in radians

Introduction:

The knowledge of Laplace transforms is essential part of mathematics required by the

engineers and scientists. The Laplace transform is an excellent tool for solving linear differential

equations with given initial values of an unknown function and its Derivatives without the

necessity of first finding the general solution (complementary function + particular integral) and

then evaluating from its the particular solution satisfying the given conditions. This technique is

useful to solve from fractional differential equations as well. This is a powerful tool in diverse

fields of engineering.

The Laplace Transformation

Pierre-Simon Laplace (1749-1827)

Laplace was a French mathematician, astronomer, and physicist who applied the

Newtonian theory of gravitation to the solar system (an important problem of his day).

He played a leading role in the development of the metric system.

The Laplace Transform is widely used in engineering applications (mechanical and

electronic), especially where the driving force is discontinuous. It is also used in process

control.

Definition of Laplace Transform of f(t)

The Laplace transform of a function f(t) for t> 0 is defined by the following integral defined over

0 to :

{ f(t)} =

The resulting expression is a function of s, which we write as F(s). In words we say

"The Laplace Transform of f(t) equals function F of s"

and write:

{f(t)} = F(s)

Similarly, the Laplace transform of a function g(t) would be written:

{g(t)} = G(s)

.

PROPERTIES:

PROPERTY 1:

If a is a constant and f(t) is a function of t, then

{a f(t)} = a {f(t)}

{7 sin t} = 7 {sin t}

PROPERTY 2: Linearity property

If a and b are constants while f(t) and g(t) are functions of t, then

{a f(t) + b g(t)} = a {f(t)} + b {g(t)}

PROPERTY 3: Change of scale property

If {f(t)} = F(s) then

PROPERTY:4:Shifting property (Shift theorem)

{eat

f(t)} = F(s a)

{e3tf(t)} = F(s 3)

PROPERTY 5:

Property 6:

The Laplace transforms of the real (or imaginary) part of a complex function is equal to the real

(or imaginary) part of the transform of the complex function.

Let Re denote the real part of a complex function C(t) and Im denote the imaginary part of C(t),

then

{Re[C(t)]} = Re {C(t)} and {Im[C(t)]} = Im {C(t)}

Properties of the unilateral Laplace transform

Time domain 's' domain Comment

Linearity

Can be proved using

basic rules of

integration.

Frequency

differentia

tion

is the first derivative

of .

Frequency

differentia

tion

More general form,

nthderivative of F(s).

Differenti

ation

is assumed to be a

differentiable function,

and its derivative is

assumed to be of

exponential type. This

can then be obtained

by integration by parts

Second

Differenti

ation

is assumed twice

differentiable and the

second derivative to be

of exponential type.

Follows by applying the

Differentiation property

to .

General

Differenti

ation

is assumed to be n-

times differentiable,

with nth derivative of

exponential type.

Follow by mathematical

induction.

Frequency

integratio

n

Integratio

n

u(t) is the Heaviside

step function. Note (u *

f)(t) is the convolution

of u(t) and f(t).

Scaling

wherea is positive.

Frequency

shifting

Time

shifting u(t) is the Heaviside

step function

Multiplica

tion

the integration is done

along the vertical line

Re() = c that lies

entirely within the

region of convergence

of F.[12]

Convoluti

on

(t) and g(t) are

extended by zero for

t < 0 in the definition of

the convolution.

z

f(t) is a periodic

function of periodT so

that

. This is the result of the

time shifting property

and the geometric

series

APPLICATIONS :

There are two (related) approaches:

1. Derive the circuit (differential) equations in the time domain, then transform these ODEs

to the s-domain;

2. Transform the circuit to the s-domain, then derive the circuit equations in the s-domain

(using the concept of "impedance").

We will use the first approach. We will derive the system equations(s) in the t-plane, then

transform the equations to the s-plane. We will usually then transform back to the t-plane.

EXAMPLE 1:

Consider the circuit when the switch is closed at t = 0 with VC(0) = 1.0 V. Solve for the current

i(t) in the circuit.

EXAMPLE 2:Solve for i(t) for the circuit, given that V(t) = 10 sin5t V, R =

4W and L=2H

Conclusion

In this article, the basic development of the NumericalLaplace Transform has been

presented. This techniquehasproven to be efficient for the analysis of electromagnetic transients

in power systems. The main advantages of the NLT aresummarized below:

The modeling of components with distributed and frequencydependentparameters can be

done in a straightforwardmanner.

