3 LAPLACE TRANSFORM part2.ppt

Excellent does not an accident, but it comes through a hard work!!

Excellent does not an accident, but it comes through a hard work!!LESSON OUTCOMES

Excellent does not an accident, but it comes through a hard work!!3.1 Definition and notation

Excellent does not an accident, but it comes through a hard work!! INTRODUCTION :

Laplace transform methods have a key role to play in the modern approach to the analysis and design of engineering systems.

The attraction of the Laplace transform is that it transforms differential equations in the t (time) domain into algebraic equations in the s (frequency) domain. Ex:

In general, the function to be transformed is denoted by a lowercase letter, while its Laplace transform will be denoted by the corresponding upper case letter. The Laplace transform is a one-to-one function.

The Laplace transform operator.

Excellent does not an accident, but it comes through a hard work!!Advantage of using the Laplace transform for solving differential equations is that initial conditions play an essential role in the transformation process, so they are automatically incorporated into the solution.

This contrasts with the classical approach considered previously, where the initial conditions are only introduced when unknown constants of integration are determined.

The laplace transform is therefore an ideal tool for solving initial-value problems such as those occurring in the investigation of electrical circuits and mechanical vibrations. INTRODUCTION :

Excellent does not an accident, but it comes through a hard work!!LAPLACE TRANSFROM via DEFINITIONDefinition and Notation:

The Laplace transform of a function

by the expression

: complex variable

: kernel of transformation

Represent the Laplace transform of a function by the corresponding capital letter

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Excellent does not an accident, but it comes through a hard work!!LAPLACE TRANSFROM via DEFINITIONExample 1: Find the Laplace transform of

Solution :

So,

By using definition, find Laplace transform of

Excellent does not an accident, but it comes through a hard work!!LAPLACE TRANSFROM via DEFINITIONExample 2

Find the Laplace transform of

Solution :

So;

Excellent does not an accident, but it comes through a hard work!!LAPLACE TRANSFROM via DEFINITIONExample 3Find the Laplace transform of

Solution :

So,



Solution :

Excellent does not an accident, but it comes through a hard work!!LAPLACE TRANSFROM via DEFINITIONSo, Identities Trigonometric Functions (Functions of Complex Variable)



Solution :

Excellent does not an accident, but it comes through a hard work!!So, LAPLACE TRANSFROM via DEFINITIONIdentities Trigonometric Functions (Functions of Complex Variable)



Solution :As we know,

So,

Excellent does not an accident, but it comes through a hard work!!3.2 Existence of Laplace Transform

Excellent does not an accident, but it comes through a hard work!!Existence of Laplace Transform

Definition :

A function

is said to be of exponential order as

if there exists a real number

and positive constants M and T such that

for all

.

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Excellent does not an accident, but it comes through a hard work!!Existence of Laplace TransformWe just use thisfor our purpose ofstudy

Theorem :

If the causal function

is piecewise-continuous on

and is of exponential order, with abscissa of convergence

, then its Laplace transform exists, with region of convergence

in the s domain; that is

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Excellent does not an accident, but it comes through a hard work!! Test your understanding.Determine the Laplace transform for the given f(t) by using table of Laplace transform and state its existence:


3.3 Properties of the Laplace transform

Excellent does not an accident, but it comes through a hard work!!PROPERTIES OF THE LAPLACE TRANSFORMExample 10Find the Laplace transform of

(a)

Solution :

Excellent does not an accident, but it comes through a hard work!!PROPERTIES OF THE LAPLACE TRANSFORM(b)

(c)Solution :Solution :Trigonometry Identity:

Excellent does not an accident, but it comes through a hard work!! Test your understanding.Find the Laplace transform of the following functions:

Excellent does not an accident, but it comes through a hard work!!PROPERTIES OF THE LAPLACE TRANSFORM

Example 11

Use the first shift theorem to find the Laplace Transform of

(a)

Solution :

Therefore

and

Excellent does not an accident, but it comes through a hard work!!PROPERTIES OF THE LAPLACE TRANSFORM

Solution :

Therefore

and

Solution :

Therefore

and

Excellent does not an accident, but it comes through a hard work!! Test your understanding.Use the table of Laplace transform and the first shift theoremto determine the Laplace transform of the following functions :Answer:

Excellent does not an accident, but it comes through a hard work!!PROPERTIES OF THE LAPLACE TRANSFORMExample 12

Solution :

Since thus by using the second shift theorem

we get

From the expression we see that d = 2

Example 13

The Laplace transform of a function is . Find the function.

