On Some Applications of New Integral Transform “ELzaki Transform”

16 Tarig. M. Elzaki et al
When a physical system is modeled under the differential sense, if finally gives a
differential equation.
Recently .Tarig ELzaki introduced a new transform and named as ELzaki
transform [1] which is defined by:
∞
−
u
(), ⎤ ( ) () , ( , )
Ε ⎡⎣f t u⎦ = T u = u∫
e f t dt u∈
k k
0
t
1 2
Or for a function f ( t ) which is of exponential order,
f
( t)
⎧
⎪Me
− t k
< ⎨
t
⎪
k 2
Me t ≥
⎩
1
, t ≤ 0
, 0
∞
() ( ) 2 −t
, ⎤
( ) , ( , )
Ε ⎡⎣f t u⎦ = T u = u ∫ e f ut dt u∈
k k
Theorem (1)
Let T( u)
is the ELzaki transform of ⎡ ( )
Proof
0
1 2
( ) = ( ) ⎤.
⎣
E f t T u
⎦ then:
T( u)
T( u)
i Ε ⎡⎣f′ t ⎤⎦ = −u f ( 0) ( ii) Ε ⎡f ( t)
f 0 uf 0
2
u
⎣
′′ ⎤⎦
= − − ′
u
n−1
( n
( ) ) T( u) 2− n+
k ( k
iii Ε ⎡f () t ⎤
u f )
⎣ ⎦
= −
( 0
n ∑
)
u
() ()
() Ε ⎡ ′() ⎤ = ′()
∞
0
−t
u
k=
0
i ⎣f t ⎦ u∫ f t e dt Integrating by parts to find that:
( )
T u
Ε ⎡f () t ⎤
⎣
′
⎦
= − u f
u
( 0)
1
⎣ ⎦
u
⎣ ⎦
( ii ) Let g ( t) = f ′( t) , then: Ε ⎡g′
() t ⎤= Ε⎡g() t ⎤ −ug( 0)
We find that, by using( i ):
T( u)
Ε ⎡⎣f′′ () t ⎤⎦
= − f 0 −uf′
0
2
u
( ) ( )
( iii ) Can be proof by mathematical induction
( ) ( )