Since its basic principles are different from those oftime domain methods, the NLT is

very useful to verifyingtime domain methods, as well as in the developmentof new

time domain models and techniques.

The application of the NLT can be very important whena high accuracy of results is

mandatory. The examplesgiven show that time domain methods may require amuch smaller

discretization step.

8.

MATHEMATICAL MODELLING TO ESTIMATE THE TIME OF

DEATH OF A MURDERED PERSON

Harshitha T and Madhavi P

I B.Tech, Electronics and Communication in Engineering

Madanapalle Institute of Technology and Science, Madanapalle, India.

Email:[email protected]

[email protected]

ABSTRACT

This paper is concerned with an exposition of the methods of solving some classes of ordinary

differential equations. An analysis is presented for solving ordinary differential equations of first order

and first degree. A mathematical model is presented to investigate the time of death of a murdered

person using the Newtons law of cooling. The time of death of a murdered person can be determined

with the help of modeling through differential equation. To formulate this process mathematically,

solved analytically and results are shown in graphical representation. It is noticed that the object

cools, the temperature difference gets smaller, and the cooling rate decreases; thus, the object

cools more and more slowly as time passes. Differential equations arise whenever we want to

represent mathematically a problem involving rate measure. Many real world phenomena can be

described through either differential equation involving ordinary derivatives/partial derivatives.

Differential equations play an important role in many applications in the fields of science and

engineering, such as (i) problems relating to motion of particles (ii) problems involving bending of beams

(iii) problems related to stability of electric system, chemical process, Economics, Anthropology and

diverse branches.

1. INTRODUCTION

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. 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.

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 non linear 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.

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.

Many of the general laws in all branches of engineering, physics, chemistry, biology, astronomy,

population studies, geometry, economics, etc. can be expressed through Mathematical Modelling in

the form of an equation connecting the variables and their rate of change, resulting in differential

equations. The time of death of a murdered person can be determined with the help of modeling

through differential equation.

The objective of the present study is to investigate the time of death of a murdered person

using the Newtons law of cooling.

2. MATHEMATICAL MODEL

When a hot object is placed in a cool room, the object dissipates heat to the

surroundings, and its temperature decreases. Newton's Law of Cooling states that the rate

at which the object's temperature decreases is proport ional to the difference between the

temperature of the object and the ambient temperature. At the beginning of the cooling

process, the difference between these temperatures is greatest, so this is when the rate of

temperature decrease is greatest. However, as the object cools, the temperature difference

gets smaller, and the cooling rate decreases; thus, the object cools more and more slowly as

time passes. To formulate this process mathematically, let T(t) denote the temperature of the

object at time t and let Ts denote the (essentially constant) temperature of the surroundings.

Newton's Law of Cooling then says,

dT

(T T )sdt (1)

i.e. K dT

(T T ) (K > 0)sdt (2)

Where K is the unknown proportionality constant, the equation (2) governing the Newtons law of

cooling, is a first order and first degree linear separable differential equation.

Since Ts

3. METHOD OF SOLUTION

We therefore present a more detailed exposition here to solve the above problem

analytically. Essentially 3 phases are central to the solution. These are:

I. Identify Ts, the temperature of the surrounding medium, so that the general

solution is given by (3).

II. Use two conditions given to determine the constant of integration C and the

unknown proportionality constant K.

III. Substituting C and K, obtained from step II, in equation (3) (a) the value of T

for a given time t or (b) the value of time t for a given temperature T can

be determined from equation (3)

Let T(t) denote the temperature of the body at time t.

Then the ( ) KtsT t T Ce since Ts =

022 C

Constants K and C can be determined provided the following information is available:

Time of arrival of the person, the temperature of the body just after his arrival, temperature of the

body after certain interval of time.

Let the officer arrived at 8.30 a.m. and the body temperature was 30 degrees. This means

that if the officer considers 8:30 a.m. as t = 0, i.e. T(0)=30 then

k*0

30 22 C e

8C

Hence k*tT t 22 8e (4)

Let the officer makes another measurement of the temperature say after 1 hour (60

minutes), that is, at 9.30 a.m. and temperature was 28 degrees. This means that T(1) = 028

,

k*128 22 8e

k*1 6

8e

ln 4 ln 3k

k = 0.2877

Now it only remains to find out when the murder occurred. At the time of death the body

was 037 C (i.e. normal human body temperature).

