3.  Óôïé÷åßá ãéá ôïí ìåôáó÷çìáôéóìü Laplace êáéôéò åöáñìïãÝò ôïõ óôçí åðßëõóç äéáöïñéêþí åîéóþóåùí.

Á.1.   BáóéêÝò Ýííïéåò

     Óôçí ðñïçãïýìåíç åíüôçôá áõôÞ, ðáñáôçñÞóáìå ìéá áíôéóôïéxßá  ìåôáîý ãñáììéêþí äéáöïñéêþí åîéóþóåùí êáé áëãåâñéêþí åîéóþóåùí (ìÝóù ôçò xáñáêôçñéóôéêÞò  åîßóùóçò,  ç ïðïßá ðñïêýðôåé êáôÜ  ôçí åðßëõóç ôùí ãñáììéêþí äéáöïñéêþí åîéóþóåùí 2  çò ôÜîçò). 
O ìåôáóxçìáôéóìüò Laplace  xñçóéìïðïéåßôáé ãéá ôïí ìåôáóxçìáôéóìü ìéáò ãñáììéêÞò äéáöïñéêÞò åîßóùóçò ìå óôáèåñïýò óõíôåëåóôÝò óå áëãåâñéêÞ åîßóùóç ç ïðïßá åðéëýåôáé .  Óôç óõíÝxåéá, ç ëýóç ôçò äéáöïñéêÞò åîßóùóçò õðïëïãßæåôáé ìå  åöáñìïãÞ ôïõ ëåãüìåíïõ áíôßóôñïöïõ ìåôáóxçìáôéóìïý Laplace.

Á.  Ïñéóìüò ôïõ Ìåôáóxçìáôéóìïý Laplace-Báóéêü èåþñçìá ' Õðáñîçò ôïõ Ìåôáóxçìáôéóìïý

Ï ìåôáóxçìáôéóìüò  Laplace åßíáé åßíáé Ýíáò ìåôáóxçìáôéóìüò  L ï ïðïßïò åöáñìüæåôáé óå ìéá óõãêåêñéìÝíç êáôçãïñßá óõíáñôÞóåùí f (t)   ïé ïðïßåò ïñßæïíôáé óôï äéÜóôçìá 0<t<+∞

 Ï ïñéóìüò ãßíåôáé ìÝóù ôïõ  áüñéóôïõ ïëïêëçñþìáôïò                                                
                                       L (f (t)) (s) = F (s) =  (∫_0^(+∞))f (t) (e^(-st)) \[DifferentialD]t = 
              (lim_(R -> +∞))   (∫_0^R) 
                    f (t) (e^(-st)) \[DifferentialD]t        (LT)                                                          
                                                                 
   ÈÅÙÑÇÌÁ :  ÕðïèÝôïõìå üôé ç óõíÜñôçóç  f (t) 

        (É)   åßíáé êáôÜ ôìÞìáôá óõíåxÞò óôï äéÜóôçìá 0 ≤ t ≤ 
          A,  ãéá êÜèå èåôéêü áñéèìü Á . 
(ÉÉ)    | 
          f (t) | ≤K(E^at)   ãéá t≥M,  üðïõ Ê, a, 
        M ,  åßíáé ðñáãìáôéêÝò óôáèåñÝò êáé Ê, Ì > 0.  

             Tüôå ï ìåôáóxçìáôéóìüò   Laplace 

                                                                                       L (f (t)) (s) = 
          F (s), 

ï oðïßïò ïñßæåôáé ìÝóù ôïõ (LT) ,    õðÜñxåé ãéá s > a.

ÐáñÜäåéãìá 1:  Ãéá ðïéÝò ôéìÝò ôïõ s, ôï ïëïêëÞñùìá 
                                                                                   L (1) (s) = (∫_0^(+∞))(E^(-st)) \[DifferentialD]t  ,                            
óõãêëßíåé .  'Ïôáí ôï ïëïêëÞñùìá óõãêëßíåé õðïëïãßóôå ôçí ôéìÞ ôïõ.

Ëýóç:  Óôçí ðåñßðôùóç üðïõ s=0, ôï ïëïêëÞñùìá ðñïöáíþò áðïêëßíåé. 
'Åôóé ãéá ôçí ðåñßðôùóç s≠0  Ýxïõìå

                                                           (∫_0^R) (E^(-st)) \[DifferentialD]t  =((1 - (E^(-sR)))/s).

               Óôçí ðåñßðôùóç üðïõ s<0, ôï ïëïêëÞñùìá áðïêëßíåé, åíþ óôçí ðåñßðôùóç  üðïõ s > 0, ôï ïëïêëÞñùìá óõãêëßíåé êáé âñßóêåôáé   

                                                               (∫_0^(+∞))(E^(-st)) \[DifferentialD]t   = (lim_(R -> +∞))  ((1 - (E^(-sR)))/s) = (1/s)   , s > 0. 

ÓõìðåñÜéíïõìå üôé ãéá s>0, ôï ïëïêëÞñùìá õðÜñxåé êáé ï ìåôáóxçìáôéóìüò Laplace åßíáé 

                                                                                                  L (1) (s) = (1/s)   ,s>0

ÐáñÜäåéãìá 2: ' Åóôù b ≠ 
              0.  Ãéá ðïéÝò ôéìÝò ôïõ s, ôï ïëïêëÞñùìá 
                                                                                   L (sinbt) (s) = (∫_0^(+∞))(E^(-st)) 
              sinbt \[DifferentialD]t  ,                              (1)   
óõãêëßíåé .  'Ïôáí ôï ïëïêëÞñùìá óõãêëßíåé õðïëïãßóôå ôçí ôéìÞ ôïõ.

Ëýóç: Óôçí ðåñßðôùóç üðïõ s=0, Ýxïõìå

                                                          (∫_0^R) 
              sinbt \[DifferentialD]t = ((1 - 
                  cosbR)/b)   .     (2)

Ôï äåîß ìÝëïò ôçò (2),  äåí Ýxåé üñéï êáèþò R -> +∞ .  
Óôçí ðåñßðôùóç üðïõ s ≠ 
            0,  åöáôìüæïíôáò  ïëïêëÞñùóç êáôÜ ðáñÜãïíôåò äõï öïñÝò,  âñßóêïõìå áðü ôç (2)   üôé ãéá s < 0 
ôï ïëïêëÞñùìá áðïêëßíåé åíþ ãéá s > 0,

                                                          (∫_0^(+∞))(E^(-st)) sinbt \[DifferentialD]t =(lim_(R -> +∞))(∫_0^R) (E^(-st)) sinbt \[DifferentialD]t
                                                                                              =(lim_(R -> +∞))(-(E^(-st)) (((s sinbt + b cosbt)/((s^2) + (b^2))))(Subscript[|, 0]^R) )
                                                                                            = (lim_(R -> +∞))(-(E^(-sR))(((s sinbR + b cosRt)/((s^2) + (b^2))))+(b/((s^2) + (b^2))))
                                                                                         = (b/((s^2) + (b^2))), s > 0. 
 ÔåëéêÜ óõìðåñÜéíïõìå üôé ãéá s > 0, ôï ïëïêëÞñùìá õðÜñxåé êáé ï ìåôáóxçìáôéóìüò Laplace åßíáé 
                                                                                                  
                                                                                 L (sinbt) (s) = (b/((s^2) + (b^2))), s>0.

