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CH6_Laplace_Transform(100.5)

# CH6_Laplace_Transform(100.5) - Chapter Chapter 6 Laplace...

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Chapter 6 1 Laplace Transforms

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The Laplace transformation is a method for solving D.E and corresponding initial and boundary value problem. The process of sol. consists of three main steps: 1st step: The given hard problem is transformed into a simple eqn. (subsidiary eqn.) 2 2nd step: The subsidiary eqn. is solved by purely algebraic manipulations. 3rd step: The sol. of the subsidiary eqn. is transformed back to obtain the sol. of the given problem.
In this way the L.T. reduces the problem of solving a D.E. to an algebraic problem . L.T. is particularly useful in problems where the during force the discontinuities, for instance, acts for short time only, or is periodic but is not merely a sine or cosine function . 3 Another advantage is that it solves problem directly. Indeed, initial value problems are solved without first determining a general sol . Similarly, nonhomogeneous eqns. Are solved without first solving the corresponding homo. eqn .

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6.1 Laplace Transform Inverse Transform Linearity Let f ( t ) be a given function which is defined for all . We multiply f ( t ) by e st and integrate with respect to t from zero to infinity. Then if the resulting integral exists, it is a function of s , say 0 t § 4 F ( s ) ; dt t f e s F st ) ( ) ( 0
The function F ( s ) is called the Laplace transform of the original function f ( t ) , and will be denoted by L ( f ) . Thus Furthermore, the original function f ( t ) in (1) is ) 1 ( ) ( ) ( ) ( 0 dt t f e f s F st L 5 called the inverse transform or inverse of F ( s ) and will be denoted by L -1 ( F ) ; i.e. ) ( ) ( 1 F t f L

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Example 1: Example 2: 0 s for 1 1 ) ( 0 when 1 ) ( Let 0 0 s s e dt e s F t f st st contant. a is , 0 , ) ( a t e t f at 6 a s for 1 F(s) 0 ) ( 0 a s s a e dt e e t s a st at
[Theorem1]: (Linearity of the Laplace transformation) For every functions f ( t ) and g ( t ) whose Laplace transform exists and any constants a and b we have g(t) b f(t) a t bg t af L L L ) ( ) ( 7 [pf]: By the definition, g(t) b f(t) a dt t g e b dt t f e a dt t bg t af e t bg t af st st st L L L 0 0 0 ) ( ) ( ) ( ) ( ) ( ) (

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Example 3: Ex: 0 a s , 1 2 1 1 2 1 2 ) ( F(s) 0 t 2 ) ( cosh ) ( 2 2 a s s a s a s e e e e at t f at at at at L  0 , ) ( t t t f a 8  0 s , 0 1 , ) 1 ( 1 ) ( ) ( 0 1 0 0 a s a du e u a s s du e s F sdt du st u Let dt e t t s F a u a u a s u st a a L
•Note that : The Gamma function Γ ( α ) is defined by the integral   0 1 ) 3 Appendix See ( 1 ) 0 ( t dt t e 9  1 1 ! 1 Integer If n n n s n s n t n L

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If f ( t ) =u ( t ) +iv ( t ) is a complex-valued function of real t
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