Inverse Laplace Transform

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# Inverse Laplace Transform - PowerPoint PPT Presentation

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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Presentation Transcript
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
Consider the Initial Value Problem:

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

Applications
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.
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
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 .
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 “ * “.
1. Writing

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

Proof of Convolution theorem can be done by
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
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
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

#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

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.

Symbolic function, generalized function, and distribution 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:

Consider the symbolic Initial Value Problem:

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.