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Applications of the Bi-Lateral Laplace-Mellin Integral Transform in the Range [0,0] to (¥,¥)

Sarita Poonia1*, Rachana Mathur2

2Department of Mathematics, Govt. Dungar College, Bikaner, Rajasthan, INDIA.
1
Research Scholar, Govt. Dungar College, Bikaner, Rajasthan, INDIA.

Corresponding Addresses:

[email protected]

Research Article


Abstract: In this paper we discuss Bi-Lateral Laplace-Mellin Integral Transform technique for solving initial value problem. This transform is studied in the range of [0,0] to (1). We investigate the properties and theorems like inversion theorem, convolution theorem, Parseval’s theorem and some properties by using Ramanujan’s formula. To illustrate the advantages and use of this transformation Cauchy’s differential equation have been solved. We have also studied graphical representation of Bi-Lateral Laplace-Mellin Integral Transform using Matlab.

Keywords: Laplace Transform, Finite Mellin Transform, Integral Transform, Double Bi Lateral Laplace Transform, Convolution and Parseval’s theorem

 

1.Introduction

The Double Bi Lateral Laplace Transform is used to find the Bi-Lateral Laplace-Mellin Integral Transform in the range  [0,0]  to  (1).We have derived the different properties like Linear property, Scaling Property, Power Property. Inversion Theorem, Convolution Theorem, Parseval  Theorem,  First and Second Shifting theorems are also obtained by using Ramanujan’s  formula. We study nth order derivative, the solution of the Cauchy’s Linear differential equation using the Bi-Lateral Laplace-Mellin Integral Transform in the range  [0,0]  to  (1) and the solution is graphically represented by using Matlab.

 

2.Preliminary Result

The Double Bi-Lateral Laplace Integral Transform in 1 to 1 is

 1dx dt    (1)

Whenever this double integral exists and s > 0, p > 0

Substitute  y = e-t , dy = - e-tdt , dy = -y dt  and z = e-x ,dz = - e-xdx , dz = -z dx

If t = -1 then y = 1  and t = 1 then y = 0 ; if x = -1 then z = 1 and x = 1  then z = 0  then

1 dz dy

Or 1dx dy    (2)

Provided this double integral exists and s > 0 , p > 0

This is the  Bi-Lateral Laplace-Mellin Integral Transform (BLLMIT) in the range  [0,0]  to  (1).

It is denoted by 1, then 

1dx dy    (3)

 

3. Result and Discussion      

3.1. Properties

3.1.1: Linearity Property

The BLLMIT is a linear operation theorem for the functions f(x,y) and g(x,y) and 1 and 1 are constant ,then the BLLMIT in [0,0] to (1).is

1dx dy                                                   

then  1 + 1   (4)

3.1.2:Scaling Property

The scaling property for BLLMIT in [0,0] to (1).is

1dx dy

Then

 1 (5)         

3.1.3:Power Property   

The power property for BLLMIT in [0,0] to (1).is

1dx dy

Then 

1      (6)


3.2.Main Results

3.2.1:Inversion Theorem

Assume that 1 is a regular function in the strips 1 < r  ( ‘r’ be real number)  of  the s-plane and  0 < c < v1 ,  c1 -i1s1 c1 + i1 where c1 is constant and  1 < q (‘q’ be real number ) of the p-plane and  0 <c <v2 , c2 -i1p1 c2 + i1 where c2 is constant. Then the BLLMIT in [0,0] to  (1).is

1dx dy

then  1 dp

Proof

If  1 , then its inverse is

1ds,


and the Mellin Transform is

1,

Then its inverse is

1.

The BLLMIT in [0,0] to (1) is

1dx dy

then

1 dp] dx dy

=1 dp [1dx dy]

=1 dp[1]

=1 dp[ [1[1]

=1 dp [1][1]

=1 dp

By assuming1=1  and  1=1


This is the Inverse of Laplace–Mellin Integral Transform.