Solution :

The expression shows that d = 3 and

We want to find the function which has the Laplace

transform . Therefore this function must be of the form

Since , thus

Hence

The function which has the Laplace transform is therefore Excellent does not an accident, but it comes through a hard work!!PROPERTIES OF THE LAPLACE TRANSFORM

Excellent does not an accident, but it comes through a hard work!! Test your understanding.Find the Laplace transform of the following functions

Excellent does not an accident, but it comes through a hard work!!PROPERTIES OF THE LAPLACE TRANSFORMExample 14:

PROPERTIES OF THE LAPLACE TRANSFORMExcellent does not an accident, but it comes through a hard work!!Example 15:

Excellent does not an accident, but it comes through a hard work!! Test your understanding.Use the table of Laplace transform and the derivative of transform to determine the Laplace transform of the following functions :

Test your understanding.Find the Laplace transform for each of the following function:Excellent does not an accident, but it comes through a hard work!!


3.4 Inverse Laplace Transform

Excellent does not an accident, but it comes through a hard work!!INVERSE LAPLACE TRANSFORMIf then Example:

Excellent does not an accident, but it comes through a hard work!!INVERSE LAPLACE TRANSFORM

Excellent does not an accident, but it comes through a hard work!!INVERSE LAPLACE TRANSFORM Examples :

Find the inverse Laplace transform of the following functions

(a) (b) (c) (d) (e)

Solutions :

Excellent does not an accident, but it comes through a hard work!!Solution for (a) Cont..Solution for (c) Cont..Solution for (b) Cont..Solution for (f) Cont..Solution for (e) Cont..Solution for (d) Cont.. Test your understanding.

Find the inverse Laplace transform of :

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Cont.. solution for (a)

Cont.. solution for (b)

Cont.. solution for (c)

Cont.. solution for (d)

Cont.. solution for (e)

Cont.. solution for (f)

Excellent does not an accident, but it comes through a hard work!!INVERSION USING LINEARITY PROPERTYExamples:

Linear Property of Inverse Laplace Transform:

If

and

are any constants, then

EMBED Equation.DSMT4

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Excellent does not an accident, but it comes through a hard work!!INVERSION USING LINEARITY PROPERTY

Excellent does not an accident, but it comes through a hard work!!INVERSION USING THE FIRST SHIFT THEOREM From the first shift theorem,

since , then Example:

Use the first shift theorem to find the inverse Laplace transform of the following functions

Solution :

Excellent does not an accident, but it comes through a hard work!!INVERSION USING THE FIRST SHIFT THEOREMb)Solution :

Excellent does not an accident, but it comes through a hard work!!INVERSION USING THE FIRST SHIFT THEOREMc)Solution :d)Solution :

Excellent does not an accident, but it comes through a hard work!!INVERSION USING THE FIRST SHIFT THEOREMe)Solution :

Excellent does not an accident, but it comes through a hard work!!INVERSION USING THE FIRST SHIFT THEOREM

Excellent does not an accident, but it comes through a hard work!!INVERSION USING THE FIRST SHIFT THEOREMSolution :e)

Excellent does not an accident, but it comes through a hard work!!INVERSION USING THE FIRST SHIFT THEOREMSolution :f)

Excellent does not an accident, but it comes through a hard work!!INVERSION USING THE FIRST SHIFT THEOREMExample:

.

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Excellent does not an accident, but it comes through a hard work!! Test your understanding.Find the inverse Laplace transform of the following functions :b)a)d)c)f)e)h)g)j)i)l)k)

Excellent does not an accident, but it comes through a hard work!!EVALUATION OF INVERSE TRANSFORM

Sometimes, it is possible to write down the inverse transform of the function

directly from the table.

But more often than not it is first necessary to carry out some algebraic manipulation on

.

For example :

In such cases, the procedure is first to resolve the function into partial fractions and then use the table.

