A New Fuzzy Technique for Drug Concentration in Blood

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2326-9865

Vol. 71 No. 3s3 (2022) 213 http://philstat.org.ph

A New Fuzzy Technique for Drug Concentration in Blood

Shaymaa Maki Kadham^#1, Ameera N. Alkiffai^*2

#1University of Kufa, College of Computer Science and Mathematics, Iraq

*2University of Kufa, College of Education for Girls, Iraq

#1[email protected] , ^*2[email protected]

Issue: Special Issue on Mathematical Computation in Combinatorics and Graph Theory in Mathematical Statistician and Engineering Applications

Article Info

Page Number: 213 - 226 Publication Issue:

Vol 71 No. 3s3 (2022)

Article History

Article Received: 30 April 2022 Revised: 22 May 2022

Accepted: 25 June 2022 Publication: 02 August 2022

Abstract

The use of integral transforms to obtain a form of solution to differential equations is a very general technique. This powerful methods allows one to turn a complicated problem into a simpler one.

In this paper, we introduce a new integral transform which is called ‘SHA- transform’ that is used to solve ordinary differential equations. Some important definitions, properties and theorems are discussed as well. The concepts of fuzzy sets are widely used in differential equations field, so in this work, a fuzzy SHA-transform, definitions and established theorems are represented. An equation related with the drug concentration in the plasmas is solved using fuzzy SHA- transform, due to integral transforms accuracy. Drug concentration drives by collecting a blood sample after taking the drug at any time, then measuring the drug’s amount in a given volume of the plasma in the sample.

Keywords: SHA-transform, Fuzzy SHA-transform, solve equation drug concentration in blood .

Mathematics subject classification 2010:41A10, 41A50

1. Introduction.

Mathematically, integral transforms converts a function from its original space to another space via integration. The transformed function generally be mapped back to its original one using the inverse transform. In this work, we derived a new integral transform that is called “SHA-transform” with its related theorems.

Fuzzy transforms is a technique belongs to fuzzy approximation models. Fuzzy derivative was first introduced by Chang and Zadeh (1972) [6,11], whereas the concept of fuzzy differential equations was introduced by Kandel and Byatt (1978-1980) [7]. After a while of time, the method of numerical solution was introduced for solving fuzzy differential equations by Abbasbandy and Allahviranloo (2002) [5]. This paper constructs a new fuzzy transform based on “SHA-transform”

for solving fuzzy differential equations due to its accurate. The measured drug concentration is demonstrated as an application for this fuzzy procedure. In fact, this drug measurement is generally known as “ plasma concentration”. An extension for the SHA-transform’s kernel is discussed.

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2326-9865

Vol. 71 No. 3s3 (2022) 214 http://philstat.org.ph 2.Main Result

Definition1:: Let ( ) is function defined for  0, let  ( )= , 0be positive real functions and  ( )=i²ⁿ + be positive complex function , the general integral transform SHA( ) of ( ) can be defined by the following formula:

 

^{( )} ⁽ⁱ² ⁾

0 0

(1) SHA  ( ),   =   ( )

^

  ( )  e

⁻^{  }

d   =

^

  ( )  e

⁻ ⁿ^{  }⁺

d  , n 1 

^.

Definition 2( The Inverse of SHA-Transform): The SHA-transform inverse of S( ) denoted by

( )

1 S

SHA⁻    is the piecewise continuous function( ) on [0, ) which satisfies:

( )

(2) SHA ( ) =S  . 3. SHA-Transform forSome Basic Functions:

Assume that for any function ( ) , the integral in equation (1) exists.SoSHA-transformwill be:

Function ^^{( )}^ ⁼^T⁻¹



^SHA^{( )}^



^SHA^{( )}^ ⁼^SHA



^^{( ),}^{ }



( ) 1

 =



^{( ),}



₂_n ^{, n} ¹

SHA

i

  

 =   

+ ( ) 

 =

 

(

²

)

²

( ), , n 1

n

SHA

i

  

 =   

+ ( ) ^

 =

  ( )

