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Consider a given function F(s), is it possible to find a function f(t) defined on [0,  ), such that If this is possible, we say f(t) is the inverse Laplace transform of F(s), and we write . Inverse Laplace Transform.

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Inverse laplace transform l.jpg

Consider a given function F(s), is it possible to find a function f(t) defined on [0,  ), such that

If this is possible, we say f(t) is the inverse Laplace transform of F(s), and we write

Inverse Laplace Transform



Applications l.jpg

Consider the Initial Value Problem: Linear Operator”.

We shall use Laplace Transform and Inverse Laplace Transform to solve this I.V.P.

Applications


Next let us consider a d e with variable coefficents l.jpg

Given the following I.V.P: (#36, P. 406) Linear Operator”.

Next let us consider a D.E. with variable coefficents


Laplace transform of discontinuous and periodic functions l.jpg

Discontinuous functions play a very important role in Engineering, for example:

This is known as the unit step function.

Laplace Transform of Discontinuous and Periodic Functions

1

t

0



Unit step functions can be used to represent any piecewise continuous function l.jpg

For example: … Engineering, for example:

What is the Laplace Transform of u(t- a), a > 0?

Unit step functions can be used to represent any piecewise continuous function.


Next what is l.jpg
Next, what is: Engineering, for example:


Some examples9 l.jpg

Let us consider the following : Engineering, for example:

Some Examples


Laplace transform of periodic functions l.jpg

Periodic functions play a very important role in the study of dynamical systems

Definition: A function f (t) is said to be periodic of period T, if for all t D(f) , we have f (t + T) = f (t).

For examples, sine waves, cosine waves and square waves are periodic functions.

What can we say about the transforms of periodic functions?

Laplace Transform of Periodic Functions


Slide11 l.jpg

It is not difficult to see that f (t) can be written as the sum of translates of f T (t). Namely,

Let f T (t) be the part of f over the basic period [0, T]. This is known as the Windowed version of the periodic function f .


Example the square wave with period 2 l.jpg

Note that in this case f sum of translates of f T (t) is given by :

f T (t) = 1 - u(t - 1) . Hence

Example: The square wave with period 2

1

t

1

2


Some problems in the exercise l.jpg

#10, 15, 31, sum of translates of f

Some problems in the exercise


Convolution operator l.jpg

Definition: sum of translates of f Given two functions f (t) and g(t) piecewise continuous on [0,). The convolution of f and g , denoted by

Covolution is 1.commutative, 2. distributive, 3. associative and with 4. existence of zero.

Convolution Operator “ * “.


Laplace transform of convolution l.jpg

An important property of convolution is the sum of translates of f

Theorem:

Laplace Transform of convolution


Proof of convolution theorem can be done by l.jpg

1 sum of translates of f . Writing

2. Then apply the Fubini’s theorem on interchanging the order of integration.

Proof of Convolution theorem can be done by


Applications17 l.jpg

Solve the initial value problem sum of translates of f

Applications


Example 2 integral differential equation l.jpg

Consider the following equation: sum of translates of f

Example 2: Integral-Differential Equation


Transfer function and impulse response function l.jpg

Consider the linear system governed by the I.V.P: sum of translates of f

Thus given g(t) we wish to find the solution y(t). g(t) is called the input function and y(t) the output. The ratio of their Laplace Transforms,

Transfer function and Impulse response function


For our example take the laplace transform of the i v p l.jpg

we get sum of translates of f

The inverse Laplace Transform of H(s), written h(t) = L-1{H(s)}(t) is called the Impulse response function for the system. Graph!!

For our example, take the Laplace transform of the I.V.P


This function h t is the unique solution to the homogeneous problem l.jpg

Namely: sum of translates of f

This can be checked easily (using Laplace transform). Now to solve a general I.V.P. such as

This is a non-homogeneous eq with non-trivial initial values.

This function h(t) is the unique solution to the homogeneous problem


We shall split the given i v p into two problems l.jpg

They are the equivalent to the original I.V.P. Namely: sum of translates of f

We shall split the given I.V.P into two problems


Theorem on solution using impulse response function l.jpg

Theorem sum of translates of f : Let I be an interval containing the origin. The unique solution to the initial value problem

Theorem on solution using Impulse Response Function


Example l.jpg
Example sum of translates of f

#24, P.428

Let a linear system be governed by the given initial value problem.

Find the transfer function H(s), the impulse response function h(t) and solve the I.V.P.

Recall: y(t) = (h*g)(t) + yk(t)


Dirac delta function l.jpg
Dirac Delta Function sum of translates of f

Paul A. M. Dirac, one of the great physicists from England invented the following function:

Definition: A function (t) having the following properties:

is called the Dirac delta function. It follows from (2) that for any function f(t) continuous in an open interval containing t = 0, we have


Remarks on theory of distribution l.jpg
Remarks on Theory of Distribution. sum of translates of f

Symbolic function, generalized function, and distribution function.


Heuristic argument on the existence of function l.jpg
Heuristic argument on the existence of -function. sum of translates of f

When a hammer strikes an object, it transfer momentum to the object. If the striking force is F(t) over a short time interval [t0, t1], then the total impulse due to F is the integral



What is the laplace transform of function l.jpg
What is the Laplace Transform of sum of translates of f -function?

By definition, we have


Application l.jpg
Application: sum of translates of f

Consider the symbolic Initial Value Problem:


Linear systems can be solved by laplace transform 7 9 l.jpg
Linear Systems can be solved by Laplace Transform.(7.9) sum of translates of f

For two equations in two unknowns, steps are:

1. Take the Laplace Transform of both equations in x(t) and y(t),

2. Solve for X(s) and Y(s), then

3. Take the inverse Laplace Transform of X(s) and Y(s), respectively.

4. Work out some examples.