VIEWS: 20 PAGES: 3 CATEGORY: Lifestyle POSTED ON: 5/25/2010 Public Domain
L17 The Inverse Laplace Transform (Section 7.4) Inverse Laplace Transform Definition: Given a function F ( s ) , if there is a function f ( t ) that The Laplace transform is defined by is continuous on [0, +∞) and satisfies ∞ L { f }( s ) = ∫ e − st f ( t ) dt , L {f}= F, 0 Then we say that f ( t ) is the inverse Laplace transform of F ( s ) that is, L : f ( t ) → F ( s ) , and use the notation then the inverse Laplace transform, L −1 : F ( s ) → f ( t ) . f = L −1{ F } . Example: y′′ + y = t , y ( 0 ) = 0, y ′ ( 0 ) = 1. To determine the inverse Laplace transform of a given function F ( s ) , we refer to the tables. Example: Determine L −1{ F } , where 6 (a) F ( s ) = 4 s s (b) F ( s ) = 2 s + 25 2 (c) F ( s ) = 2 s − 2s + 5 The linearity of the inverse Laplace transform L −1 is inherited from the linearity of the operator L. Note: y ( t ) = t is not the only function whose Laplace transform is 1 ⎧t , t ≠ 3 1 Theorem: Assume that L −1{ F } , L −1{ F1} , and L −1{ F2 } exist and , for instance, if g ( t ) = ⎨ , then L{ g} = 2 . Thus, the ⎩0, t = 3 2 s s are continuous and let c be any constant. Then, inverse Laplace transform is not defined uniquely, but among all (1) L −1{ F1 + F2 } = L −1{ F1} + L −1{ F2 } those functions there is only one which is continuous on [0, +∞ ) . (2) L −1{cF } = c L −1{ F } 132 133 ⎧ 3 2s 4 ⎫ The Inverse Laplace Transform of Rational Functions Example: Determine L −1 ⎨ + 2 + 2 ⎬ ⎩ s − 5 s + 16 s + 2s + 5 ⎭ P(s) R(s) = , where P and Q are polynomials and deg P < deg Q. Q(s) Method of Partial Fractions 1. Nonrepeated linear factors 2. Repeated linear factors 3. Quadratic factors Case 1. Nonrepeated linear factors: ⎧ 7 ⎫ ⎪ ⎪ Q ( s ) = ( s − r1 )( s − r2 ) ...( s − rn ) , ri ≠ rj (i ≠ j ) . Example: Determine L −1 ⎨ 4⎬ ⎪ ( s − 3) ⎪ ⎩ ⎭ The partial fraction expansion has the form P(s) A A A = 1 + 2 + ... + n Q ( s ) s − r1 s − r2 s − rn 2s − 1 Example: Determine L −1{ F } , where F ( s ) = . ( s − 1)( s + 2 )( s + 3) ⎧ 2s − 3 ⎫ Example: Determine L −1 ⎨ 2 ⎬ ⎩ s − 4 s + 20 ⎭ 134 135 Case 2. Repeated linear factors: Case 3. Quadratic factors: P(s) P(s) P(s) P(s) = = , Q ( s ) Q1 ( s )( s − r )m Q ( s ) Q ( s ) ⎡( s − α ) 2 + β 2 ⎤ m 1 ⎣ ⎦ The portion of partial fraction expansion that corresponds to where ( s − α ) + β cannot be reduced to a product of linear 2 2 ( s − r ) is m factors. A1 A2 Am + + ... + The portion of the partial fraction expansion that corresponds to s − r (s − r) 2 (s − r) m [( s − α ) + β 2 ]m is 2 3s 2 + 4s + 7 C1s + D1 C2 s + D2 Cm s + Dm Example: Determine L −1{ F } , where F ( s ) = . + + ... + . ( s − 2 )( s + 1) 2 ( s − α ) + β [( s − α ) + β ] [( s − α ) + β 2 ]m 2 2 2 2 2 2 It is more convenient to express it in the equivalent form A1 ( s − α ) + β B1 A2 ( s − α ) + β B2 A ( s − α ) + β Bm + + ... + m . (s −α ) + β [( s − α ) + β ] [( s − α ) + β 2 ]m 2 2 2 2 2 2 7 s 2 − 41s + 84 Example: Determine L −1{ F } , where F ( s ) = . ( s − 1) ( s 2 − 4s + 13) 136 137