Måôáóxçìáôéóìüò Laplace êáé ÌATHEMATICA.   To ÌÁÔÇÅÌÁÔÉCA äéáèÝôåé åíôïëÝò  êáé ìéá ðïëý éóxõñÞ âéâëéïèÞêç ãéá ôçí åýñåóç ôïõ ìåôáóxçìáôéóìïý Laplace .  Äßíïõìå ìåñéêÜ  ðñþôá ðáñáäåßãìáôá :

Ãéá ôéò óõíáñôÞóåéò   f (t) =  
          Cosbt,  f (t) = Sinbt,  f (t) = 1,  
        f (t) = Sinhbt,  
        f (t) = 
          Coshbt,   âñßóêïõìå

LaplaceTransform[Cos[b*t], t, s]

LaplaceTransform[Sin[b*t], t, s]

LaplaceTransform[1, t, s]

LaplaceTransform[Sinh[b*t], t, s]

LaplaceTransform[Cosh[b*t], t, s]

LaplaceTransform[(E^(-t)), t, 
      s]

LaplaceTransform[(E^t), t, 
      s]

Óôçí åðßëõóç äéáöïñéêþí åîéóþóåùí ìå xñÞóç ôïõ ìåôáóxçìáôéóìïý Laplace,  
      xñçóéìïðïéïýìå ôá áêüëïõèá ðïëý âáóéêÜ áðïôåëÝìáôá

Â.  BáóéêÝò  éäéüôçôåò ôïõ ìåôáóxçìáôéóìïý Laplace

          1.   (Ãñáììéêüôçôá)   Áí f êáé g åßíáé óõíáñôÞóåéò ãéá ôéò ïðïßåò  ïé ìåôáóxçìáôéóìïß Laplace L (f) êáé L (g) õðÜñxïõí  êáé a, 
        b,  åßíáé ðáñáãìáôéêÝò óôáèåñÝò,  ôüôå ï ìåôáóxçìáôéóìüò Laplace L (af + 
              bg) õðÜñxåé êáé éóxýåé

                                                                        L (af + bg) (s) = 
        aL (f) (s) + 
            bL (g) (s) . 

2.     (ÐáñÜãùãïò)      ÕðïèÝôïõìå üôé ïé óõíáñôÞóåéò  f êáé f' åßíáé áíôßóôïéxá óõíåxÞò êáé êáôÜ ôìÞìáôá óõíåxÞò óå êÜèå ðåðåñáóìÝíï äéÜóôçìá  0 ≤ t ≤ 
          A,   êáé üôé éêáíïðïéïýí ôç óõíèÞêç (ÉÉ) ãéá êÜðïéï åêèÝôç a .  Ôüôå ï ìåôáóxçìáôéóìüò Laplace L (f') (s) õðÜñxåé 
ãéá s > 
        a  êáé éóxýåé

                                                                     L (f' (t)) (s) = 
        sL (f (t)) (s) - 
          f (0) . 

3.  (ÐáñÜãùãïò áíþôåñçò ôÜîçò)  YðïèÝôïõìå üôé ïé óõíáñôÞóåéò f, 
      f', ... (f^((n - 
                  1) )) åßíáé óõíåxåßò êáé üôé ç óõíÜñôçóç (f^((n))) åßíáé óõíåxÞò óõíåxÞò óå êÜèå ðåðåñáóìÝíï äéÜóôçìá  0 ≤ t ≤ 
        A,   êáé üôé éêáíïðïéïýí ôç óõíèÞêç (ÉÉ) ãéá êÜðïéï åêèÝôç a .   Ôüôå ï ìåôáóxçìáôéóìüò Laplace L ((f^((n)))) (s) õðÜñxåé 
ãéá s > 
        a  êáé éóxýåé

                                              L ((f^((n))) (t)) (s) = (s^n) L (f (t)) (s) - (s^(n - 1)) 
                f (0) -  ...  - (sf^((n - 
                    2))) (0) - (f^((n - 
                    1))) (0) . 

4.   (Ãéíüìåíï (t^n) 
                f (t))   YðïèÝôïõìå üôé ï ìåôáóxçìáôéóìüò Laplace ôçò óõíÜñôçóçò f (t) õðÜñxåé .  
              Tüôå ï ìåôáóxçìáôéóìüò Laplace L ( (t^n) 
                f (t)) (s)  õðÜñxåé êáé éóxýåé

                                                  L ((t^n) f (t)) (s)=(((-1))^n)((d^n)/(ds^n))(L (f (t)) (s))

ÐáñÜäåéãìá 3: ' Åóôù b≠0 êáé s>0. Õðïëïãßóôå ôï ìåôáóxçìáôéóìü  Laplace   L(tsinbt)(s)

Ëýóç: Eöáñìüæïíôáò ôçí éäéüôçôá 4.  âñßóêïõìå 

                                            L (tsinbt) (s)=-(d/ds)(L(sinbt))(s)=-(d/ds)((b/((b^2) + (s^2))))=((2  bs)/((((b^2) + (s^2)))^2)).

Xñçóéìïðïéþíôáò ôçí åíôïëÞ ôïõ Mathematica, âñßóêïõìå

LaplaceTransform[t*Sin[b*t], t, s]

ÐáñÜäåéãìá 4:  Õðïëïãßóôå ôï ìåôáóxçìáôéóìü  Laplace   L((E^at))(s), t ≥ 0

Ëýóç: Må âÜóç ôïí ïñéóìü, âñßóêïõìå
 
                                    L ((E^at)) (s)=(∫_0^(+∞))(E^(-st)) (E^at) \[DifferentialD]t = (∫_0^(+∞))(E^(-(s - a) t)) \[DifferentialD]t = 
                (1/(s - a)),  

ãéá s>a.

ÓõíÜñôçóç ÃÜììá êáé ìåôáóxçìáôéóìüò Laplace.    Óôïí õðïëïãéóìü ôïõ ìåôáóxçìáôéóìïý  Laplace ãéá ôç óõíÜñôçóç f(t)=(t^p), åìðëÝêåôáé ç óõíÜñôçóç ÃÜììá .  Ç óõíÜñôçóç ÃÜììá,  ïñßæåôáé áðü ôç óxÝóç  

                                                                                                   
                                                                    Ã (p + 1)=(∫_0^(+∞))(t^p) (e^(-t)) \[DifferentialD]t, 
üðïõ  ôï ïëïêëÞñùìá óõãêëßíåé üôáí p > -1.    

H óõíÜñôçóç ÃÜììá Ýxåé ôéò áêüëïõèåò éäéüôçôåò    
                                    
                                                                                                Ã (p + 1) = 
          pÃ (p),                                      (Ã  .1)
                                                                                                      
                                                                                                Ã (1) = 
          1                                                             (Ã  .2)

                                                                                                Ã (p + 1) = p!,   p ≠ 
          0,   öõóéêüò          (Ã  .3)

   H óõíÜñôçóç Ã (p),    áðïôåëåß  ãåíßêåõóç ôïõ ðáñáãïíôéêïý
(èåùñïýìå ôç óýìâáóç p != ∞,  óôçí ðåñßðôùóç üðïõ p åßíáé áñíçôéêüò áêÝñáéïò) .  ÓxåäiÜæïõìå  ìå ôç âïÞèåéá ôïõ MATHEMATICA  ôç ãñáöéêÞ ðáñÜóôáóç ãéá  -4.5<p<5.

