The Laplace Transform

Transcription

1 The Laplace Transform Laplace Transform Philippe B. Laval KSU Today Philippe B. Laval (KSU) Definition of the Laplace Transform Today 1 / 16

2 Outline General idea behind the Laplace transform and other transforms. Definition of the Laplace transform Examples Some properties of the Laplace Transform Existence of the Laplace transform What next? Philippe B. Laval (KSU) Definition of the Laplace Transform Today 2 / 16

3 General Idea The Laplace transform belongs to a family of operators called integral transforms. An operator is simply a rule which transforms a function into another function. An integral transform is an operator of the form T {f } (s) = b a f (t) K (s, t) dt where K is called the kernel of the operator. K is chosen depending on which properties we need the transform to have. A transform maps the domain of a function into another domain in which we hope the problem we are solving will be easier. Other transforms include the Fourier transform, the Hankel transform, the Hough transform, the Radon transform,... Philippe B. Laval (KSU) Definition of the Laplace Transform Today 3 / 16

4 General Idea We can use the Laplace transform as a technique to solve differential equations. It is an analytic technique, very different from the ones we have used so far. It is particularly effective on linear differential equations with constant coeffi cients. The Laplace transform allows us to replace the calculus operation of differentiation in the t domain with the algebraic operation of multiplication by s in the s domain. In other words, it turns a problem in calculus into a problem in algebra. Philippe B. Laval (KSU) Definition of the Laplace Transform Today 4 / 16

5 Definition Definition Let f (t) be a function on [0, ), The Laplace transform of f is the function F defined by Let us make a few remarks: L {f } (s) = F (s) = 1 You will recall that 0 f (t) e st dt = lim 0 f (t) e st dt N N 0 f (t) e st dt. 2 As you remember from calculus II, not every improper integral converge. So, it is possible that the Laplace transform may not exist. We will discuss conditions under which we can be garanteed the Laplace transform exists further down. For now, in the nest few examples, let us assume the functions in the examples below have a Laplace transform. Philippe B. Laval (KSU) Definition of the Laplace Transform Today 5 / 16

6 Examples Example Find L {f } (s) for f (t) = 1, t 0 (answer: 1 s for s > 0) Example Find L {f } (s) for f (t) = e at, where a is a constant. (answer: s > a) 1 s a for Example Find L {f } (s) for f (t) = sin bt, where b is a nonzero constant. (answer: b b 2 for s > 0) + s2 Philippe B. Laval (KSU) Definition of the Laplace Transform Today 6 / 16

7 Examples Example 2 for 0 < t < 5 Find L {f } (s) for f (t) = 0 for 5 < t < 10 e 4t for t > 10 2 s 2e 5s s + e 10(s 4) s 4 for s > 4) (answer: Philippe B. Laval (KSU) Definition of the Laplace Transform Today 7 / 16

8 Properties of the Laplace Transform The Laplace transform is a linear operator, in other words, we have the following theorem: Theorem Let f, f 1, and f 2 be functions whose Laplace transform exists for s > α and let c be a constant. Then, for s > α Example Find L { e 4t 6 sin 2t } (s) L {f 1 + f 2 } = L {f 1 } + L {f 2 } L {cf } = cl {f } Philippe B. Laval (KSU) Definition of the Laplace Transform Today 8 / 16

9 Existence of the Transform We discuss properties which will ensure (collectively) the existence of the transform. Definition A function f (t) is said to be piecewise continuous on a finite interval [a, b] if f (t) is continuous at every point in [a, b] except possibly for a finite number of points at which f (t) has a jump discontinuity. Definition A function f (t) is said to be piecewise continuous on [0, ) if it is piecewise continuous on [0, N] for all N > 0. Philippe B. Laval (KSU) Definition of the Laplace Transform Today 9 / 16

10 Existence of the Transform Example Which function is piecewise continuous? t for 0 < t < 1 1 f (t) = 2 for 1 < t < 2 (t 2) 2 for 2 t 3 2 f (t) = 1 t Note: A function which is piecewise continuous on a finite interval is necessarily integrable on that interval. However, piecewise continuity is not enough on [0, ), we need more. We need to consider the growth of the function for large t. This is captured in the next definition. Philippe B. Laval (KSU) Definition of the Laplace Transform Today 10 / 16

11 Existence of the Transform Definition A function f (t) is said to be of exponential order α if there exists positive constants T and M such that f (t) Me αt, for all t T We say that a function f is of exponential order to mean that for some α it satisfies the definition. Notes: To verify this condition, one can do it different ways. 1 Do it directly, that is show the definition is satisfied. 2 If f (t) Me αt f (t) then e αt M for all t T that is in particular, f (t) lim is finite. t e αt We illustrate this with some examples. Philippe B. Laval (KSU) Definition of the Laplace Transform Today 11 / 16

12 Existence of the Transform Example Which of the following functions are of exponential order? 1 e 5t sin t for t > 0. 2 t 3 sin t for t > t 2 for t > e t2 for t > 0. 5 If f is of exponential order α and β > α, is it true that f is also of exponential order β and why? Philippe B. Laval (KSU) Definition of the Laplace Transform Today 12 / 16

13 Existence of the Transform We can now list and prove the existence theorem for the Laplace transform Theorem If f (t) is piecewise continuous on [0, ) and of exponential order α, then L {f } (s) exists for s > α. The functions in the examples at the beginning of this section, f (t) = 1, 2 if 0 < t < 5 f (t) = e at, f (t) = sin bt, and f (t) = 0 if 5 < t < 10 all satisfy e 4t if t > 10 the conditions of the theorem. The first three are continuous (hence piecewise continuous) and the last one is piecewise continuous. Therefore, they have a Laplace transform, as we noticed since we were able to compute it. Philippe B. Laval (KSU) Definition of the Laplace Transform Today 13 / 16

14 Table of transforms f (t) F (s) = L {f } (s) 1 1 s, s > 0 e at 1 s a, s > a t n n!, n = 1, 2, 3,... s n+1, s > 0 b sin bt s 2 + b 2, s > 0 s cos bt s 2 + b 2, s > 0 e at t n n!, n = 1, 2, 3,... (s a) n+1, s > a e at b sin bt (s a) 2 + b, s > a 2 e at s a cos bt (s a) 2 + b, s > a 2 Philippe B. Laval (KSU) Definition of the Laplace Transform Today 14 / 16

15 What Next? Laplace transform of derivatives Inverse Laplace transform Philippe B. Laval (KSU) Definition of the Laplace Transform Today 15 / 16

16 Exercises Do the problems at the end of section 4.1 in my notes on the definition of the Laplace transform. Philippe B. Laval (KSU) Definition of the Laplace Transform Today 16 / 16