Laplace transform (TOÁN kỹ THUẬT SLIDE)

Chap Laplace Transform Outline ■ Basic Concepts ■ Laplace Transform ■ Definition, Theorems, Formula ■ Inverse Laplace Transform ■ Definition, Theorems, Formula ■ Solving Differential Equation ■ Solving Integral Equation Page Basic Concepts 微微微微微 微微微微微 Laplace Transform Differential Equation f(t) L{ f(t)} = F(s) Algebra Equation F(s) Inverse Laplace Transform Solution of Differential Equation f(t) L-1{F(s)} = f(t) Solution of Algebra Equation F(s) Page Basic Concepts Laplace Transform y ′ − y ′ + y = 4t y (0) = 1, y′(0) = −1 L{ f(t)} = F(s) s3 − 4s2 + F ( s) = s − 3s + s Inverse Laplace Transform y = f (t ) = + 2t − e t − e t L-1{F(s)} = f(t) −1 −1 F ( s) = + + + s s s −1 s − Page Laplace Transform ■ Definition The Laplace transform of a function f(t) is defined as ■ ■ Converges: F ( s ) =L{f(t)} L { f exists (t )} = ∫ ∞ e − st f (t )dt Diverges: L{f(t)} does not exist Page Laplace Transform s=0.125 e-st s=0.25 s=0.5 s=1 s=2 s=4 s=8 t Page Laplace Transform ■ Example : Find L{ } Sol: L{1} = = ∫ ∞ ∫ ∞ 0 − st e 1dt e − ( s )t dt − ( s )t ∞ e = −s = s Page Laplace Transform ■ Example : Find L{ eat } Sol: ∞ L {e } = ∫ e e dt at − st at ∞ =∫ e −( s − a ) t −( s − a ) t dt ∞ e = − (s − a) = s−a Page Laplace Transform ■ Example 4-2 : Find L{ tt } Sol: ∞ L {t } = ∫ e t dt t − st t =∞ ∴L{ tt } does not exist Page Laplace Transform ■ Exercise 4-1 : ■ Find L{2t + 6} ■ Find L {sin πt} ■ Find L {(at + b)2 } ■ Find L {eat +b } Page 10 ■ Example Find the inverse transform of the function w ln(1 + ) s Page 39 Convolution Integration Equation ■ Convolution t ■ Properties f (t ) ∗ g (t ) = ∫ f (τ )g (t − τ )dτ ■ ■ f ∗g = g∗ f ■ f ∗ ( g1 + g ) = f ∗ g1 + f ∗ g ■ ( f ∗ g ) ∗ h = f ∗ ( g ∗ h) f ∗0 = 0∗ f = Page 40 ■ Example1 Using the convolution, find the inverse h(t) of H ( s) ■ Example = (s2 H ( s) + 1) = s3 ■ Example H ( s) = , find h ( t ) s (s − a) Page 41 Laplace Transform L { f (t ) ∗ g (t )} = F ( s )G ( s ) ■ Example 4-7 : Prove Proof: ∞ L { f (t ) ∗ g (t )} = ∫ e =∫ ∞ =∫ ∞ 0 ∫ t ∫ ∞ 0 − st ∫ t f (τ )g (t − τ )dτdt e − st f (τ )g (t − τ )dτdt e − s ( v +τ ) f (τ )g ( v )dτdv, Let v = t − τ ∞ ∞ − sτ − sv    = ∫ e f (τ )dτ ∫ e g ( v )dv      = F ( s )G ( s ) Page 42 Differential Equation y′′ + ay′ + by = r (t ) y (0) = 0, y ' (0) = ( s + as + b) L( y ) = L(r ) let Q ( s ) = /( s + as + b), R( s ) = L( r ) L( y ) = Q ( s ) R( s ) y (t ) = t ∫ q(t − τ )r(τ )dτ Page 43 Integration Equations y (t ) = t + ■ Example ∫ t y (τ ) sin(t − τ )dτ t y (t ) = t + ∫ y (τ ) sin(t − τ )dτ = t + y ∗ sin t Y = L { y (t )} = L {t + y ∗ sin t} 1 = +Y s s +1 s2 + 1 Y = = 2+ 4 s s s 1 t ∴ y (t ) = L-1{Y } = L-1{ } + L-1{ } = t + s s Page 44 Homeworks ■ Section 5-4 ■ #1,#13 ■ Section 5-5 ■ #7, #14, #27 Page 45 Laplace Transform ■ Formula f(t) t n , n = 1,2,3, t , p > −1 p e at cos ωt sin ωt F(s) = L {f(t)} s n! s n +1 Γ( p + 1) s p +1 s−a s s2 + ω ω s2 + ω Page 46 Laplace Transform ■ Formula f(t) cosh ωt sinh ωt e cos ωt at e sin ωt at t n e at , n = 1,2, t p e at , p > −1 F(s) = L {f(t)} s s2 − ω ω s2 − ω s−a ( s − a )2 + ω ω ( s − a )2 + ω n! ( s − a )n +1 Γ( p + 1) ( s − a ) p +1 Page 47 Inverse Laplace Transform ■ Definition The Inverse Laplace Transform of a function F(s) is defined as a +i∞ st f (t ) = L {F ( s )} = e F ( s )ds ∫ 2πi a −i∞ -1 Page 48 Inverse Laplace Transform ■ Theorems Theorem Inverse Laplace Transform Linear Property Derivatives Integrals First Shifting Property Second Shifting Property Description a +i∞ st -1 f (t ) = L {F ( s )} = e F ( s )ds ∫ a − i ∞ 2πi L-1{aF ( s ) + bG ( s )} = af (t ) + bg (t ) L-1{s n F ( s) − s n−1 f (0) −  − f ( n−1) (0)} = f ( n ) (t ) t -1 L { F ( s )} = ∫ f (τ )dτ s L-1{F ( s − a )} = e at f (t ) L-1{e − as F ( s )} = f (t − a )u(t − a ) Page 49 Inverse Laplace Transform ■ Theorems Theorem Change of Scale Property Multiplication by tn Division by t Unit Impulse Function Unit Step Function Convolution Theorem Description s L-1{F ( )} = af ( at ) a L-1{F ( n ) ( s )} = ( −1) n t n f (t ) ∞ L {∫ F (u )du} = -1 s f (t ) t L-1{e − as } = δ (t − a ) e − as L { } = U (t − a ) s -1 L-1{F ( s )G ( s )} = f (t ) ∗ g (t ) Page 50 Inverse Laplace Transform ■ Formula F(s) s s2 s n +1 , n = 0,1,2, s−a s s2 + ω ω s2 + ω f(t) = L -1{F(s)} t tn n! at e cos ωt sin ωt Page 51 Inverse Laplace Transform ■ Formula F(s) s s2 − ω ω s2 − ω s−a ( s − a )2 + ω ω ( s − a )2 + ω n! ( s − a )n +1 Γ( p + 1) ( s − a ) p +1 f(t) = L -1{F(s)} cosh ωt sinh ωt e cos ωt at e sin ωt at t n e at , n = 1,2, t p e at , p > −1 Page 52 Solving Differential Equation y ′′ + ay ′ + by = r (t ) Let Y = L { y (t )} [ s Y − sy (0) − y′(0)] + a[ sY − y (0)] + bY = R( s) ⇒ ( s + as + b)Y = ( s + a ) y (0) + y ′(0) + R ( s ) ( s + a ) y (0) + y ′(0) R( s) ⇒Y = + 2 ( s + as + b) ( s + as + b) Assume Q ( s ) = ( s + as + b) ⇒ Y = [ ( s + a ) y (0) + y ′(0]Q ( s ) + R ( s )Q ( s ) Page 53 ... exist Page Laplace Transform ■ Exercise 4-1 : ■ Find L{2t + 6} ■ Find L {sin πt} ■ Find L {(at + b)2 } ■ Find L {eat +b } Page 10 Laplace Transform ■ Theorems Theorem Definition of Laplace Transform. .. Inverse Laplace Transform ■ Definition The Inverse Laplace Transform of a function F(s) is defined as a +i∞ st f (t ) = L {F ( s )} = e F ( s )ds ∫ 2πi a −i∞ -1 Page 48 Inverse Laplace Transform. .. L(f) from L(1) ■ Example 2: Derive the Laplace transform of cos wt Page 20 Differential Equations, Initial Value Problem ■ How to use Laplace transform and Laplace inverse to solve the differential