(

²

)

¹

( ), 1 , 0 ,n 1.

n

SHA

i ^

 

  

  ⁺

 +

 =  

+ ( ) sin

 =



^{( ),}



₂ ₂ ^{, n 1}

(i ⁿ ) 1

SHA   

 =   

+ +

( ) cos

 =



^{( ),}



₂⁽ⁱ² ^{+ )}₂ ^{, n} ¹

(i ) 1

n

SHA   n  

 =   

+ −

( ) sinh

 =



^{( ),}



₂ ₂ ^{, n 1}

(i ⁿ ) 1

SHA   

 =   

+ −

( ) cosh

 =



^{( ),}



⁽ⁱ₂ ² ₂⁾² ^{, n} ¹

(i ) 1

n

SHA    n  

 

 = + 

+ −

( ) e^

 =



^{( ),}



₂ ^{, n} ¹

n 1

SHA i

  

 =   

+ − Following are proofs for some SHA-transform of some functions:

1- If ( ) =1, then ^SHA



^{( ),}



₂_n ^{, n} ¹

i

  

 =   

+

  ⁽ⁱ² ⁾ ₂ ⁽ⁱ² ⁾ ² ₂ ⁽ⁱ² ⁾ ₂

0 0 0

( ), 1 ⁿ = ⁿ (i ⁿ + ) d ⁿ = .

n n n

SHA e d e e

i i i

           

      

     

  

− + − +  − + 

 =



− +



− =− +   +

2- If ( ) =sin then

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2326-9865

 

²

2 2

2

(i )

0

(i ) (i )

( )

2 2

0 0 0

( ), sin

(3) sin sin cos

(4)

:

n

n n

ni

n n

SHA e d

e e

e

integrating by parts

Integr

d d

i ating by parts

i

  

     

  

    

    

   



− +

 − +   − +

− +

 =

 −  −

=  −

+ +

 

 



 

 

2 2 2

2

(i ) (i ) (i )

2 2 2

0 0 0

(i )

2 2 0

(i )

0

(4)

cos con -sin

sin

(i ) 1

then ( ), s

(3)

in

n n n

n

n n n

W e compe n

e e e

d d

i i i

e d

SHA e d

nsate in

        

  

    

     

   

 

    

 − + − +   − +



− +



− +

 

− − −

=  −

+  +  +

= + +

 =

 



⁼_(i²ⁿ_{ }₊^ ₎²₊₁

3- If ( ) =e^ then



^{( ),}



₂ ^{, n} ¹

n 1 SHA

i

  

 =   

+ −

 

⁽² ⁾ ²

0

( ), e =

1

n

i

SHA e d

i

e

   

 

  

^ ⁻ ⁺

 

₊

 =

 −

^.

Theorem1:Let( ) is a differentiable function for 0and ( )= , 0be real positive function, ( )=i²ⁿ  + , 0be positive complex function then :

 

2

2 2 2

1

( ) 2 2 1

0

( ) ( ) ', (i ) ( ) (0) , n 1 ,

( ) ( )", (i ) ( ) (i ) (0) (0) ' , n 1 ,

( ) ( ) , (i ) ( ) (i ) (0) , n 1 .

n

n n

n

n n n n n k k

k

SHA SHA

     

        

      ⁻   ^{− −}

=

  = + −  

  = + − +  −  

  = + −



+  

Proof :

 

2 2

2

(i ) (i ) 2

0 0

2

(i )

0

(i )

0

( ) since ( )', ,

( )', ( ) ( ) (i ) ( ) (0) , n 1.

( ) since ( )",

( )' :

( )"

n n

n

integrating by part SHA

SHA e e d SHA

SHA

e d s

e d

     

  

 

         

 

  

 

− + − +

 − +

 

 

 =    +  = + −  

 

= 



 

² ^\

2 2 2

,

(i ) ( )', (0)' (i ) ( ) (i ) (0) (0) , n 1,

:

n n n

integrating SH

by p

A SH

arts

       A     

= +  −  = + − +  −  

( 𝛱𝛪 ) For this n-th derivative can be prove using mathematical induction.