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3.5 Finding The Inverse Laplace Transform using Partial Fractions

Excellent does not an accident, but it comes through a hard work!!USING PARTIAL FRACTION TO FIND THE INVERSE LAPLACE TRANSFORMRecall back: Partial FractionAny rational function of s, for example

where the degree of P(s) is less than the degree of Q(s), could be expressed as a sum of relatively simpler rational functions, called partial fractions.The form of the partial fractions depends on the kind of factors in Q(s) which are real quadratic and linear.Before we deal with some examples in finding the inversion, let us recall those rules of partial fractions.

Excellent does not an accident, but it comes through a hard work!!Recall back: Partial FractionCase 1: The denominator Q (s) is a product of distinct linear factor Examples

ALGEBRAIC FRACTIONPARTIAL FRACTIONS

Excellent does not an accident, but it comes through a hard work!!Recall back: Partial FractionCase 2: The denominator Q (s) is a product of repeated linear factors Examples


Excellent does not an accident, but it comes through a hard work!!Recall back: Partial FractionCase 3: The denominator Q (s) contains irreducible quadratic factors Examples


Excellent does not an accident, but it comes through a hard work!!Recall back: Partial FractionCase 4: The denominator Q (s) contains repeated irreducible quadratic factors Example:


Excellent does not an accident, but it comes through a hard work!!USING PARTIAL FRACTION TO FIND THE INVERSE LAPLACE TRANSFORMExamples:

Find the inverse Laplace transform of the following expressions.a)Solution :b)We express the given expression as partial fractions

coeff. coeff.

Excellent does not an accident, but it comes through a hard work!!USING PARTIAL FRACTION TO FIND THE INVERSE LAPLACE TRANSFORMHenceThereforeb) We express the given expression as partial fractions

coeff. coeff.

Excellent does not an accident, but it comes through a hard work!!USING PARTIAL FRACTION TO FIND THE INVERSE LAPLACE TRANSFORM

Example:

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Excellent does not an accident, but it comes through a hard work!!USING PARTIAL FRACTION TO FIND THE INVERSE LAPLACE TRANSFORMSolution for (a)Solution for (b)Solution for (c) Solution for (d)

Find the inverse Laplace transform of :

EMBED Equation.DSMT4

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EXCELLENT DOES NOT AN ACCIDENT, BUT IT COMES THROUGH A HARD WORK!*Cont.. solution for (a)

EXCELLENT DOES NOT AN ACCIDENT, BUT IT COMES THROUGH A HARD WORK!

EXCELLENT DOES NOT AN ACCIDENT, BUT IT COMES THROUGH A HARD WORK!*Cont.. solution for (b)


EXCELLENT DOES NOT AN ACCIDENT, BUT IT COMES THROUGH A HARD WORK!*Cont.. solution for (c)


EXCELLENT DOES NOT AN ACCIDENT, BUT IT COMES THROUGH A HARD WORK!*Cont.. solution for (d)


Excellent does not an accident, but it comes through a hard work!! Test your understanding.Express the following expressions as partial fractions and hence find their inverse Laplace transform.b)a)d)c)f)e)h)g)

3.6 The Convolution Theorem


Excellent does not an accident, but it comes through a hard work!!The Convolution Theorem

Excellent does not an accident, but it comes through a hard work!!The Convolution TheoremExample:

Excellent does not an accident, but it comes through a hard work!!The Convolution TheoremBy using convolution theorem, find the inverse Laplace transforms of the followingfunctions:(a)(b)Solution for (a)Solution for (b)

Excellent does not an accident, but it comes through a hard work!!Cont.. solution for (a)

Excellent does not an accident, but it comes through a hard work!!Cont.. solution for (b)

Excellent does not an accident, but it comes through a hard work!! Test your understanding.Find the inverse of the given Laplace Transformby using Convolution theorem.


3.6 Transform of derivatives

Excellent does not an accident, but it comes through a hard work!!TRANSFORM OF DERIVATIVESIf we use Laplace transform methods to solve the differential equations, we need to find convenient expressions for the Laplace transforms of derivative.