Plot[Gamma[p], {p, -4.5, 5}, 
      PlotStyle -> {RGBColor[0, 0, 
            1]}]

Må ôç âïÞèåéá ôçò óõíÜñôçóçò ÃÜììá,  õðïëïãßæïõìå ôï ìåôáóxçìáôéóìü Laplace ôçò óõíÜñôçóçò f (t) = (t^p),  
      p > -1.  Ðñáãìáôïðïéþíôáò áëëáãÞ ìåôáâëçôÞò óôïí ïñéóìü ôïõ ìåôáóxçìáôéóìïý Laplace ãéá ôçí  f (t) = 
        (t^p),  Ýxïõìå ôçí éäéüôçôá

5.  (Ìåôáóxçìáôéóìüò Laplace ôçò f (t) = (t^p))       

                                    L ((t^p)) (s) = (∫_0^∞) 
                      (e^(-st)) (t^p) \[DifferentialD]t = (1/(s^(p + 1))) (∫_0^∞) 
                          (e^(-u)) (u^p) \[DifferentialD]u = ((Ã (p + 1))/(s^(p + 1))),  p>-1

Xñçóéìïðïéïýìå êáé ôçí áíôßóôïéxç åíôïëÞ ôïõ MATHEMATICA

LaplaceTransform[(t^p), t, s]

Ã.  Ïñéóìüò ôïõ Áíôßóôñïöïõ Ìåôáóxçìáôéóìïý Laplace-Báóéêü èåþñçìá ' Õðáñîçò ôïõ  Áíôßóôñïöïõ Ìåôáóxçìáôéóìïý

Ãéá ìéá óõíÜñôçóç F (s),  õðÜñxåé ôï ðïëý ìéá  óõíåxÞò óõíÜñôçóç f (t),  
      t ≥ 
        0,  ãéá ôçí ïðïßá éóxýåé   L (f (t)) (s) = 
        F (s)        
Áí õðÜñxåé,  ç ìïíáäéêÞ óõíåxÞò óõíÜñôçóç  ãéá ôçí ïðïßá éóxýåé  L (f (t)) (s) = F (s),  
        oíïìÜæåôáé áíôßóôñïöïò ìåôáóxçìáôéóìüò  Laplace ôçò F (s) .  Óõìâïëßæïõìå ìå 

                                                                                                f (t) = (L^(-1)) (F (s)).

Ç ìÝèïäïò åðßëõóçò äéáöïñéêþí åîéóþóåùí ìå xñÞóç ôïõ ìåôáóxçìáôéóìïý Laplace,  èá ìðïñïýóå íá ðåñéãñáöåß ìå ôá áêüëïõèá âÞìáôá:

Ä.     ÌÝèïäïò åðßëõóçò äéáöïñéêþí åîéóþóåùí ìå xñÞóç  ôïõ ìåôáóxçìáôéóìïý Laplace

ÂÇÌÁ 1.    Ìå åãáñìïãÞ ôïõ ìåôáóxçìáôéóìïý Laplace ìåôáôñÝðïõìå ôç äéáöïñéêÞ åîßóùóç óå áëãåâñéêÞ åîßóùóç

ÂÇÌÁ 2.   Åðéëýïõìå ôçí áëãåâñéêÞ åîßóùóç

ÂÇÌÁ 3.   Åöáñìüæïõìå ôïí áíôßóôñïöï ìåôáóxçìáôéóìü Laplace óôç ëýóç ôçò áëãåâñéêÞò åîßóùóçò,  ï ïðïßïò áðïäßäåé ôç ëýóç ôçò äéáöïñéêÞò åîßóùóçò

ÐáñÜäåéãìá 5: Íá åðéëõèåß ôï ðñüâëçìá áñxéêþí ôéìþí

                                                                        {({{\(\(y\^''\) - \(y\^'\) - 2  y = 0\)}, {\(y \((0)\) = 1, \ \(y\^'\) \((0)\) = 0\)}})
Ëýóç:    ÂÇÌÁ 1:  Åöáñìüæïõìå ôï ìåôáóxçìáôéóìü Laplace óôç äéáöïñéêÞ åîßóùóç .  Áðü ôçí éäéüôçôá 1. , ëáìâÜíïõìå

                                                                            \(L \((\(y\^''\) \((t)\))\) - L \((\(y\^'\) \((t)\))\) - 
            2  L \((y \((t)\))\) = 
          0. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((1)\)\)

Xñçóéìïðïéþíôáò ôéò éäéüôçôåò 2.  êáé 3.  ãéá ôéò ðáñáãþãïõò (ãéá ôç äåýôåñç ðáñÜãùãï,  åöáñìüæïõìå ôçí éäéüôçôá 3.  ãéá n = 
            2) êáé Ýxïõìå  áðü ôçí (1)

\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\(s\^2\) L \((y)\) - sy \((0)\) - \(y\^'\) \((0)\) - {sL \((y)\) - y \((0)\)} - 
          2  L \((y)\) = 
        0\  . \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((2)\)\)\(
\)\)
Xñçóéìïðïéþíôáò ãéá äéåõêüëõíóç ôï óõìâïëéóìü L (y) = 
        Y (s),  ç (2) ãñÜöåôáé

                                                                     ((s^2) - s - 2) Y (s) + (1 - s) y (0) - y' (0) = 
        0.                                     (3)

Xñçóéìïðïéïýìå ôþñá ôéò áñxéêÝò óõíèÞêåò,  áðü ôçí (3) ëáìâÜíïõìå ôçí áëãåâñéêÞ åîßóùóç 

                                                                    ((s^2) - s - 2) Y (s) + (1 - s) = 
        0.                                                                    (4)

                        ÂÇÌÁ 2 :   Ôçí áëãåâñéêÞ åîßóùóç (4)  ïðïßá ìðïñïýìå íá  ôçí åðéëýóïõìå ùò ðñïò  Õ (s), ëáìâÜíïíôáò

                                                                             Õ (s) = ((s - 1)/((s^2) - s - 2)) = ((s - 1)/((s - 2) (s + 
                      1)))   .                                          (5)

H (5) åßíáé ï ìåôáóxçìáôéóìüò Laplace ôçò ëýóçò y(t) ôïõ ðñïâëÞìáôïò áñxéêþí  ôéìþí.

                  ÂÇÌÁ 3:   Ç ëýóç    y(t)  ôïõ ðñïâëÞìáôïò áñxiêþí ôéìþí, èá âñåèåß ìå åöáñìïãÞ ôïõ áíôßóôñïöïõ ìåôáóxçìáôéóìïý Laplace óôçí  Õ (s) .  ÄçëáäÞ  Ýxoõìå 
 y (t) = (L^(-1)) (Y (s)) .    Ãéá íá åöáñìüóïõìå ôç ãñáììéêüôçôá ôïõ áíôßóôñïöïõ ìåôáóxçìáôéóìïý,  áíáëýïõìå ôçí Ýêöñáóç (5) ìå ôç ìÝèïäï ôùí ìåñéêþí áèñïéóìÜôùí,   ãñÜöïíôáò

                                  Õ (s) = ((s - 1)/((s - 2) (s + 1)))   = (a/((s - 2))) + 
                (b/((s + 1))) =  ((a (s + 1) + b (s - 2))/((s - 2) (s + 1)))            (6)

Aðü ôçí åîßóùóç (6), áðáéôþíôáò ôçí éóüôçôá ôùí áñéèìçôþí, ðñïêýðôåé ç óxÝóç

                                                                                        s - 1 = a (s + 1) + b (s - 2), 

ç ïðïßá èá ðñÝðåé íá éóxýåé ãéá êÜèå s .     Ãéá  s = 2, âñßóêïõìå a = 
            1/3.  Áíôßóôïéxá, ãéá s = -1, âñßóêïõìå b = 
            2/3.   EðïìÝíùò Ýxoõìå áðü ôçí (6)    üôé

                                                                                        Õ (s) = ((1/3)/((s - 2))) + ((2/
                    3)/((s + 
                      1))) .                                                    (7)