So, equation (4) k*t

37 22 8e

k*t 15

8e

k*t ln 1.875

1

t *ln 1.875k

2.19t Hours

Here sign minus represents the time before found the victim.

From this, we can conclude that the murder occurred about 2 hours and 19 minutes before

the body was found.

The death occurred approximately 139 minutes before the first measurement at 8.30a.m that is

at 6:19a.m approximately

4. RESULTS AND DISCUSSIONS

The present analysis integrates the equations by the analytical method. The details of the

solution method is shown and the variation of the temperature T of the body before identifying

and after for different time t are also shown graphically.

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. The time of death of a murdered person is

determined with the help of modeling through differential equation.

According to Newtons law of cooling, the body will radiate heat energy into the room at

a rate proportional to the difference in temperature between the body and the room.

For some time after the death, the body will radiate heat into the cooler room, causing the

bodys temperature to decrease assuming that the victims temperature was normal 37oC (98.6oF) at the

time of death. Forensic expert will try to estimate this time from bodys current temperature and

calculating how long it would have had to lose heat to reach this point.

The following figure display the temperature of the body of victim for different time. It is

found that temperature decrease monotonically from normal human body temperature (37oC) for

increasing time because the body will radiate heat into the cooler room, causing the bodys

temperature to decrease. This is owing to the fact that the body temperature suddenly falls in the

first 10 minutes after murdered and then slowly decrease with increasing time.

The death occurred approximately 139 minutes before the first measurement at 8.30a.m

that is at 6:19a.m approximately

TABLE AND GRAPH:

t (time in minutes) T(t) (temperature in

degrees) t (time in minutes)

T(t) (temperature in

degrees)

-130 37 0 30

-120 32.8 10 29.8353

-110 32.1 20 29.674

-100 31.8497 30 29.516

-90 31.6469 40 29.361

-80 31.4483 50 29.209

-70 31.2539 60 29.0613

-60 31.0634 70 28.916

-50 30.9124 80 28.7736

-40 30.6941 90 28.634

-30 30.5151 100 28.497

-20 30.3398 110 28.364

-10 30.1681 120 28.2329

130 28.104

5. CONCLUSIONS

The present paper analytically studied to investigate the time of death using the Newtons law of cooling. The time of death of a murdered person is determined with the help of modeling through differential equation. To formulate this process mathematically, solved analytically and results are shown in graphical representation. It is noticed that the murdered body cools, the temperature difference gets smaller, and the cooling rate decreases; thus, the object cools more and more slowly as time passes. The death occurred approximately 139 minutes before the first measurement at 8.30a.m that is at 6:19a.m approximately. The problem of solving differential equations is a natural goal of differential and integral calculus. Further many of the general laws of nature in Physics, Chemistry, Biology and Astronomy can be expressed in the language of differential equations and hence the theory of differential equations is the most important part of mathematics for understanding Physical sciences. 6. REFERENCES 1. Engineering mathematics volume-1, Dr.T.K.V.Iyengar, Dr.B.krishnagandhi, S.Ranganatham,M.V.S.S.N.Prasad 2. AText book of EngineringMathematics, B.V.Ramana. 3. A Text Book of Engineering Mathematics, Thomson Book Collection.

9.

CONCEPT ON

MATRICES MATHEMATICS

MOULA ALI COLLEGE OF ENGINEERING AND TECHNOLOGY

PREPARED BY

T.GOWSIA B.PRIYANKA

REG.NO:10F51A0424 10F51A0460

E.C.E E.C.E

I-BTECH I-BTECH

Email:[email protected]

ABSTRACT

DEFINATION

NOTATION

BASIC OPERATORS

MATRICES MULTIPLICATION,LINEAR EQUATIONS

RANK OF MATRICES

SQUARE MATRICES

INVERSE MATRICES

TRACE

TRIANGULAR MATRICES

DIAGONAL MATRICES

DETERMINENT

EIGENVECTORS AND EIGENVALUES

SYMMETRY

MATRICES DECOMPOSITION METHODS(LU-decomposition method)

MATRICES:

Definition

A matrix is a rectangular arrangement of numbers.