Such as or in general By definition

Integrating parts, we have

That is

Excellent does not an accident, but it comes through a hard work!!TRANSFORM OF DERIVATIVES And, this also gives

That is

Excellent does not an accident, but it comes through a hard work!!TRANSFORM OF DERIVATIVES

Excellent does not an accident, but it comes through a hard work!!TRANSFORM OF DERIVATIVES And, this procedure may be extended to obtain the Laplace transform of

in the form

Excellent does not an accident, but it comes through a hard work!!TRANSFORM OF DERIVATIVESEXAMPLE(1):The Laplace transform of is, .Find the Laplace transforms of the following expressions if given :(a)

(b)

(c)

Excellent does not an accident, but it comes through a hard work!!TRANSFORM OF DERIVATIVESEXAMPLE(2):

The Laplace transform of w(t) is W(s), w(0) = 3, w(0) = 1. Find the Laplace transform of the following expressions.

Excellent does not an accident, but it comes through a hard work!!TRANSFORM OF DERIVATIVESGiven problemUsing Laplace transform convert the equation intoInvert the transformThe solution y(t)is foundSteps for solving first order linear differential equationsSource: Abd. Wahid Md. Roji and Mohd Nor Mohamad Differential Equations for Engineering Students Solving First ODE by using Laplace Transform

TRANSFORM OF DERIVATIVESExample:Solve

Solution: Excellent does not an accident, but it comes through a hard work!!

TRANSFORM OF DERIVATIVESSolve the differential equations :

(a)

(b) Solution : aSolution : bExcellent does not an accident, but it comes through a hard work!!

EXCELLENT DOES NOT AN ACCIDENT, BUT IT COMES THROUGH A HARD WORK!*Solution : a


EXCELLENT DOES NOT AN ACCIDENT, BUT IT COMES THROUGH A HARD WORK!*Solution : b


TRANSFORM OF DERIVATIVESExcellent does not an accident, but it comes through a hard work!!Given problemUsing Laplace transform convert the equation intoInvert the transformThe solution y(t)is foundSteps for solving first order linear differential equationsSource: Abd. Wahid Md. Roji and Mohd Nor Mohamad Differential Equations for Engineering Students Solving Second ODE by using Laplace Transform

TRANSFORM OF DERIVATIVESExcellent does not an accident, but it comes through a hard work!!Example:Solve

Solution:

TRANSFORM OF DERIVATIVESExcellent does not an accident, but it comes through a hard work!!Solve the differential equations :

(a)

(b)

TRANSFORM OF DERIVATIVESExcellent does not an accident, but it comes through a hard work!!Simultaneous differential equations: In engineering we frequently encounter systems whose characteristics are modeled by a set of simultaneous linear differential equations with constants coefficients.

The method of solution is essentially the same as that adopted for solving a single differential equation in one unknown.

Now, let us discuss how to solve simultaneous differential equation by looking at application problems.

APPLICATION OF LAPLACE TRANSFORMExcellent does not an accident, but it comes through a hard work!!Glyn James, Modern Engineering Mathematics 3rd edFigure 1 : Cylindrical Tanks Two cylindrical water tanks are connected as shown in Figure 1. Initially there are 250 litres in the top tank and 50 litres in the bottom tank. At time the valve between the two tanks are opened. The flow rate through each of these valves is proportional to the volume of the water in the tank immediately in the valve. The differential equations of volume for both tanks are denoted as follows

where and denote the volume in the top tankand the volume in the bottom tank respectively. By using inverse Laplace transforms, find

APPLICATION OF LAPLACE TRANSFORMExcellent does not an accident, but it comes through a hard work!!Glyn James, Modern Engineering Mathematics 3rd ed

Excellent does not an accident, but it comes through a hard work!! Test your understanding.Solve the system (simultaneous equation)

with the initial conditions

Answer :

Excellent does not an accident, but it comes through a hard work!!From the previous lecture, we knowSo, obviouslyTRANSFORM OF INTEGRAL

Excellent does not an accident, but it comes through a hard work!!TRANSFORM OF INTEGRALExample 1 :

Excellent does not an accident, but it comes through a hard work!!TRANSFORM OF INTEGRALApplying Kirchhoffs second law to the circuit in the previous figure :

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3 LAPLACE TRANSFORM part2.ppt

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