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2326-9865

Theorem2(Convolution): Suppose that₁( ) and₂( ) are integrable functions with SHA- transforms on [0, +∞),and there are SHA-transforms defined when 0, SHA



1( )



= ,



2( )



SHA   =H,Then the SHA- transform of their convolution₁( )  ₂( ) can defined by:

(5) SHA



 1 2



( ) = ¹ ( ) ( ) 

 H Proof:

2 2

2

(i ) (i )

1 2 1 2 1 2

0 0 0

(i )

1 2 1 2

0 0

( ( ) * ( )) ( ( ) * ( )) ( ) ( )

( ) ( ) 1 ( ) ( ).

n n

n

SHA e e d

e d SHA SHA

     

  

         

      



  

− + − +

 

− +

  =   =   −

=   − =

  

 

Next theorem shows the relationship between SHA-transform and Laplace transform.

Theorem3(Duality between Laplace Transform and SHA-Transform): Let ( ) be a differentiable function if Fis Laplace transform of ( ) and ^SHA

( )

^ is SHA-Transform of ( ) , then:

(6) SHA

( )

 =F i

(

²ⁿ +

)

Proof:

From Definition 1:

( ) ( )

⁽ ² ⁾

( )

0

[ ; ]

i n

SHA  =SHA    =^



e⁻ ^{  }⁺   d , since Laplace transform is

denoted by

( ) ( ) ( )

0

[ ; ] ^pt

F p f t p e f t dt



= =



− ^{, Then}^SHA

^{( )}

^ ⁼^^{F i}

(

²ⁿ^{ }⁺

)

_.

Theorem4.(Linearity property):Let ₁( ), ₂( ),..., _n( ) are continuous functions have SHA- transforms and  ₁, ₂,..._n are arbitrary constants ,then linearity property is defined by



1 1( )

 

2 2( )



...



_n _n( )



1



1( )



2



2( ) +...



_n



_n( )



SHA   +    + +    =SHA   +SHA   + SHA   Proof:

Let ₁( ) and ₂( ) are continuous functions for  0:

   

     

2

2 2 2

(i )

1 1 2 2 1 1 2 2

0

(i ) (i ) (i )

1 1 2 2

0 0 0

1 1 2 2

( ) ( ) ... ( ) ( ) ( ) ... ( ) e

( ) e ( ) e ... ( ) e

( ) ( ) ... ( )

n

n n n

n n n n

n n

SHA d

d d d

SHA SHA SHA

  

        

             

           

     



− +

  

− + − + − +

 +  + +  =  +  + + 

=  +  + + 



  

Illustrative Examples : The following examples usingSHA-transform to solve initial value problems described by ordinary differential equation.

Example 1: Solve ^y^{ +}²^y ⁼^{0 ; y 0}

( )

⁼¹

Applying SHA-transform for both sides of original equation:

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2326-9865

   

(

²

) ^{ } ^{( )} ^{ }

2 0

ⁿ 0 2 0 ,using the initial , :

SHA y SHA y

i   SHA y y SHA y condition to obtain

 + =

 + − + =

(

²

) ^{ }

2

^{ }

0

( (

²

)

2

) ^{ } ^{ } ₍

2

₎

.

2

n n

i SHA y SHA y i SHA y SHA y n

i

      

 + + − =  + + =  =  

+ +

Applying the inverse of SHA-transform for the last equation:y =e⁻²^ . Example 2: Solve ^y^^{+ =}^y ^{0 ; y 0}

( )

⁼^{0, y 0}^

( )

⁼¹

Applying SHA-transform for both sides of original equation:

   

(

²

)

²

^{ } (

²

) ^{( )} ^{( )} ^{ }

0

ⁿ ⁿ 0 0 0 :

SHA y SHA y

i  SHA y  i  y y SHA y Using the initial conditions

 + =

 + − + −  + =

(

²

)

²

^{ } ^{ } ( (

²

)

²

) ^{ } ^{ } ₍

2

₎

²

0 1

1

n n

n

i SHA y SHA y i y SHA y

i

      

 + + − =  + + =  =  

+ + Applying inverse of SHA-transform for the last equation, we get: y =sin.

