Nothing Special   »   [go: up one dir, main page]
Scribd Logo
Upload

Welcome to Scribd!

  • 17 views

    Inverse Transform

    Uploaded by

    Lief Deltora
    The document discusses the inverse Laplace transform and methods for evaluating inverse Laplace transforms. It provides examples of taking the inverse Laplace transform of various functions. It also discusses using partial fractions and Laplace transforms to solve differential equations, providing examples of using Laplace transforms to solve first-order and second-order initial value problems.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd

    Inverse Transform

    Uploaded by

    Lief Deltora
    17 views11 pages
    The document discusses the inverse Laplace transform and methods for evaluating inverse Laplace transforms. It provides examples of taking the inverse Laplace transform of various functions. It also discusses using partial fractions and Laplace transforms to solve differential equations, providing examples of using Laplace transforms to solve first-order and second-order initial value problems.

    Original Description:

    Original Title

    6. Inverse Transform.pptx

    Copyright

    © © All Rights Reserved

    Available Formats

    PPTX, PDF, TXT or read online from Scribd

    Did you find this document useful?

    Is this content inappropriate?

    The document discusses the inverse Laplace transform and methods for evaluating inverse Laplace transforms. It provides examples of taking the inverse Laplace transform of various functions. It also discusses using partial fractions and Laplace transforms to solve differential equations, providing examples of using Laplace transforms to solve first-order and second-order initial value problems.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    17 views11 pages

    Inverse Transform

    Uploaded by

    Lief Deltora
    The document discusses the inverse Laplace transform and methods for evaluating inverse Laplace transforms. It provides examples of taking the inverse Laplace transform of various functions. It also discusses using partial fractions and Laplace transforms to solve differential equations, providing examples of using Laplace transforms to solve first-order and second-order initial value problems.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    of 11

    The Laplace Transform

    Inverse Laplace Transform


    The Inverse Problem
    If F(s) represents the Laplace transform of a function f(t); that is
    ℒ{f(t)} = F(s), we then say that f(t) is the Inverse Laplace Transform of
    F(s) and write that f(t) = ℒ-1{F(s)}.

    For example,

    ℒ-1   1
    1
    s
    ℒ-1 s1  t
    2 ℒ-1   e
    1
    s 3
    3t
    Some Inverse Transforms
    =

    = = =

    = =

    = =

    When evaluating inverse transforms, it often happens that a function of s


    under consideration does not match exactly the form of Laplace transform F(s)
    given in the table. It may be necessary to “fix up” the function of s by multiplying
    and dividing by an appropriate constant.
    Example 1
    Evaluate ℒ-1{1/s5}

    Ans: f(t) = t4/24


    Example 2
    Evaluate ℒ-1{1/(s2 + 7)}

    7 sin 7t
    Ans: f(t) =
    7
    Example 3
     2s  6 
    Evaluate ℒ 
    -1 
     s 4 
    2

    Ans: f(t) = -2 cos 2t + 3 sin 2t


    Partial Fractions
    Partial fractions play an important role in finding inverse Laplace
    Transforms. We will review some of the basic algebra in the important
    cases in which the denominator of a Laplace Transform F(s) contains
    distinct linear factors, repeated linear factors, and quadratic
    polynomials with no real factors.

    For example:

    F(s) = 3/(s2 - 3s + 2); F(s) = 3/(s2 - 4s + 4); F(s) = 3/[(s2 +4)(s-3)]


    Example 4
     s 2  6s  9 
    Evaluate ℒ-1  ( s  1)( s  2)( s  4) 
     

    16et 25e2t e4t


    Ans: f(t) =   
    5 6 30
    Solving ODE's
    REMEMBER: Transform of a Derivative

    ℒ{f'(t)} = sF(s) - f(0)

    Similarly, ℒ{f''(t)} = sℒ{f'(t)} - f'(0) = s2F(s) - sf(0) - f'(0)

    In like manner, we can show that


    ℒ{f(n)(t)} = snF(s) - sn-1f(0) - sn-2f'(0) - ... - sf(n-2)(0) - f(n-1)(0)
    Example 5
    Use Laplace transform to solve the first-order IVP
    dy/dt + 3y = 13 sin 2t, y(0) = 6

    Ans: y(t) = 8 e-3t - 2 cos 2t + 3 sin 2t


    Seatwork
    Use Laplace transform to solve the 2nd-order IVP
    y'' - 3y' + 2y = e-4t, y(0) = 1, y'(0) = 5

    16et 25e2t e4t


    Ans: y(t) =   
    5 6 30

    You might also like