It is denoted by 1

1=1 dp

Or         1 dp  (7)

 

3.2.2.Convolution Theorem      

The BLLMIT in [0,0] to (1)  is

1dx dy

Then    1

1 dp     


Proof

The BLLMIT in [0,0]  to   (1)  .is

1dx dy  then

1dx dy

1 dp]dx dy

=1 dp[1]

=1 dp

1

1 dp     (8)


3.2.3:Parsavals Theorem          

The BLLMIT in [0,0] to   (1) .is

1dx dy

Then 
1

1 dp           


Proof

The BLLMIT in [0,0] to   (1) .is

1dx dy    then

1dx dy

=1 dp]dx dy

=1 dp[1]

=1 dp

1

1 dp    (9)


3.2.4:Definitions 

(a) Unit Step Function :-

If  H(t) = U(t) = 1, when t > 0

                     = 0, when t < 0    

Then  H(t) = U(t) is known as the unit step function.

(b) Heviside Unit Step Function

If H(t-a) = U(t-a) = 1, when t > a

                          = 0, when t < a

Then H(t-a) = U(t-a) is known as the Heviside Unit Step Function.


3.2.5: First Shifting Theorem

The BLLMIT in [0,0] to   (1) .is

1dxdy    then1

Proof

The BLLMIT in [0,0] to   (1) .is

1dxdy   then

1dxdy

                                          =1dxdy

= 1          

1          (10)


3.2.6:Second Shifting Theorem

The BLLMIT in [0,0] to   (1) .is

1dxdy  then

1 

Proof

The BLLMIT in [0,0] to   (1) .is

1dxdy  then

1dxdy 

Substitute x-c=u, dx=du , if x=0 then u=-c and if x=1 then u=1

1dudy

                                                      1dudy

                                                      1

1      (11)

 

1dxdy 

Substitute x-c=u, dx=du , if x=0 then u=-c and if x=1 then u=1

1dudy

                                                      1dudy

                                                      1

1      (11)

 

3.2.7:Theorem ( Ramanujan’s Formula)                          

If 1

Then   1

1

1

1

1(12)

Where  1


This is the Laplace transform of f (x,-p) w.r.t. parameter s > 0 , denoted by L[f(x,-p),s] .

 

3.3.Derivatives

Theorem: Suppose that 1 is continuous for all t ≥ 0 and z ≥ 0 satisfying (1.2) for some value 1,1 and m has a derivative 1 which is piecewise continuous on every finite interval in the range of t ≥ 0 and z ≥ 0.Then by using the Bi-Lateral Laplace-Mellin Integral transform, the derivative of 1 exists when s >1 and p >1 and 1 for all  t ≥ 0, z ≥ 0 for some constants.

3.3.1: BLLMIT of first order partial derivative of 1 w.r.t. 1 

The BLLMIT in [0,0] to   (1) .is

1dxdy     then

1dxdy

                                   1

                                   1

                                   1

     By assuming 1     

                                     1     

        1    (13)

Where 1

 

3.3.2: BLLMIT of second order partial derivative of 1 w.r.t. 1

1 

                                        1

  1

1   1

1(By Using DUIS)

                                        1

                                        1]

                                        1

1      (14)

Where  1


3.4: Applications:

The Cauchy’s linear Differential Equation is

1  ,x is constant variable.

The BLLMIT of  1 

1

                                         1

=1                                      

1

                                            1

    If   1     then

         1

1 

                               1         (15)

         Where   1   

This is the required BLLMIT of Cauchy’s Linear Differential Equation.


3.5.  Graphical Representation    of BLLMIT of Cauchy’s Linear Differential Equation

                                 1                                                                                      where  1       

1

Plot of (y,p): We consider 1 i.e. BLLMIT in [0,0] to (1) on x-axis and p(parameter) on y-axis

4. Conclusion:

We have obtained interesting results of BLLMIT by using Ramanujan’s formula .To illustrate the advantages and use of these transforms, some important differential equation has been solved. We have also studied graphical representation of the solution using matlab.

 

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