4. Fuzzy SHA-Transform.

In the last decades, fuzzy differential equations have been used in many fields due to their numerous and important applications in a wide range of fields[7,9]. In order to keep pace with the rapid development and progress in the field of fuzzy differential equations, we presented this paper that contains a new method for solving this type of equations[5,6], In this section, we convert the SHA-Transform into a fuzzy SHA-transformand introduce some properties as well as fuzzy theories of the first and second orderthen use these formulas to solve realistic problem which is drug concentration in blood.

Definition 3[4]:A fuzzy number in parametric form is a pair

( )

^{ }^, of functions

( ) ( )

^, ^{, 0} ¹

      , which satisfy the following requirements:

1. ^{ }

( )

is a bounded non-decreasing left continuous function in

(

^0,1



,and right continuous at0 . 2. ^{ }

( )

is a bounded non-increasing left continuous function in

(

^0,1



,and right continuous at 0 . 3. ^{ }

( )

^^{ }

( )

^{, 0}^{ }^ ¹. For arbitrary^{    }⁼

( ) ( )

^, ^for^{    }⁼

( ) ( )

^, ^and  0we

define addition  ,subtraction   and scalar multiplication by  0 as following (a) Addition:          =

( )

+

( ) ( )

^, +

( )

.

(b) Subtraction:           =

( ) ( ) ( ) ( ) ( )

− ^, ^, − . (c) Scalar multiplication:

( )

, 0 , 0

  

    

  

 

=  

  

 

.

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Definition4.[4] :Let  , ( E the set of all fuzzy number) If there exists  such that

  + = then is named Hukuhara difference of  , and is identify by  . Note: The sign " " always stands for Hukuhara difference.

Definition 5.[7]:Let ^{ }

( ) ( )

^: ^{a b}^, ^{→ }( E the set of all fuzzy number) continuous fuzzy- valued function and 0

(

a b,

)

,it has been that is strongly generalized differential at ₀if an aspect exists an element  ^\

( )

0 such that :

1- For all ^ h0sufficiently small 

(

0 +h

)

 

( )

0 , 

( )

0  

(

0−h

)

and the

( ) ( ) ( ) ( ) ( )

\ 0 0 0 0

0 0 0

lim li m .

h h

im t i h

l i s         h

 

₊ ₊

→ →

+ −

= =

Or

2- For all ^ h0sufficiently small  

( )

0  

(

0 +h

)

, 

(

0−h

)

 

( )

0 and the

( ) ( ) ( ) ( ) ( )

\ 0 0 0 0

0 0 0

lim lim .

h h

lim h h

h h

it is        

 

₊ ₊

→ →

+ −

= =

− −

Or

3- For allh 0sufficiently small  

(

0+h

)

 

( )

0 , 

(

0−h

)

 

( )

0 and the limitis

( ) ( ) ( ) ( ) ( )

\ 0 0 0 0

0 0 0

lim lim .

h h

       

 

₊ ₊

→ →

+ −

= =

−

Or

4- For allh 0sufficiently small  

( )

0  

(

0+h

)

, 

(

0 −h

)

 

( )

0 and the limit is

( ) ( ) ( ) ( ) ( )

\ 0 0 0 0

0 0 0

lim lim .

h h

       

 

₊ ₊

→ →

+ −

= =

−

Theorem 5. [5]: Assume that :R →  (E: the set of all fuzzy numbers)be a differentiable function and indicate^{ }

( )

=

(

     

(

^;

) (

^, ^;

) )

for each ^^

 

^0,1 ^{. Then:}

1- If is the first form, then^{  }

(

^;

)

^and^{  }

(

^;

)

are differentiable functions and

( ) ( ) ( )

\ ; , ;

  =      .

2- If is the second form, then^{  }

(

^;

)

^and^{  }

(

^;

)

are differentiable functions and

( ) ( ) ( )

\ ; , ;

  =      .