For example A= 43

21

The horizontal and vertical lines in a matrix are called rows and columns, respectively.

The numbers in the matrix are called its entries or its elements.

To specify the size of a matrix, a matrix with m rows and n columns is called an m-by-n matrix or m n matrix, while m and n are called its dimensions.

ROW MATRICES:

A matrix with one row is called a row matrix.

For example: A= 21

COLUMN MATRICES:

A matrix with one column is called acolumn matrix.

For example: A=3

2

TRANSPOSE MATRICES:

The rows of a matrix are equal to the corresponding columns of itsmatrix, the matrix is transpose

matrix.

Most of this article focuses on real and complex matrices, i.e., matrices whose entries are real or complex numbers.

Notation

The entry that lies in the i-th row and the j-th column of a matrix is typically referred to as the

(i,j), or (i,j)th entry of the matrix.

For example, the (2, 2) entry of the above matrix A is 4.

The (i, j) th

entry of a matrix A is most commonly written as ai,j.

Alternative notations for that entry are A [i,j] or Ai,j.

Letai, refers to the ith row of A, and a,j refers to the j

th column of A. The set of all m-by-n

matrices is denoted (m, n).

A = [a, j]i=1,...,m; j=1,...,n or more briefly A = [ai,j]mn

To define an m n matrix A. Usually the entries ai,j are defined separately for all integers 1 i m and 1 j n.

Basic operations

Matrix addition

Scalar multiplication

Transpose

Row operations

There are a number of operations that can be applied to modify matrices called matrix addition,

scalar multiplication and transposition. These form the basic techniques to deal with matrices.

Operation Definition

Addition

The sumA+B of two m-by-n matrices A and B is calculated entrywise:

(A + B)i,j = Ai,j + Bi,j, where 1 i m and 1 j n.

Scalar

multiplication

The scalar multiplication cA of a matrix A and a number c is given by

multiplying every entry of A by c:

(cA)i,j = c Ai,j.

Transpose

The transpose of an m-by-n matrix A is the n-by-m matrix AT (also denoted

Atr or

tA) formed by turning rows into columns and vice versa:

(AT)i,j = Aj,i.

Addition of matrices iscommutative,

A + B = B + A.

The transpose is compatible with addition and scalar multiplication

(cA)T = c (A

T) and (A + B)

T = A

T + B

T. Finally, (A

T)

T = A.

Row operations

There are three types of row operations: row switching, that is interchanging two rows of a

matrix; row multiplication, multiplying all entries of a row by a non-zero constant; and finally

row addition, which means adding a multiple of a row to another row. These row operations are

used in a number of ways including solving linear equations and finding inverses

Matrix multiplication, linear equations

Matrix multiplication

Schematic depiction of the matrix product AB of two matrices A and B.

Multiplication of two matrices is defined only if the number of columns of the left matrix is the

same as the number of rows of the right matrix.

If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix productAB is the m-by-p

matrix whose entries are given by dot-product of the corresponding row of A and the

corresponding column of B:

Matrix multiplication satisfies the rules (AB) C = A (BC) (associativity),

And (A+B) C = AC+BC as well as C(A+B) = CA+CB (left and right distributivity),

Whenever the size of the matrices is such that the various products are defined.

IfA and B are m-by-n and n-by-k matrices, respectively, and m k. Even if both products are defined, they need not be equal, i.e. generally one has

AB BA,

i.e., matrix multiplication is not commutative,

The identity matrixIn of size n is the n-by-n matrix in which all the elements on the main

diagonal are equal to 1 and all other elements are equal to 0,

It is called identity matrix because multiplication with it leaves a matrix unchanged: MIn = ImM

= M for any m-by-n matrix M.

Linear equations

Linear equation and System of linear equations

A particular case of matrix multiplication is tightly linked to linear equations: if x designates a

column vector (i.e. n1-matrix) of n variables x1, x2, ..., xn, and A is an m-by-n matrix, then the

matrix equation

Ax = b,

Whereb is some m1-column vector, is equivalent to the system of linear equations

A1,1x1 + A1, 2x2 + ... + A1,nxn = b1

...

Am, 1x1 + Am, 2x2 + ... + Am, nxn = bm.

This way, matrices can be used to compactly write and deal with multiple linear equations, i.e.

systems of linear equations.

