Boye/DiPrima 9 h ed, Ch 6.: Definiion of Laplae Transform Elemenary Differenial Equaions and Boundary Value Problems, 9 h ediion, by William E. Boye and Rihard C. DiPrima, 2009 by John Wiley & Sons, In. Many praial engineering problems involve mehanial or elerial sysems aed upon by disoninuous or impulsive foring erms. For suh problems he mehods desribed in Chaper 3 are diffiul o apply. In his haper we use he Laplae ransform o onver a problem for an unknown funion f ino a simpler problem for F, solve for F, and hen reover f from is ransform F. Given a known funion Ks,, an inegral ransform of a funion f is a relaion of he form F s K s, f d,
Improper Inegrals The Laplae ransform will involve an inegral from zero o infiniy. Suh an inegral is a ype of improper inegral. An improper inegral over an unbounded inerval is defined as he limi of an inegral over a finie inerval d lim a f f d A where A is a posiive real number. If he inegral from a o A exiss for eah A > a and if he limi as A exiss, hen he improper inegral is said o onverge o ha limiing value. Oherwise, he inegral is said o diverge or fail o exis. a A
Example Consider he following improper inegral. 0 e d
Example 2 Consider he following improper inegral. / d.
Example 3 Consider he following improper inegral. p From Example 2, his inegral diverges a p = We an evaluae his inegral for p as follows: A p lim p d d lim A A A p The improper inegral diverges a p = and If If p, p, lim A lim A d p p p A p p A p
Pieewise Coninuous Funions A funion f is pieewise oninuous on an inerval [a, b] if his inerval an be pariioned by a finie number of poins a = 0 < < < n = b suh ha f is oninuous on eah k, k+ 2 lim k f, k 0,, n 3 lim k f, k,, n In oher words, f is pieewise oninuous on [a, b] if i is oninuous here exep for a finie number of jump disoninuiies.
The Laplae Transform Le f be a funion defined for 0, and saisfies erain ondiions o be named laer. The Laplae Transform of f is defined as an inegral ransform: s L f F s e f d The kernel funion is Ks, = e -s. Sine soluions of linear differenial equaions wih onsan oeffiiens are based on he exponenial funion, he Laplae ransform is pariularly useful for suh equaions. Noe ha he Laplae Transform is defined by an improper inegral, and hus mus be heked for onvergene. On he nex few slides, we review examples of improper inegrals and pieewise oninuous funions. 0
Theorem 6..2 Suppose ha f is a funion for whih he following hold: f is pieewise oninuous on [0, b] for all b > 0. 2 f Ke a when M, for onsans a, K, M, wih K, M > 0. Then he Laplae Transform of f exiss for s > a. L s f F s e f d finie 0 Noe: A funion f ha saisfies he ondiions speified above is said o o have exponenial order as.
Example 4 Le f = for 0. Then he Laplae ransform Fs of f is:
Example 5 Le f = e a for 0. Then he Laplae ransform Fs of f is:
Example 6 Consider he following pieewise-defined funion f, f k, 0 0 where k is a onsan. This represens a uni impulse. Noing ha f is pieewiseoninuous, we an ompue is Laplae ransform s s s e L { f } e f d, 0 0 e d s 0 s Observe ha his resul does no depend on k, he funion value a he poin of disoninuiy.
Example 7 Le f = sina for 0. Using inegraion by pars wie, he Laplae ransform Fs of f is found as follows:
Lineariy of he Laplae Transform Suppose f and g are funions whose Laplae ransforms exis for s > a and s > a 2, respeively. Then, for s greaer han he maximum of a and a 2, he Laplae ransform of f + 2 g exiss. Tha is, wih finie is 0 2 2 d g f e g f L s 2 0 2 0 2 g L f L d g e d f e g f L s s
Example 8 Le f = 5e -2-3sin4 for 0.