Theorem 6. [3,6]: Let:R →  (E: the set of all fuzzy numbers)and is represented by

(

^;

) (

^, ^;

)

     

 

  . For any fixed ^^

(

^0,1



assume that ^{  }

(

^;

)

^and ^{  }

(

^;

)

are Riemann- integrable functions on [a, b] for everyba, and assume there are two positiveM_ andM_ such that^b

(

^;

)

a

d M_

    



^and ^b

⁽

^;

⁾

a

d M_

    



for every b a.Then, ^{ }

( )

is an improper

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2326-9865

fuzzy Riemann-integrable on



^a^,^

)

^. Furthermore, we have:

( ) (

^;

)

^,

(

^;

)

a a a

d d d

          

   

=  

 

  

^.

Definition6: Let^{ }

( )

be a continuous fuzzy-valued function Suppose that ⁽ⁱ² ⁾

( )

0

e n^{  } d

^



⁻ ⁺    ^is improper fuzzy Riemann-integrable on [0,), then ⁽ⁱ² ⁾

( )

0

e n^{  } d

^



⁻ ⁺    is called SHA^· -transform and it denoted by:

·

( )

^·

( ) (

²

)

⁽ⁱ² ⁾

^{( )}

0 n n

SHA  =SHA  = i  + ^



e⁻ ^{  }⁺   d

( ) ( ) ( )

2 2 2

(i ) (i ) (i )

0 0 0

; ,

2 : ⁿ ⁿ ⁿ ;

Sine from theorem to get ^



e⁻ ^{  }⁺    d = ^



e⁻ ^{  }⁺     d ^



e⁻ ^{  }⁺    d Using the definition of classic SHA-transform :

·

( ) ( )

·

( ) ( )

2

(i )

0

(i )

0

; ; ,

; ;

n

SHA e d

  

       



− +



− +

  =

 

  =

 



So: ^SHA·   

(

^;

)

=^SHA^  

(

^;

)

^^,SHA^  

(

^;

)

^ .

Theorem7. Duality Between Fuzzy Laplace – SHA^· transforms

If 𝐹(p) is fuzzy Laplace transform of  ( )and ^SHA

( )

^ ^is ^SHA^· -transformof  ( )then

( ) (

²ⁿ

)

SHA  =F  + .

Proof: Let  ( )E then: ( ² ) ( ² )

0 0

( ) ( ; ) , ( ; )

n n

i i

SHA  _{= }^   ^ e⁻ ^{  }⁺ d    ^ e⁻ ^{  }⁺ d^_



 



Since fuzzy Laplace transform denoted by:

0 0

( ) ( ; ) ^pt , ( ; ) ^pt

F p _{= }^^f t  e⁻ dt ^f t  e⁻ dt^_



 



( ² ) ( ² )

0 0

( ) ( ; ) , ( ; )

n n

i i

SHA  ^   ^ e⁻ ^{  }⁺ d    ^ e⁻ ^{  }⁺ d^

 =  



 

 ^Thus^SHA^{( )}^ ⁼^^{F i}

(

²ⁿ^{ }⁺

)

Theorem 8. Let 1

( ) ( )

, 2 ,..., ( )

     n be continuous fuzzy- valued functions and suppose that

1, 2,..., _n

   are constant, then

·

( ( ) ) ( ( ) ) ( ( ) ) (

^·

( ) ) (

^·

^{( )} )

·

( )

1 1 2 2 2 1 1 2 2

2

...

n

SHA SHA SHA

SHA

              

  

    =     

 

   

e e e e e

e

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2326-9865

Vol. 71 No. 3s3 (2022) 220 http://philstat.org.ph proof:

·

( ( ) ) ( ( ) )

( ( ) ) ( ^{( )} ) ( ( ) ) ( ( ) )

( ( ) ) ( ^{( )} ) ( ^{( )} ) ( ^{( )} ) ( ^{( )} ) ( ^{( )} )

( ( ) ) ( ^{( )} ) ( ^{( )} ) ( ^{( )} )

2

1 1 2 2 (i )

1 1 2 2 2

2 0

(i )

1 1 2 2 1 1 2 2

0

1 1 1 1 2 2 2 2

(i )

...