RANK OF MATICES:

The rank of a matrixA is the maximum number of linearly independent row vectors of the

matrix, which is the same as the maximum number of linearly independent column vectors.

Square matrices

A square matrix is a matrix which has the same number of rows and columns.

For example: A= 32

21

An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same

order can be added and multiplied.

A square matrix A is called invertible or non-singular if there exists a matrix B such that

AB = In.

INVERSE MATRICES:

This is equivalent to BA = In. Moreover, if B exists, it is unique and is called the inverse matrix

of A, denoted A1

.

TRACE:

The entries Ai,i form the main diagonal of a matrix. The trace, tr(A) of a square matrix A is the

sum of its diagonal entries,

the trace of the product of two matrices is independent of the order of the factors:

tr(AB) = tr(BA).

Also, the trace of a matrix is equal to that of its transpose, i.e. tr(A) = tr(AT).

DIAGONAL MATRICES:

If all entries outside the main diagonal are zero, A is called a diagonal matrix.

TRIANGULAR MATRICES:

If only all entries above (below) the main diagonal are zero, A is called a lower triangular matrix

(upper triangular matrix, respectively). For example, if n = 3, they look like (Diagonal), (lower)

and (upper triangular matrix).

Lower triangular matrix: A= 30

21

Upper triangular matrix: A= 43

01

Determinant

The determinantdet (A) or |A| of a square matrix A is a number encoding certain properties of the

matrix. A matrix is invertible if and only if its determinant is nonzero.

The determinant of a product of square matrices equals the product of their determinants: det

(AB) = det (A) det (B).

Adding a multiple of any row to another row, or a multiple of any column to another column,

does not change the determinant.

To solve linear systems using Cramer's rule, where the division of the determinants of two

related square matrices equates to the value of each of the system's variables.

Eigenvalues and eigenvectors

A number and a non-zero vector v satisfying

AX = X

are called an eigenvalue and an eigenvector of A, respectively. The number is an eigenvalue of an nn-matrix A if and only if AIn is not invertible.

The function pA(t) = det(AtI) is called the characteristic polynomial of A, its degree is n. Therefore pA(t) has at most n different roots, i.e., eigenvalues of the matrix

They may be complex even if the entries of A are real. According to the Cayley-Hamilton

theorem, pA(A) = 0, that is to say, the characteristic polynomial applied to the matrix itself yields

the zero matrix.

SYMMETRY:

SYMMETRIC MATRICES:

A square matrix A that is equal to its transpose, i.e. A = AT, is a symmetric matrix.

SKEW-SYMMETRICE MATRICES:

If instead, A was equal to the negative of its transpose, i.e. A = AT, then A is a skew-symmetric matrix.

HERMITIAN MATRICES:

In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which

satisfy A = A, where the star or asterisk denotes the conjugate transpose of the matrix, i.e. the

transpose of the complex conjugate of A.

Matrix decomposition methods

Matrix decomposition

Matrix diagonalization

Gaussian elimination

Montante's method

There are several methods to render matrices into a more easily accessible form. They are

generally referred to as matrix transformation or matrix decomposition techniques.

LU decomposition:

The LU decomposition factors matrices as a product of lower (L) and an upper triangular

matrices (U).

Problem on LU-decomposition method:

Using LU-decomposition method solve x+y+z=1, 3x+y-3z=5, x-y-5z=10.

Solution:

521

313

111

z

y

x

=

10

5

1

i.e., AX=B

Let A=LU

Where L=

13231

012

001

ll

l

U=

3300

23220

131211

u

uu

uuu

A=LU

=

13231

0121

001

ll

l

=

3323321331223212311131

2313212212211121

131211

uululululul

uuluulul

uuu

Compare the corresponding elements of LU and the elements of A

u11=1 ,

u12=1 ,

u13=1,

l21u11=3 ,

l21=3

l21u12+u22=1

u22=-2

l21u13+u23=-3

u23=-6

l31u11=1

l31=1

l31u12+l32u22=-2

l32=3/2

l31u13+l32u23+u33=-5

u33=3

12/31

013

001

300

620

111

AX=B

LUX=B

Let UX=Y

LY=B

12/31

013

001

3

2

1

Y

Y

Y

=

10

5

1

322/31

213

1

YYY

YY

Y

=

10

5

1

Y1=1

3Y1+Y2=5

Y2=2

Y1+3/2Y2+Y3=10

Y3=6

32

1

Y

Y

Y

=

6

2

1

Since UX=Y

300

620

111

Z

Y

X

6

2

1

Z

ZY

ZYX

3

62

=

6

2

1

X+Y+Z=1

-2Y-6Z=2

3Z=6

Z=2

-2Y-6Z=2

Y=-7

X=6

X =

z

y

x

=

2

7

6

Therefore the problem is sloved.