Xñçóéìïðïéþíôáò ôï ÐáñÜäåéãìá 4, ðñïêýðôåé üôé ç óõíÜñôçóç (Subscript[F, 1])(s)=((1/3)/((s - 2))), åßíáé ï ìåôáóxçìáôéóìüò Laplace 
ôçò óõíÜñôçóçò    (Subscript[f, 1])(t)=1/3(e^(2  t)). ÅðïìÝíùò  (L^(-1))((Subscript[F, 1]) (s))=(Subscript[f, 1])(t).  Åðßóçò, ç óõíÜñôçóç   (Subscript[F, 2])(s)=((2/3)/((s + 1))) åßíáé ï ìåôáóxçìáôéóìüò Laplace 
ôçò óõíÜñôçóçò  (Subscript[f, 2])(t)=2/3(e^(-t)).      ÄçëáäÞ ç ëýóç ôïõ ðñïâëÞìáôïò áñxéêþí ôéìþí  åßíáé

                                                                               y (t) = (L^(-1)) (Y (s)) = 
          1/3  (e^(2  t)) + 2/3 
               (e^(-t))                                    (8)

Áíôßóôñïöïò Måôáóxçìáôéóìüò Laplace êáé ÌATHEMATICA.      To ÌÁÔÇÅÌÁÔÉCÁ, äéáèÝôåé åðßóçò åíôïëÞ ãéá ôïí õðïëïãéóìü ôïõáíôßóôñïöïõ ìåôáóxçìáôéóìïý  Laplace .   Ùò ðáñÜäåéãìá ãéá ôçí xñÞóç ôçò åíôïëÞò, åðéóôñÝöïõìå óôï ÐáñÜäåéãìá 5, õðïëïãßæïíôáò ôïí áíôßóôñïöï ìåôáóxçìáôéóìü Laplace ãéá ôçí (7)

InverseLaplaceTransform[(((1/3))/((s - 2))) + (((2/3))/((s + 1))), s, t]

äçëáäÞ åðéóôñÝöåé ôçí Ýêöñáóç ôçò ëýóçò  (8) .  Ôï MATHEMATICA,  ðáñÜãåé åðßóçò ôéò éäéüôçôåò ôïõ ìåôáóxçìáôéóìïý Laplace 3. , 
      4. ,  ãéá ôéò ðáñáãþãïõò .  Ãíùñßæïíôáò ôç âáóéêÞ èåùñßá ãéá ôçí åðßëõóç äéáöïñéêþí åîéóþóåùí ìå xñÞóç ôïõ ìåôáóxçìáôéóìïý Laplace,  ïé åíôïëÝò áõôÝò,  ðáñÝxïõí; Ýíá ðïëý óçìáíôéêü âïÞèçìá ãéá ôçí áðëïðïßçóç ôùí õðïëïãéóìþí .  Óôï ðáñÜäåéãìá ðïõ áêïëïõèåß,  èá åðéëýóïõìå Ýíá ìç - ïìïãåíÝò ðñüâëçìá áñxéêþí ôéìþí,  áêïëïõèþíôáò ôá âÞìáôá åðßëõóçò êáé xñçóéìïðïéþíôáò  to ÌÁÔÇÅÌÁÔÉCA,  ãéá ôçí åêôÝëåóç ôùí õðïëïãéóìþí.

ÐáñÜäåéãìá 6: Íá åðéëõèåß ôï ðñüâëçìá áñxéêþí ôéìþí

                                                                        {({{\(\(y\^''\) + y = sin2t\)}, {\(y \((0)\) = 2, \ \(y\^'\) \((0)\) = 1\)}})
Ëýóç:  ÂÇÌÁ 1:  Åöáñìüæïõìå ôï ìåôáóxçìáôéóìü Laplace óôç äéáöïñéêÞ åîßóùóç .   Ðñþôá ïñßæïõìå ôçí åîßóùóç ìÝóù ôïõ ðáêÝôïõ.

diafexis1 = y''[t] + y[t] == 
        Sin[2  t]

Óôç óõíÝxåéá,  åöáñìüæïõìå ôïí ìåôáóxçìáôéóìü Laplace óôçí äéáöïñéêÞ åîßóùóç.

metasx1 = LaplaceTransform[diafexis1, t, s]

Ðáñáôçñïýìå üôé ôï ÌÁÔÇÅÌÁÔÉCÁ,  åðÝóôñåøå ôï ìåôáóxçìáôéóìü ôçò åîßóùóçò,  åöáñìüæïíôáò ôçí éäéüôçôá 4. , äçëáäÞ
                                                            L (y (t))+(s^2) L (y (t))-s y (0)-y'(0)=(2/(4 + (s^2))),

äçëáäÞ ôçí áíôßóôïéxç áëãåâñéêÞ åîßóùóç.
(Xñçóéìïðïéþíôáò ãéá äéåõêüëõíóç ôï óõìâïëéóìü L(y)=Y (s), ç áëãåâñéêÞ åîßóùóç åßíáé                    
Õ(s)+(s^2) Y (s)-s y (0)-y'(0)=(2/(4 + (s^2))))

ÂÇÌÁ 2: Åðéëýïõìå ôçí áëãåâñéêÞ åîßóùóç ùò ðñïò Õ(s) (ìå ôï óõìâïëéóìü åîüäïõ ôïõ 

              MATHEMATICA, ≪Y (s)=LaplaceTransform[y[t], t, s]≫)
ãéá íá ðÜñïõìå ôï ìåôáóxçìáôéóìü Laplace ôçò ëýóçò y (t).

lapy = Solve[metasx1, LaplaceTransform[y[t], t, s]]

Må áíôéêáôÜóôáóç ôùí áñxéêþí  óõíèçêþí ëáìâÜíïõìå

lapy=lapy/.{y[0] -> 2,y'[0]->1}

Må ôçí áíôéêáôÜóôáóç ôùí áñxéêþí óõíèçêþí,  âñßóêïõìå üôé
Y (s)=LaplaceTransform[y[t], t, s]=-((-2 s - (2/(4 + (s^2))))/(1 + (s^2)))

ÂÇÌÁ 3: H  ëýóç ôïõ ðñïâëÞìáôïò áñxéêþí ôéìþí åßíáé ï áíôßóôñïöïò
 ìåôáóxçìáôéóìüò  Laplace ôçò Õ (s) .  
            Eöáñìüæïõìå ôçí áíôßóôïéxç åíôïëÞ

invlapy = InverseLaplaceTransform[lapy, s, t]

Ùò Üóêçóç ìðïñïýìå íá åðéâåâáéþóïõìå ôç ëýóç ôïõ ðñïâëÞìáôïò, xùñßò xñÞóç ôïõ MATHEMATICA

ÐáñÜäåéãìá  7(åöáñìïãÞ ): Eöáñìüóôå ôï ìåôáóxçìáôéóìü Laplace, ãéá ôïí õðïëïãéóìü ôçò  ëýóçò  ôçò ãåíéêÞò ãñáììéêÞò ìç - ïìïãåíïýò äéáöïñéêÞò åîßóùóçò 2  çò ôÜîçò ìå óôáèåñïýò óõíôåëåóôÝò
                                                                                              
                                                                                              {({{\(\(ay\^''\) + by' + cy = f \((t)\)\)}, {\(y \((0)\) = y\_0, \ \(y\^'\) \((0)\) = y\_1\)}})
Ëýóç:   ÂÇÌÁ 1:  Åöáñìüæïõìå ôï ìåôáóxçìáôéóìü Laplace óôç äéáöïñéêÞ åîßóùóç .  
Må xñÞóç ôùí éäéïôÞôùí 1. ,2.,3.  (ãéá n = 
                2)  ëáìâÜíïõìå

                                             \(a[\(s\^2\) Y \((s)\) - sy \((0)\) - \(y\^'\) \((0)\)] + 
              b[sY \((s)\) - y \((0)\)] + cY \((s)\) = F \((s)\)\), 

üðïõ ìå F (s), óõìâïëßæïõìå ôï ìåôáóxçìáôéóìü Laplace ôçò f(t).  