= ... , ...

, ,

=

.

n

n n

n n n n

SHA e d

e d

e

  

     

          

  

                   

           





− +



− +

  

 

   =     

 

 

 

 + + +  + + +

   

  + 

   

+



e e

e e e

e

( ( ) ) ( ^{( )} )

( ( ) ) ( ^{( )} ) ( ^{( )} ) ( ^{( )} )

( ( ) ) ( ^{( )} ) (

^·

^{( )} ) (

^·

^{( )} )

·

( )

2 2

2

0

(i ) (i )

1 1 1 1 2 2 2 2

0 0

(i )

1 1 2 2

0

2

.. ,

= , ,

... , d =

...

n n

n

n n n n

n

d

e d e d

e SHA SHA

SHA

     

  

      

               

             

  



 

− + − +



− +

 

 

 +  

   

 

  +  

   

 

+ +       

   



 



^e ^e

e

Theorem 9. Assume that ^{ }^\

( )

be continuous fuzzy-valued function and ^{ }

( )

the primitive of

( )

 \ on



^0,^

)

^{, then:}

1. ^SHA· ^_^{ }^\

( )

^{ =}_ ⁽ⁱ²ⁿ^{ }⁺ ⁾^SHA ^_^{ }

( )

^_ ^

( )

⁰ ^,where^is differentiable in the first form

2. ^SHA· ^^{ }^\

( )

^{ = −} ^

( )

⁰

(

⁻ⁱ²ⁿ^{ }⁺

)

^SHA^^{ }

^{( )}

^, where is differentiable in the second form

proof: Since ^{ }^\

( )

is continuous fuzzy-valued function then there are two cases as following:

Case 1.Ifis the first form, for any arbitrary^^

 

^0,1 ^,

· ^\

( )

^\

(

^,

)

^, ^\

(

^,

)

SHA    = SHA     SHA ^   ^ From Theorem 1/1:

( )

( ) ( )

( ) ( ) ( )

\ 2

(i ) , 0,

n

SHA SHA

         

 = +  −  

     

 = +  −  

 

 

By Theorem 2 : ^SHA· ^_^{ }^\

( )

^{ =}_ ⁽ⁱ²ⁿ^{ }⁺ ⁾^SHA^_^{ }

( )

^_ ^

( )

⁰

Case 2.

If is the second form, for any arbitrary ^^

 

^0,1 ^,

·

( ) (

^,

)

^,

(

^,

)

SHA  =SHA    SHA^   ^ From Theorem 1/1:

( ) ( ) ( )

\ 2 2

(i ) , (i ) 0,

n n

SHA SHA

           

 = +  − +  

 

 

 = +  − +  

     

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2326-9865

By Theorem 2 :^SHA· ^_^{ }^\

( )

^{ = −}_ ^⁽ⁱ²ⁿ^{  }⁺ ⁾

( )

⁰ ⁽ⁱ²ⁿ^{ }⁺ ⁾^SHA ^_^{ }

( )

^_

Theorem10. Assume that,^{ }

( )

^,^{ }^\

( )

are continuous fuzzy-valued functions on



^0,^

)

^{, fuzzy}

derivative of Fuzzy SHA-transform about second order it will be:

1. If  , ^\ are first form then:

· ^\\

( ) (

²ⁿ

)

²

^{( )} (

²ⁿ

) ^{( )}

⁰ ^\

^{( )}

⁰

SHA

 

 = i

 

+ SHA

 





i

  

+



2. If  is first form and ^\second form then:

· ^\\

( ) (

²ⁿ

)

⁽⁰⁾

(

²ⁿ

)

²

^

^{( )}

^

^\

^{( )}

⁰

SHA

 

 = −



i

  

+ i

 

+ SHA

  

