1. 1 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

2. 2 First we note that the Inverse Laplace Transform is a “ Linear Operator”. Some Examples.

3. 3 Consider the Initial Value Problem: We shall use Laplace Transform and Inverse Laplace Transform to solve this I.V.P. Applications

4. 4 Given the following I.V.P: (#36, P. 406) Next let us consider a D.E. with variable coefficents

5. 5 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 t 0 1

6. 6 Shifts and scalar multiples of the Unit Step Functions

7. 7 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.

8. 8 Next, what is:

9. 9 Let us consider the following : Some Examples

10. 10 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

11. 11 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.

12. 12 Note that in this case f T (t) is given by : f T (t) = 1 - u(t - 1). Hence Example: The square wave with period 2 t 1 2 1

13. 13 #10, 15, 31, Some problems in the exercise

14. 14 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 “ * “.

15. 15 An important property of convolution is the Theorem: Laplace Transform of convolution

16. 16 1. Writing 2. Then apply the Fubini’s theorem on interchanging the order of integration. Proof of Convolution theorem can be done by

17. 17 Solve the initial value problem Applications

18. 18 Consider the following equation: Example 2: Integral-Differential Equation

19. 19 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

20. 20 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

21. 21 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

22. 22 They are the equivalent to the original I.V.P. Namely: We shall split the given I.V.P into two problems

23. 23 Theorem: Let I be an interval containing the origin. The unique solution to the initial value problem Theorem on solution using Impulse Response Function

24. 24 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) + y k (t)

25. 25 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

26. 26 Remarks on Theory of Distribution. Symbolic function, generalized function, and distribution function.

27. 27 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 [t 0, t 1 ], then the total impulse due to F is the integral

28. 28 This heuristic leads to conditions 1 and 2.

29. 29 What is the Laplace Transform of  -function? By definition, we have

30. 30 Application: Consider the symbolic Initial Value Problem:

31. 31 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.