               ÂÇÌÁ 2:        Áíôéêáèéóôþíôáò ôéò áñxéêÝò  óõíèÞêåò  êáé åðéëýïíôáò ôçí áíùôÝñù åîßóùóç ùò ðñïò Õ(s), âñßóêïõìå üôé ï ìåôáóxçìáôéóìüò Laplace Õ (s) ôçò ëýóçò , åßíáé

                                                                      Õ (s)=\(\(\((as + b)\) y \((0)\) + \(ay\^'\) \((0)\)\)\/\(as\^2 + bs + c\) + \(F \((s)\)\)\/\(as\^2 + bs + c\)\)
                                                                            =(((as + b) (Subscript[y, 0]) + (Subscript[ay, 1]))/((as^2) + bs + c)) + ((F (s))/((as^2) + bs + c))
                     
                      ÂÇÌÁ 3: Ç ëýóç ôïõ ðñïâëÞìáôïò åßíáé ï áíôßóôñïöïò ìåôáóxçìáôéóìüò (áí õðÜñxåé) ôçò  Y (s)
äçëáäÞ   y(t)=(L^(-1)) (Y (s)) (t)

Må xñÞóç ôïõ MATHEMATICA,  áêïëïõèþíôáò ôç äéáäéêáóßá ôïõ ðáñáäåßãìáôïò 6. ,

diafgen = a*y''[t] + b*y'[t] + c*y[t]==f[
          t]

metasxgen = LaplaceTransform[diafgen, t, s]

lapgen = Solve[metasxgen, LaplaceTransform[y[t], t, s]]

ðáñÜãïõìå ôçí Ýêöñáóç ãéá ôïí ìåôáóxçìáôéóìü Laplace ôçò 
ëýóçò ôçò ãåíéêÞò ãñáììéêÞò äéáöïñéêÞò åîßóùóçò 2çò ôÜîçò ôçò åöáñìïãÞò. 

Méá ñïõôßíá óôï ÌÁÔÇÅÌÁÔÉCA, ãéá ôçí åðßëõóç ôçò ãåíéêÞò åîßóùóçò: 
Mðïñïýìå ôþñá íá êáôáóêåõÜóïõìå Ýíá ≪ðñüãñáììá≫ ôï ïðïßï åðéëýåé ôç ãåíéêÞ ãñáììéêÞ äéáöïñéêÞ åîßóùóç(ìÝxñé 2  çò ôÜîçò)  ìå óôáèåñïýò óõíôåëåóôÝò. Ôï 
≪ðñüãñáììá≫, äéáâÜæåé ôïõò óõíôåëåóôÝò (a, b, c), ôéò áñxéêÝò óõíèÞêåò (y (0) = d, 
            y' (0) = 
              z) êáé ôïí ìç ïìïãåíÞ üñï  f (t) êáé åðéóôñÝöåé ùò Ýîïäï, ôç ëýóç ìå xñÞóç toõ ìåôáóxçìáôéóìïý Laplace.

lyshlap[a, b, c, d, z, f_][t_]:=
Module[{diafgen, metasxgen, lapgen,  invlapgen},
diafgen = 
              a*y''[t] + b*y'[t] + c*y[t] == f[t];
             metasxgen = LaplaceTransform[diafgen, t, s];
           lapgen=Solve[metasxgen, 
                  LaplaceTransform[y[t], t, s]]

                           lapgen=lapgen/.{y[0] -> d,y'[0]->z};
  invlapgen = InverseLaplaceTransform[lapgen, s, t]]

Xñçóéìïðïéïýìå ôï ðñüâëçìá ôïõ ðáñáäåßãìáôïò 6. ,  ãéá ôïí Ýëåãxo ôïõ ðñïãñÜììáôïò ìáò

a = 1

b = 1

c = 1

d = 2

z = 1

f[t_] = Sin[2  t]

Ðáßñíïõìå ôç ëýóç ôïõ ðñïâëÞìáôïò:

lyshlap[a, b, c, d, z, f][t]

ÐáñÜäåéãìá:  Ùò Ýíá Üëëï  áðëü ðáñÜäåéãìá, ìðïñïýìå íá åðéëýóïõìå ôï ðñüâëçìá áñxéêþí  ôéìþí
                                                                                           
                                                                                            {({{\(\(y\^'\) + 4  y = cost\)}, {y (0) = 1,}})

ìå xñÞóç ôïõ ðñïãñÜììáôïò (ç åîßóùóç åßíáé ðñþôçò ôÜîçò êáé èÝôïõìå áõèáßñåôá  z = 
                  0).

a = 0

b = 1

c = 4

d = 1

f[t_] = Cos[t]

z = 0

lyshlap[a, b, c, d, z, f][t]

Mðïñåßôå ùò Üóêçóç,  íá åðéâåâáéþóåôå ôçí ïñèüôçôá ôïõ áðïôåëÝóìáôïò ðïõ äßíåé ôï MATHEMATICA.

Á.2.   ÅéäéêÝò ðåñéðôþóåéò åîùôåñéêÞò äéÝãåñóçò

Ïëïêëçñþíïõìå ôç óýíôïìç áõôÞ åéóáãùãÞ óôï ìåôáóxçìáôéóìü Laplace,  ìå ôç èåþñçóç ìç - ïìïãåíþí äéáöïñéêþí åîéóþóåùí,  üðïõ ï üñïò f (t) ìðïñåß íá åßíáé áóõíåxÞò Þ óôéãìéáßáò åðßäñáóçò êáé ìåãÜëïõ ìåãÝèïõò.

H óõíÜñôçóç ìïíáäéáßïõ âÞìáôïò ôïõ Çeaviside .  
                                                    EöáñìïãÞ óôéò äéáöïñéêÝò åîéóþóåéò.

Méá Üëëç óðïõäáßá åöáñìïãÞ ôïõ ìåôáóxçìáôéóìïý Laplace, áðáíôÜôáé óôçí åðßëõóç äéáöïñéêþí åîéóþóåùí óôçí ðåñßðôùóç ðïõ ç óõíÜñôçóç äéÝãåñóçò åßíáé áóõíåxÞò .   ' Åíá ðïëý óçìáíôéêü ðáñÜäåéãìá ìéáò ôÝôïéáò óõíÜñôçóçò ðïõ óõíáíôÜôáé óôéò äéáöïñéêÝò åîéóþóåéò, åßíáé ç óõíÜñôçóç ìïíáäéáßïõ âÞìáôïò.

Á.  Ïñéóìüò ôçò óõíÜñôçóçò ìïíáäéáßïõ âÞìáôïò

 H óõíÜñôçóç  ìïíáäéáßïõ âÞìáôïò ìå Üëìá óôï 0,  ïñßæåôáé ùò  

                                                                                         H (t)={({{0,  áí t < 0}, {1,  áí t ≥ 0}})

H óõíÜñôçóç ìïíáäéáßïõ âÞìáôïò ìå Üëìá óôïí áñéèìü a oñßæåôáé ùò

                                                                                      (Subscript[Ç, a]) (t):=  H (t - a)={({{0,  áí t < a}, {1,  áí t ≥ a}})

Ãéá ôç óõíÜñôçóç ìïíáäéáßïõ âÞìáôïò,  ôï MATHMATICA  xñçóéìïðïéåß ôçí åíôïëÞ UnitStep[
              t] .  
ÓxåäéÜæïõìå ôç ãñáöéêÞ ðáñÜóôáóç ôçò âçìáôéêÞò

             óõíÜñôçóçò     Ç (t - π/2)

Plot[
      UnitStep[t - (π/2)],  {t, -12, 12}, 
      PlotStyle -> {Thickness[0.005], RGBColor[1, 0, 0]},  
      PlotRange -> All]

