inverse laplace transform l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Inverse Laplace Transform PowerPoint Presentation
Download Presentation
Inverse Laplace Transform

Loading in 2 Seconds...

play fullscreen
1 / 31

Inverse Laplace Transform - PowerPoint PPT Presentation


  • 149 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Inverse Laplace Transform' - graham


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
inverse laplace transform
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
Consider the Initial Value Problem:

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

Applications
unit step functions can be used to represent any piecewise continuous function
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.
laplace transform of periodic functions
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
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 .
convolution operator
Definition: 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 “ * “.
proof of convolution theorem can be done by
1. Writing

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

Proof of Convolution theorem can be done by
transfer function and impulse response function
Consider the linear system governed by the I.V.P:

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
we get

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
Namely:

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
example
Example

#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
Dirac Delta Function

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
Remarks on Theory of Distribution.

Symbolic function, generalized function, and distribution function.

heuristic argument on the existence of function
Heuristic argument on the existence of -function.

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

application
Application:

Consider the symbolic Initial Value Problem:

linear systems can be solved by laplace transform 7 9
Linear Systems can be solved by Laplace Transform.(7.9)

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.