3. If  is second form and ^\first form then:

· ^\\

( ) (

²ⁿ

)

⁽⁰⁾

(

²ⁿ

)

²

^

^{( )}

^

^\

^{( )}

⁰

SHA

 

 = −



i

  

+ −i

 

+ SHA

 

−



4. If  , ^\ are second form then:

· ^\\

( ) (

²ⁿ

)

²

^

^{( )}

^ (

²ⁿ

)

⁽⁰⁾ ^\

^{( )}

⁰

SHA

 

 = i

 

+ SHA

  

−i

  

+ −



Proof:

1. ,^\are first form and for any arbitrary ^^

 

^0,1 ^{, then:}

· ^{\ \}

( )

^{\ \}

(

^,

)

^, ^{\ \}

(

^,

)

SHA ^   = SHA^   ^ SHA^   ^Form Theorem 1/2:

( ) ( ) ⁽ ⁾ ( ) ^{( )} ^{( )}

\\ 2 2 2 \

, 0, 0,

n n

SHA i SHA i

             

 = +  − + −

 

 = +  − + −

 

 

By Theorem 2:^SHA· ^

^{ }

^\\

( )

^{ =}

(

ⁱ²ⁿ

^{ }

⁺

)

²^SHA^

^{ } ^{( )}

^

^ (

ⁱ²ⁿ

^{  }

⁺

) ^{( )}

⁰

^

^\

^{( )}

⁰

2. is first form and^\ second form, then:

· ^{\ \}

( )

^{\ \}

(

^,

)

^, ^{\ \}

(

^,

)

SHA ^   = SHA^   ^ SHA^   ^Form Theorem 1/2:

( ) ( ) ⁽ ⁾ ( ) ^{( )} ^{( )}

\\ 2 2 2 \

, 0, 0,

n n

SHA i SHA i

             

 = +  − + −

 

 = +  − + −

 

 

By Theorem 2 :^SHA· ^

^{ }

^\\

( )

^{ = −}

^ (

ⁱ²ⁿ

^{  }

⁺

)

⁽⁰⁾

(

⁻ⁱ²ⁿ

^{ }

⁺

)

²^SHA^

^{ } ^{( )}

^

^

^\

^{( )}

⁰ 3. is second form and^\ first form, then:

· ^{\ \}

( )

^{\ \}

(

^,

)

^, ^{\ \}

(

^,

)

SHA    = SHA^   ^ SHA^   ^Form Theorem 1/2:

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2326-9865

( ) ( ) ⁽ ⁾ ( ) ^{( )} ^{( )}

\\ 2 2 2 \

, 0, 0,

n n

SHA i SHA i

             

 = +  − + −

 

 = +  − + −

 

 

^{ }

^\\

( )

^{ = −}

^ (

ⁱ²ⁿ

^{  }

⁺

)

⁽⁰⁾

(

⁻ⁱ²ⁿ

^{ }

⁺

)

²^SHA^

^{ } ^{( )}

^⁻

^

^\

^{( )}

⁰ 4. ,^\ are second form and for any arbitrary ^r^

 

^0,1 ^{, then:}

· ^{\ \}

( )

^{\ \}

(

^,

)

^, ^{\ \}

(

^,

)

SHA    = SHA^   ^ SHA^   ^ Form Theorem 1/2:

( ) ( ) ⁽ ⁾ ( ) ^{( )} ^{( )}

\\ 2 2 2 \

, 0, 0,

n n

SHA i SHA i

             

 = +  − + −

 

 = +  − + −

 

 

^{ }

^\\

( )

^{ = −}

(

ⁱ²ⁿ

^{ }

⁺

)

²^SHA^

^{ } ^{( )}

^

^ (

ⁱ²ⁿ

^{  }

⁺

)

⁽⁰⁾⁻

^

^\

^{( )}

⁰

Application: The following real word example is illustrated the usage ofSHA^· -transform.