10. Moula ali college of engineering and

technology,

Anantapur.

Prepared by

B.Peddy reddy B.Aravind kumar

ECE(I B.TECH) ECE(I B.TECH)

REG.NO 10F51A0458 REG.NO 10F51A0408

EMAIL.ID [email protected]

ABSTRACT

Applications of Differential Equations

Application 1 : Exponential Growth - Population

Application 2 : Exponential Decay - Radioactive Material

Application 3 : Falling Object

MATRICES

Definition of a Matrix

Introduction to Matrix:

Other Types of Matrix

Transpose Matrix:

Definition

Adjoint Matrix

Minors

Co-factors

Properties of Determinants

Applications of Differential Equations

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

Application 1 : Exponential Growth - Population

Let P(t) be a quantity that increases with time t and the rate of increase is proportional to the same quantity P as follows

d P / d t = k P where d p / d t is the first derivative of P, k > 0 and t is the time. The solution to the above first order differential equation is given by

P(t) = A ek t where A is a constant not equal to 0. If P = P0 at t = 0, then P0 = A e

0 which gives A = P0 The final form of the solution is given by

P(t) = P0 ek t

Assuming P0 is positive and since k is positive, P(t) is an increasing exponential. d P / d t = k P is also called an exponential growth model.

Application 2 : Exponential Decay - Radioactive Material

Let M(t) be the amount of a product that decreases with time t and the rate of decrease is proportional to the amount M as follows

d M / d t = - k M where d M / d t is the first derivative of M, k > 0 and t is the time.

Solve the above first order differential equation to obtain

M(t) = A e- k t where A is non zero constant. It we assume that M = M0 at t = 0, then M0 = A e

0 which gives A = M0 The solution may be written as follows

M(t) = M0 e- k t

Assuming M0 is positive and since k is positive, M(t) is an decreasing exponential. d M / d t = - k M is also called an exponential decay model.

Application 3 : Falling Object

An object is dropped from a height at time t = 0. If h(t) is the height of the object at time t, a(t) the acceleration and v(t) the velocity. The relationships between a, v and h are as follows: a(t) = dv / dt , v(t) = dh / dt. For a falling object, a(t) is constant and is equal to g = -9.8 m/s. Combining the above differential equations, we can easily deduce the follwoing equation d 2h / dt 2 = g Integrate both sides of the above equation to obtain dh / dt = g t + v0 Integrate one more time to obtain h(t) = (1/2) g t + v0 t + h0 The above equation describes the height of a falling object, from an initial height h0 at an initial velocity v0, as a function of time. Substitute c in the solution - (L / R) ln(E - R i) = t + (-L/R) ln (E)

which may be written (L/R) ln (E)- (L / R) ln(E - R i) = t ln[E/(E - Ri)] = t(R/L) Change into exponential form [E/(E - Ri)] = et(R/L) Solve for i to obtain i = (E/R) (1-e-Rt/L) The starting model for the circuit is a differential equation which when solved, gives an expression of the current in the circuit as a function of time.

Definition of a Matrix

A rectangular array of entries is called a Matrix. The entries may be real, complex or functions.

The entries are also called as the elements of the matrix.

The rectangular array of entries are enclosed in an ordinary bracket or in square bracket.

Matrices are denoted by capital letters.

Example:

(i)

A matrix having m rows and n columns is called as matrix of order mxn. Such a matrix has mn

elements.

In general, an mxn matrix is in the form

Where aij represents the element in ith column.

The above matrix may be denoted as A = [aij]mxn.

Introduction to Matrix:

A Matrix is a rectangular arrangement of number in rows and columns and enclosed by

Parenthesis (or) Brackets. Matrices are denoted by A, B, C Matrix is a way of organizing the

data in order to rows and columns. it can be w