On Some Applications of New Integral Transform “ELzaki Transform”

18 Tarig. M. Elzaki et al
Proof
−at
We have, by definition ⎡ ( )
∞
0
( 1+
)
au t
u
( )
Ε ⎣e f t ⎤
⎦ = u∫ e f t dt . Let
−
u
w = or
1 + au
w
u = , we have:
1 − aw
∞ a
∞ t
−
w −
1 1
u
w
⎡ u ⎤
u∫e f () t dt = e f () t dt = T ( w)
= T and
1−aw ∫
1− aw au ⎢1
0 0
1
au ⎥
− ⎣ + ⎦
1+
au
−at
u
Ε ⎡
⎣e f () t ⎤
⎦ = ( 1+
au)
T
1+
au
Theorem (5)
−
u
If Ε ⎡⎣f ( t) ⎤⎦ = T ( u),
then: Ε⎡⎣f ( t −a) H ( t − a) ⎤⎦
= e T ( u)
Where H ( t − a)
is Heaviside unit step function.
Proof
It follows from the definition that:
∞
a
−
−
u
u
( ) ( ) ⎤ ( ) ( ) ( )
Ε⎡f t−a H t− a = u e f t−a H t− a dt = u e f t−a dt
⎣ ⎦ ∫ ∫
Let t a τ ,
0 0
a ∞ t a
− − −
u
u
τ
− = then we have: e u e f ( τ) dτ
= e T( u)
∫
0
2
In particular if f ( t ) = 1, then: H ( t a) u e − u
Ε⎡⎣ − ⎤⎦
=
Also we can prove by mathematical induction that:
n−1
⎡
a
( t−
a)
⎤
−
n+
1 u
Ε⎢
H ( t− a)
⎥ = u e
⎢ Γ( n
⎣
) ⎥⎦
Example (1) (Linear dynamical systems and signals)
In physical and engineering sciences, a large number of linear dynamical systems
with a time dependent input signal f ( t ) that generates an output signal x ( t ) can be
described by the ordinary differential equation with constant coefficients.
n n−
( 1 m m−
D + a ) ( ) ( 1
n−1D + ... + a0 x t = D + bm−
1D + .... + b0) f ( t)
(1)
d
Where D = , a0, a1,... an−
1
, b0, b1,...
bm−
1
are constants
dx
We apply ELzaki transform to find the output x ( t ) so that (1) becomes.
P ( u) x ( u) R ( u) q ( u) f ( u) s ( u)
− = − (2)
n n−1 m m−1
a
a
∞
t