Â.  Ìåôáóxçìáôéóìüò Laplace ôçò óõíÜñôçóçò ìïíáäéáßïõ âÞìáôïò

 Ãéá ôï ìåôáóxçìáôéóìü Laplace ôçò âçìáôéêÞò óõíÜñôçóçò éóxýåé

                                                                                         L( (Subscript[Ç, a]) (t))=((E^(-as))/s)                                                                         (MB1)

H ìåôáöïñÜ ìéáò óõíÜrôçóçò f  óå ìéá áðüóôáóç a,  åßíáé ç óõíÜñôçóç 
                                                                                 
                                                                                g (t)={({{f (t - a),  áí     t < 
                            a}, {0,                áí     t ≥ a}})    =  (Subscript[Ç, a]) (t) 
              f (t - 
                  a)               (MTÖ)

Aí ï ìåôáóxçìáôéóìüò Laplace L (f (t)) (s) õðÜñxåé ãéá s > c >  ≥ 
          0 êáé áí a åßíáé ìéá èåôéêÞ óôáèåñÜ,  ôüôå 

                                                                       L (g (t)) (s)=L( (Subscript[Ç, a]) (t) f (t - a))(s)=(E^(-as)) 
              L (f (t)) (s)           (MB2)

Â.1   Èåþñçìá ýðáñîçò ëýóçò ãéá ôçí ðåñßðôùóç áóõíåxïýò åîùôåñéêÞò äéÝãåñóçò

 Èåùñïýìå ôï äéÜóôçìá  a=(Subscript[t, 0]) < (Subscript[t, 1]) < (Subscript[t, 2]) < ⋯ < (Subscript[t, n - 1]) < 
              (Subscript[t, n]) = b êáé üôé ç óõíÜñôçóç f (t) åßíáé óõíåxÞò óå êÜèå äéÜóôçìá ((Subscript[t, i - 1]), 
                (Subscript[t, i]))  êáé
 ôá üñéá 
                                                                                              \(lim\+\(t -> \(t\_i\^+\)\)f \((t)\), \ \ lim\+\(t -> \(t\_i\^-\)\)f \((t)\), \ 
        i = 1, \(\(...\) \(n\)\), 

õðÜñxïõí . \ Ôüôå\ ôï\ ðñüâëçìá\ áñxéêþí\ ôéìþí\ Ýxåé\ ìïíáäéêÞ\ ëýóç\ ìå\ ôéò\ áêüëïõèåò\ \(éäéüôçôåò : 

\ \ \ \ \ \ \ \ 1. \ \ \ Ïé\ y \((t)\)\ êáé\ y' \((t)\)\ åßíáé\ óõíåxåßò\ óôï\ a < t < 
              b . 
\ 
                  2. \ \ \ H\ \ y'' \((t)\)\ åßíáé\ óõíåxÞò\ óå\ êÜèå\ äéáóôçìá\ \((t\_\(i - 1\), t\_i)\)\)\) 
             3.  Ç y (t) éêáíïðïéåß ôï (Ð . Á . Ô),        
óå êÜèå äéÜóôçìá ((Subscript[t, i - 1]), (Subscript[t, i]))

ÐáñÜäåéãìá  9:  Íá åðéëõèåß ôï  ðñüâëçìá  áñxéêþí ôéìþí 

                                                                                            {({{\(\(y\^''\) + y = f \((t)\)\)}, {\(y \((0)\) = 1, \ \(y\^'\) \((0)\) = 0\)}})


ãéá ôçí ðåñßðôùóç üðïõ  f(t)={({{1,  áí t < ð, }, {0, áí t ≥ ð.}})
Ëýóç: Eöáñìüæïõìå ôï ìåôáóxçìáôéóìü Laplace óôç äéáöïñéêÞ åîßóùóç .  Âñßóêïõìå

                                                                            ((s^2) + 1) Y (s)=(E^(-sπ/2))=∫(E^(-st)) \[DifferentialD]t=((1 - (E^(-πs)))/s).

Eðéëýïíôáò ôçí ðáñáðÜíù áëãåâñéêÞ åîßóùóç ùò ðñïò Õ(s), âñßóêïõìå üôé                                                   

                                                                                             Y (s) = L (y (t)) (s) = (1/((s^2) + 
                                    1)) (((1 - (E^(-πs)))/s))
             
                                                                                                      =     (1/(s ((s^2) + 1)))   - ((E^(-πs))/(s ((s^2) + 1)))                                      (9.1)

Xñçóéìïðïéþíôáò ôç ìÝèïäï ôùí ìåñéêþí êëáóìÜôùí, ðáñáôçñïýìå üôé ï üñïò  (1/(s ((s^2) + 1))), áíáëýåôáé ùò
                                                                                                         (1/(s ((s^2) + 1))) = 
                      (1/s) - (s/(1 + (s^2)))

êáé åðïìÝíùò ç (9.1), ãñÜöåôáé

                                                        Y (s) = L (y (t)) (s) = (1/s) - 
                          (s/(1 + (s^2))) - (E^(-πs)) 
                            (1/s) + (E^(-πs)) 
                            (s/(1 + (s^2)))

Ðáñáôçñïýìå xñçóéìïðïéþíôáò ôç   (ÂÌ2)

                                                                        ((s (E^(-πs)))/( (s^2) + 1)) = (E^(-πs)) L (cos t) (s) = 
                                L ((Subscript[H, π]) (t) 
                                      cos (t - π))                   (9.2)

Åðßóçò, Ýxïõìå ôïõò  áíßóôñïöïõò ìåôáóxçìáôéóìïýò 

                                                                                           (1/s)=L (1)    êáé    ((E^(-πs))/s)=L((Subscript[H, π]) (t)) .                                      (9.3)

ÔåëéêÜ ç ëýóç ôïõ ðñïâëÞìáôïò åßíáé üðùò ðñïêýðôåé ìå xñÞóç ôùí (9.1)-(9.3) , åßíáé   ç  

                                                                  y (t)=(1-cost)-(Subscript[H, π]) (t) (1 - 
                                cos (t - π)).

Eðéëýïõìå ôï ðñüâëçìá êáé ìå ôéò áíôßóôïéxåò åíôïëÝò ôïõ MATHEMATICA

F[t_] := 
      1 - UnitStep[t - π]

Óôç óõíÝxåéá ïñßæïõìå ôç äéáöïñéêÞ åîßóùóç

parad9 = y''[t] + y[t] == F[t]

Âñßóêïõìå ôï ìåôáóxçìáôéóìü Laplace ôçò åîßóùóçò ...

metparad9 = LaplaceTransform[parad9, t, s]

êáé åðéëýïõìå ôçí áíôßóôïéxç  áëãåâñéêÞ åîßóùóç.

alparad9 = Solve[metparad9, LaplaceTransform[y[t], t, s]]

Må áíôéêáôÜóôáóç ôùí áñxéêþí óõíèçêþí ...

alparad9 = alparad9 /. {y[0] -> 0, y'[0] -> 0}

êáé ìå xñÞóç ôïõ áíôßóôñïöïõ ìåôáóxçìáôéóìïý Laplace

antistparad9 = InverseLaplaceTransform[alparad9, s, t]

âñßóêïõìå ôç ëýóç ôïõ ðñïâëÞìáôïò.

FullSimplify[%]

ÔÝëïò,  óxåäéÜæïõìå ôç ãñáöéêÞ ôçò ðáñÜóôáóç.

lyshparad9 = %[[1, 1, 2]]

Plot[lyshparad9, {t, -10, 10}, 
      PlotStyle -> {Thickness[0.005], RGBColor[1, 0, 0]}, 
      PlotPoints -> 500]

H óõíÜñôçóç äÝëôá ôïõ Dirac -EöáñìïãÞ óôéò äéáöïñéêÝò åîéóþóåéò.

