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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.

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 (). 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  ().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  () and the solution is graphically represented by using Matlab.

2.Preliminary Result

The Double Bi-Lateral Laplace Integral Transform in  to  is

dx 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 = - then y =   and t =  then y = 0 ; if x = - then z =  and x =   then z = 0  then

dz dy

Or dx 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  ().

It is denoted by , then

dx 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  and  are constant ,then the BLLMIT in [0,0] to ().is

dx dy

then   +    (4)

3.1.2:Scaling Property

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

dx dy

Then

(5)

3.1.3:Power Property

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

dx dy

Then

(6)

3.2.Main Results

3.2.1:Inversion Theorem

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

dx dy

then   dp

Proof

If   , then its inverse is

ds,

and the Mellin Transform is

,

Then its inverse is

.

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

dx dy

then

dp] dx dy

= dp [dx dy]

= dp[]

= dp[ [[]

= dp [][]

= dp

By assuming=1  and  =1

This is the Inverse of Laplace–Mellin Integral Transform.

It is denoted by

= dp

Or          dp  (7)

3.2.2.Convolution Theorem

The BLLMIT in [0,0] to ()  is

dx dy

Then

dp

Proof

The BLLMIT in [0,0]  to   ()  .is

dx dy  then

dx dy

dp]dx dy

= dp[]

= dp

dp     (8)

3.2.3:Parsavals Theorem

The BLLMIT in [0,0] to   () .is

dx dy

Then

dp

Proof

The BLLMIT in [0,0] to   () .is

dx dy    then

dx dy

= dp]dx dy

= dp[]

= dp

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   () .is

dxdy    then

Proof

The BLLMIT in [0,0] to   () .is

dxdy   then

dxdy

=dxdy

=

(10)

3.2.6:Second Shifting Theorem

The BLLMIT in [0,0] to   () .is

dxdy  then

Proof

The BLLMIT in [0,0] to   () .is

dxdy  then

dxdy

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

dudy

dudy

(11)

dxdy

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

dudy

dudy

(11)

3.2.7:Theorem ( Ramanujan’s Formula)

If

Then

(12)

Where

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  is continuous for all t ≥ 0 and z ≥ 0 satisfying (1.2) for some value , and m has a derivative  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  exists when s > and p > and  for all  t ≥ 0, z ≥ 0 for some constants.

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

The BLLMIT in [0,0] to   () .is

dxdy     then

dxdy

By assuming

(13)

Where

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

(By Using DUIS)

]

(14)

Where

3.4: Applications:

The Cauchy’s linear Differential Equation is

,x is constant variable.

The BLLMIT of

=

If        then

(15)

Where

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

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

where

Plot of (y,p): We consider  i.e. BLLMIT in [0,0] to () 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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