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CHAPTER 4

CHAPTER 4. The Laplace Transform. Contents. 4.1 Definition of the Laplace Transform 4.2 The Inverse Transform and Transforms of Derivatives 4.3 Translation Theorems 4.4 Additional Operational Properties 4.5 The Dirac Delta Function. 4.1 Definition of Laplace Transform. DEFINITION 4.1.

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CHAPTER 4

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  1. CHAPTER 4 The Laplace Transform

  2. Contents • 4.1 Definition of the Laplace Transform • 4.2 The Inverse Transform and Transforms of Derivatives • 4.3 Translation Theorems • 4.4 Additional Operational Properties • 4.5 The Dirac Delta Function

  3. 4.1 Definition of Laplace Transform DEFINITION 4.1 If f(t)is defined for t  0,then (2)is said to be the Laplace Transform of f. Laplace Transform • Basic DefinitionIf f(t)is defined for t  0,then the improper integral(1)

  4. Example 1 Evaluate L{1} Solution: Here we keep that the bounds of integral are 0 and  in mind.From the definition , s>0Since e-st 0as t ,for s > 0.

  5. Example 2 Evaluate L{t} Solution , s>0

  6. Example 3 Evaluate L{e-3t} Solution

  7. Example 4 Evaluate L{sin2t} Solution

  8. Example 4 (2) Laplace transform of sin 2t ↓

  9. L.T. is Linear • We can easily verify that(3)

  10. THEOREM 4.1 (a) (b) (c) (d) (e) (f) (g) Transform of Some Basic Functions

  11. DEFINITION 4.2 A function f(t)is said to be of exponential order, if there exists constants c>0, M > 0, and T > 0,such That |f(t)|  Mect for all t > T. See Fig 4.2, 4.3. Exponential Order

  12. Fig 4.2

  13. Examples See Fig 4.3

  14. Fig 4.4 • A function such as is not of exponential order, see Fig 4.4

  15. THEOREM 4.2 If f(t) is piecewise continuous on [0, ) and of exponential order, then L{f(t)}exists for s > c. Sufficient Conditions for Existence

  16. Fig 4.1

  17. Example 5 Find L{f(t)}for Solution

  18. 4.2 If F(s)=L(f(t)), then f(t) is the inverse Laplace transform of F(s) and f(t)=L(F(s)) THEOREM 4.3 (a) (b) (c) (d) (e) (f) (g) Some Inverse Transform

  19. Example 1 Find the inverse transform of (a) (b) Solution(a)(b)

  20. L -1 is also linear • We can easily verify that(1)

  21. Example 2 Find Solution(2)

  22. Example 3: Partial Fraction Find SolutionUsing partial fractionsThen (3)If we set s = 1, 2, −4,then

  23. Example 3 (2) (4)Thus (5)

  24. Uniqueness of L -1 • Suppose that the functions f(t) and g(t) satisfy the hypotheses of Theorem 4.2, so that their Laplace transform F(s) and G(s) both exist. If F(s)=G(s) for all s>c (for some c), then f(t)=g(t) whenever on [0, + ) both f and g are continuous.

  25. Transform of Derivatives , s>0 (6) , s>0 (7) (8)

  26. THEOREM 4.4 If are continuous on [0, ) and are of Exponential order and if f(n)(t) is piecewise-continuous On [0, ), thenwhere Transform of a Derivative

  27. Solving Linear ODEs • Then(9)(10)

  28. We have(11)where

  29. Example 4: Solving IVP Solve Solution(12)(13)

  30. Example 4 (2) We can find A = 8, B = −2, C = 6Thus

  31. Example 5 Solve Solution(14)Thus

  32. 4.3 Translation Theorems THEOREM 4.5 Proof If f is piecewise continuous on [0, ) and of exponential order, then limsL{f} = 0. Behavior of F(s) as s → 

  33. THEOREM 4.6 ProofL{eatf(t)} = e-steatf(t)dt = e-(s-a)tf(t)dt = F(s – a): replacing all s in F(s) by s-a If L{f} = F(s) and a is any real number, then L{eatf(t)} = F(s – a), See Fig 4.10. Translation on the s-axis

  34. Fig 4.10

  35. Example 1 Find the L.T. of(a) (b) Solution(a)(b)

  36. Inverse Form of Theorem 4.6 • (1)where

  37. Parttial Fraction • To perform the inverse transform of R(s)=P(s)/Q(s): • Rule 1: Linear Factor Partial Fractions • Rule 2: Quadratic Factor Partial Fractions

  38. Example 2 Find the inverse L.T. of(a) (b) Solution(a) we have A = 2, B = 11 (2)

  39. Example 2 (2) And(3)From (3), we have(4)

  40. Example 2 (3) (b) (5) (6) (7)

  41. Example 3 Solve Solution

  42. Example 3 (2) • (8)

  43. Example 4 Solve Solution

  44. Example 4 (2)

  45. DEFINITION 4.3 The Unit Step Function U(t – a) defined for is Unit Step Function See Fig 4.11.

  46. Fig 4.11

  47. Fig 4.12 Fig 4.13 • Fig 4.12 shows the graph of (2t – 3)U(t – 1).Considering Fig 4.13, it is the same as f(t) = 2 –3U(t – 2) +U(t – 3),

  48. Also a function of the type(9)is the same as(10)Similarly, a function of the type(11)can be written as (12)

  49. Example 5 Express in terms of U(t). See Fig 4.14. SolutionFrom (9) and (10), with a = 5, g(t) =20t, h(t) = 0 f(t) =20t – 20tU(t – 5)

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