Óå ðïëëÜ   öáéíüìåíá   ðáñáôçñïýíôáé åîùôåñéêÝò åðéäñÜóçò  ìåãÜëçò Ýíôáóçò êáé óôéãìéáßáò äéÜñêåéáò, ïé ïðïßåò êáëüõíôáé óõxíÜ, åîùôåñéêÝò åðéäñÜóåéò  óôéãìéáßáò þèçóçò : xáñáêôçñéóôéêü ðáñÜäåéãìá,  
áðïôåëïýí  äõíÜìåéò ìåãÜëïõ ìåãÝèïõò   ïé ïðïßåò åðéäñïýí ãéá ðïëý ìéêñÜ xñïíéêÜ äéáóôÞìáôá .   
H ìáèçìáôéêÞ ìïíôåëïðïßçóç öáéíïìÝíùí áõôïý ôïõ xáñáêôÞñá, âáóßæåôáé óôç óõíÜñôçóç äÝëôá ôïõ Dirac.

Áò èåùñÞóïõìå ðñþôá ôç óõíÜñôçóç 

                                                                              (Subscript[f, å]) (t)={({{(1/(2  å)),         ãéá     å < t < -å}, {0,                 äéáöïñåôéêÜ }})

êáé áò óxåäéÜóïõìå ôç ãñáöéêÞ ðáñÜóôáóç ôçò (Subscript[f, å]) (t),  ãéá äéÜöïñåò ôéìÝò ôïõ å .

g1 = 
        Which[-1 < t < 1, (1/2),  Abs[t] ≥ 1, 
          0];

g03 = 
        Which[-0.3 < t < 0.3, (1/(2*0.3)),  Abs[t] ≥ 0.3, 
          0];

g06 = 
        Which[-0.6 < t < 0.6, (1/(2*0.6)),  Abs[t] ≥ 0.6, 
          0];

g08 = 
        Which[-0.8 < t < 0.8, (1/(2*0.8)),  Abs[t] ≥ 0.8, 
          0];

g14 = 
        Which[-1.4 < t < 1.4, (1/(2*1.4)),  Abs[t] ≥ 1.4, 
          0];

g01 = 
        Which[-0.1 < t < 0.1, (1/(2*1.4)),  Abs[t] ≥ 0.1, 
          0];

f008 = 
        Which[-0.08 < t < 0.08, (1/(2*0.08)),  
          Abs[t] ≥ 0.08, 0];

Plot[{g03, g06, g08, g1, g14, g01, f008}, {t, -2, 2}, 
      PlotRange -> {0, 7}, 
      PlotStyle -> {RGBColor[0, 0, 1]}]

Eßíáé öáíåñü üôé ãéá ôç óõíÜñôçóç (Subscript[f, å]) (t) ,  ôï ïëïêëÞñùìá 
   
                                                                                 É ((Subscript[f, å]) (t)) = (∫_(-∞)^(+∞))(Subscript[f, å]) (t) \[DifferentialD]t = (∫_(-å)^å) 
                      (Subscript[f, å]) (t) \[DifferentialD]t = 
              1.        (Ðä1)
EðéðëÝïí

                                                                                                   (Underscript[(  lim), å -> 
                  0]) (Subscript[f, å]) (t) = 
          0,  ãéá t ≠ 
          0,                         (Ðä2)

êáé                                                                                                    (Underscript[(  lim), å -> 0]) É ((Subscript[f, å]) (t)) = 
          1  .                                (Ðä3)

Ç óõíÜñôçóç (Subscript[f, å]) (t), ìå ôéò éäéüôçôåò (Ðä1) - (Ðä3), áðïôåëåß ðñïóÝããéóç ìéáò ðåñßöçìçò óõíÜñôçóçò, ôçò óõíÜñôçóçò äÝëôá ôïõ Dirac.

Ã.  Ïñéóìüò ôçò óõíÜñôçóçò äÝëôá ôïõ Dirac

 H óõíÜñôçóç  äÝëôá ôïõ Dirac  åßíáé ç óõíÜñôçóç ìå ôéò éäéüôçôåò  

                                                                                         ä (t) = 
          0,  ãéá t ≠ 
          0,                  (ä1)

                                                                                  É (ä (t)) = (∫_(-∞)^(+∞))ä (t) \[DifferentialD]t = 
            1 ,     (ä2)

Áðü ôéò óõíèÞêåò (ä1) - (ä2),  ðñïêýðôåé üôé

                                                                                           ä (t - (Subscript[t, 0])) = 
          0,  ãéá t ≠ 
          (Subscript[t, 0]),       (ä3)
                                                      
                                                                                         (∫_(-∞)^(+∞))ä (t - (Subscript[t, 0])) \[DifferentialD]t = 
          1   .             (ä4)

Ôï ïëïêëÞñùìá ìéáò óõíÜñôçóçò ç ïðïßá åßíáé ìçäåíéêÞ åêôüò áðü Ýíá óçìåßï, ðñÝðåé íá åßíáé 0.  ÅðïìÝíùò ç óõíÜñôçóç ä (t), äåí ìðïñåß íá õðÜñxåé ùò ìáèçìáôéêÞ óõíÜñôçóç ìå ôç óõíÞèç Ýííïéá!!   Ç óõíÜñôçóç ä(t) áðïôåëåß  ìéá ãåíßêåõóç ôçò óõíÞèïõò Ýííïéáò ôçò óõíÜñôçóçò êáé áðïôåëåß xáñáêôçñéóôéêü ðáñÜäåéãìá  ãåíéêåõìÝíçò óõíÜñôçóçò. Åßíáé ìéá óõíÜñôçóç ðïëý ìåãÜëçò óçìáóßáò ãéá ôç öõóéêÞ áëëÜ êáé ãéá Üëëåò åöáñìïãÝò, êáèþò áíáðáñéóôÜ ìéá þèçóç ðïëý ìåãÜëïõ ìåãÝèïõò ç ïðïßá äñá óôéãìéáßá,ôç xñïíéêÞ  óôéãìÞ t = 
            0 (åßíáé ìçäÝí ïðïéáäÞðïôå Üëëç xñïíéêÞ  óôéãìÞ).

Ï ìåôáóxçìáôéóìüò Laplace ôçò óõíÜñôçóçò ä (t),  ìðïñåß íá õðïëïãéóôåß (ôïõëÜxióôïí ôõðéêÜ) .   Ïñßæïõìå ôï ìåôáóxçìáôéóìü Laplace,  ùò ôï üñéï 
                                                                                                   L (δ (t - (Subscript[t, 0]))) = 
          (lim_(å -> 0))L ((Subscript[f, å]) (t))

ÐáñÜäåéãìá  8(åöáñìïãÞ ):  Eáí g (t) åßíáé ìéá óõíåxÞò óõíÜñôçóç ïñéóìÝíç óôï -∞<t<∞, ôüôå ãéá êÜèå a,b, ôÝôïéá þóôå  - ∞ ≤ 
            a ≤ (Subscript[t, 0])  ≤ 
            b ≤ +∞, Ýxïõìå 
                                                                                                              (∫_a^b) 
              g (t) δ (t - (Subscript[t, 0])) \[DifferentialD]t = 
          g ((Subscript[t, 0])) .     (Lä)
Ëýóç: Óýìöùíá ìå ôï èåþñçìá ìÝóçò ôéìÞò ôïõ áðåéñïóôéêïý ëïãéóìïý ãéá ôï ïëïêëÞñùìá, õðÜñxåé êÜðïéï (Subscript[t, 1]), (Subscript[t, 0]) - ϵ < (Subscript[t, 1]) < (Subscript[t, 0]) + ϵ, Ýôóé þóôå 
              