Example3[15]: For an oral dosage of a drug as tablet or capsule is determined by the amount of drug present in the plasma at a given time. Thus if we let ^^\

( )

^t to be the rate of drug from a oral dosage that is released and ^

( )

^t is the response of the human body to a unit input of drug which represents a function of the time. The equation will be:

( ) ( ) ( ) ( ) ( )

\ 1

, 0 =[ 0; , ; 0 ]

t t

 =t     

There are two cases:

1. If ^

( )

^t is the first form, then:

fuzzy SHA-transform for both sides of original equation, we get:

· ^[ ^\

( )

^] ^· ^[

( )

^]

SHA t =SHA t Last equation becomes:

(

ⁱ²ⁿ^{ }⁺

)

^SHA^[^{ }

^{( )}

^t; ^]⁻^{ }

^{( )}

^0; ⁼^SHA^[^{ }

^{( )}

^t; ^],

(

ⁱ²ⁿ^{ }⁺

)

^SHA^[^{ }

^{( )}

^t; ^]⁻^{ }

^{( )}

^0; ⁼^SHA^[^{ }

^{( )}

^t; ^],

So:

( )

(

²

) ^{( )} ^{( )} (

²

) ^{( )}

[ t; ] 0; , [ t; ] 0;

1 1

n n

SHA SHA

i i

 

       

   

= =

+ − + −

Now we use inverse FSHA-Transform for last equations:

( )

^t; ^e^

( ) ( )

^0; ^, ^t; ^e^

( )

^0;

  =     =   2. If ^{ }

( )

is second form, then:

By using FSHA-Transform for both sides of original equation, we get:

· ^[ ^\

( )

^] ^· ^[

( )

^]

SHA  =SHA 

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2326-9865

Vol. 71 No. 3s3 (2022) 223 http://philstat.org.ph Last equation becomes:

( ) ^{( )} ( ) ^{( )} ^{( )}

2 2

[ t; ] 0; [ t; ],

[ t; ] 0; [ t; ]

n n

i SHA i SHA

          

          

+ − + =

By solve above equation:

( ) ( )

( ) ^{( )} ( )

( ) ^{( )}

( ) ( )

( ) ^{( )} ( )

( ) ^{( )}

2 2

2 2 2

2 2

[ t; ] 0; 0;

1 1

[ t; ] 0; 0;

1 1

n n

i i

SHA

i i

SHA

i i

     

     

   

     

     

   

+ +

= −

+ − + −

+ +

= −

+ − + −

Now we use inverse FSHA-Transform for last equations:

( )

^0; ^{cosh t}

( )

^0; ^{sinh t}

( ) ( )

^0; ^, ^0; ^{cosh t}

( )

^0; ^{sinh t}

( )

^0;

  =   −     =   −  

5- Kernel of SHA-Transform Expansion:

In this section, we are working on expanding the kernel of SHA -transform to bein the interval-   n ,

 

⁽ⁱ² ⁾

0

( ), ( ) ⁿ ,-

SHA   =^



 e⁻ ^{  }⁺ d    n First Case: If n =0then

(7)

 

⁽ ² ⁾

 

0 0

( ), ( ) , then ( ), ( )

i n

SHA   =^



 e ⁻ ^{  }⁺ d SHA   =^



 e⁻^d A special case which is of Laplace multiplied by a constant or Natural transform

Assume that for any function ( ) , the integral in equation (7) exists. So SHA-transform will be:

Function ^^{( )}^ ⁼^T ⁻¹



^SHA^{( )}^



^SHA^{( )}^ ⁼^SHA



^^{( ),}^{ }



( ) 1

 = ^SHA



^{( ),} 



 ¹^{, n=0}

 =  ( ) 

 =

 

( )

²

( ), , n 0

SHA   

 =  =

( ) ^

 =

  ( )

( )

¹

( ), 1 , 0 ,n 0.

SHA  _ 

  

 ⁺

 +

 =  =

( ) sin

 =



^{( ),}



₂ ^{, n=0}

( ) 1

SHA   

 =  + ( ) cos

 =



^{( ),}



^{( )}₂ ^{, n} ⁰

( ) 1 SHA    

 =  =

−