                                                                                                                (∫_((Subscript[t, 0]) - ϵ)^((Subscript[t, 0]) + ϵ))g (t) \[DifferentialD]t = 
          2  ϵg ((Subscript[t, 1])) .                      (8.1)

ÄéáëÝãïõìå ôï  ∈  áñêåôÜ ìéêñü Ýôóé þóôå a < (Subscript[t, 0]) - ϵ < (Subscript[t, 0]) + ϵ < 
            b .  Tüôå áðü ôçí (8.1),  ëáìâÜíïõìå üôé

                                                                                     (∫_a^b) (Subscript[f,  ∈])  (t) 
              g (t) \[DifferentialD]t = (1/(2  ϵ)) (∫_((Subscript[t, 0]) - ϵ)^((Subscript[t, 0]) + ϵ))g (t) \[DifferentialD]t = g ((Subscript[t, 1])) .       (8.2)

Ôüôå,  êáèþò (Subscript[t, 1]) -> 
        (Subscript[t, 0]),  üôáí ϵ -> 
        0,  èåùñþíôáò ôï üñéï ôçò (8.2) êáèþò ϵ -> 
        0,  ëáìâÜíïõìå ôçí (Lä).

Ä.    Ìåôáóxçìátióìüò Laplace  ôçò óõíÜñôçóçò äÝëôá ôïõ Dirac

 Ï ìåôáóxçìáôéóìüò Laplace ôçò óõíÜñôçóçò   ä(t - (Subscript[t, 0]))g(t), äßíåôáé áðü ôç óxÝóç

                                                                                           L (ä (t - (Subscript[t, 0])) g (t))=(E^(-s (Subscript[t, 0])))g((Subscript[t, 0]))                      (ä5)
                                                      
                                                                                         
 Ï ìåôáóxçìáôéóìüò Laplace ôçò óõíÜñôçóçò  äÝëôá ôïõ Dirac,  äßíåôáé áðü ôç óxÝóç 
  
                                                                                          L (ä (t - 
                    (Subscript[t, 0]))) = (E^(-s (Subscript[t, 0])  )) êáé   L (ä (t)) = 
            1       (ä6)

ÈÝôïíôáò a = 0, 
        b = ∞ êáé f (t) = (e^(-st)) 
              g (t) óôçí (Lä),  ëáìâÜíïõìå ôçí (ä5).

ÐáñÜäåéãìá  10:  Íá åðéëõèåß ôï  ðñüâëçìá  áñxéêþí ôéìþí 

                                                                                            {({{\(\(y\^''\) + 4  y = ä \((t - π/2)\)\)}, {\(y \((0)\) = 0, \ \(y\^'\) \((0)\) = 0\)}})

Ëýóç: Eöáñìüæïõìå ôï ìåôáóxçìáôéóìü Laplace óôç äéáöïñéêÞ åîßóùóç .  Âñßóêïõìå

                                                                            ((s^2) + 4)Y(s)=(E^(-sπ/2)) . 

ÄçëáäÞ                                                       Y (s) = L (y (t)) (s) = ((E^(-sπ/2))/( ((s^2) + 
                            4)))  .                     (10.1)


                  Ðáñáôçñïýìå üôé 


                                                                                          (( (E^(-sπ/2)))/( ((s^2) + 4))) = ((E^(-sπ/2))/( 2)) 
                      L (sin2t) (s) .                             (10.2)


                  
                  Xñçóéìïðïéþíôáò ôþñá ôç (ÌÂ2), ç  (9.2)  ãßíåôáé
      

                                                                                      ((E^(-sπ/2))/( ((s^2) + 4))) = ((E^(-sπ/2))/( 2)) 
                  L (sin2t) (s) = (1/2) 
                    L ((Subscript[H, π/2]) (t) 
                        sin (2 (t - π/
                                  2))) (s)
                                                                                          
                                                                                      = -(1/2) 
                    L ((Subscript[H, π/2]) (t) 
                        sin (2  
                            t)) (s)          (10.3)  


                  Áðü ôçí  (9.1) êáé ôçí (9.3)  âñßóêïõìå üôé ç ëýóç åßíáé ç


                                                                                                 y (t) = -(((Subscript[H, π/2]) (t) 
                            sin (2  t))/2)

Ëýíïõìå ôþñá ôï ðñüâëçìá, xñçóéìïðïéþíôáò ôéò åíôïëÝò ôïõ MATHEMATICA .  
              To MATHEMATICA xñçóéìïðïéåß ãéá ôç óõíÜñôçóç äÝëôá ôïõ Dirac, ôçí åíôïëÞ DiracDelta[t]. Oñßæïõìå ôï äåîß ìÝëïò ôïõ ðñïâëÞìáôïò

h[t_] := 
      DiracDelta[t - (π/2)]

Óôç óõíÝxåéá ïñßæïõìå ôç äéáöïñéêÞ åîßóùóç

ex9 = y''[t] + 4  y[t] == h[t]

Âñßóêïõìå ôï ìåôáóxçìáôéóìü Laplace ôçò åîßóùóçò ...

metex9 = LaplaceTransform[ex9, t, s]

êáé åðéëýïõìå ôçí áíôßóôïéxç  áëãåâñéêÞ åîßóùóç.

algex9 = Solve[metex9, LaplaceTransform[y[t], t, s]]

Må áíôéêáôÜóôáóç ôùí áñxéêþí óõíèçêþí ...

algex9 = algex9 /. {y[0] -> 0, y'[0] -> 0}

êáé ìå xñÞóç ôïõ áíôßóôñïöïõ ìåôáóxçìáôéóìïý Laplace

antistex9 = InverseLaplaceTransform[algex9, s, t]

âñßóêïõìå ôç ëýóç ôïõ ðñïâëÞìáôïò.

FullSimplify[%]

ÔÝëïò,  óxåäéÜæïõìå ôç ãñáöéêÞ ôçò ðáñÜóôáóç.

lysh9 = %[[1, 1, 2]]

Plot[lysh9, {t, 0, 10}, 
      PlotStyle -> {Thickness[0.005], RGBColor[0, 0, 1]}]

l = (1/(s ((s^2) + 1)))

Apart[l]

Áîßæåé íá áíáöåñèåß üôé ç åéóáãùãÞ ôçò óõíÜñôçóçò äÝëôá áðü ôïí Dirac,óôá ôÝëç ôçò äåêáåôßáò ôïõ 1920, áìöéóâçôÞèçêå Ýíôïíá áðü ôïõò ìáèçìáôéêïýò (êáé xñçóéìïðïéÞèçêå êáôÜ êüñïí áðü ôïõò öõóéêïýò!) .  Ç óõíÜñôçóç äÝëôá ïñßóôçêå ìå áõóôçñÜ ìáèçìáôéêü ôñüðï óôç èåùñßá ôùí ãåíéêåõìÝíùí óõíáñôÞóåùí(Þ êáôáíïìþí )ðïõ áíÝðôõîå ï  ìáèçìáôéêüò Laurent Schwartz 20 xñüíéá áñãüôåñá .  Ï  Paul Dirac âñáâåýôçêå ìå ôï âñáâåßï Nobel ÖõóéêÞò (ìáæß ìå ôïí Årwin Schr  (Overscript[o, ¨]) 
              dinger) ôï 1933. Ï Laurent Schwartz âñáâåýôçêå ìå ôï  ìåôÜëëéï Fields (ôçí áíôßóôïéxç åðéóôçìïíéêÞ äéÜêñéóç ãéá ôïõò ìáèçìáôéêïýò) ôï 1950.