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 Representationof 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.
References
A .E. Gracc and M. Spann, A comparision between Fourier-Mellin descriptors and Moment based feactures for invariant object recognition using neural network, Pattern Recog. Lett.,12 (1991), 635-643.
A. H. Zemanian, Generalized Integal Transformation, Interscience Publication, New York (1968)
A. Z .Zemanian, The distributional Laplace and Mellin transformations, J. SIAM, 14(1) (1908)
Bruce Littlefield, Mastering Matlab, Prentice Hall ,Upper saddle River, New York.
C. Fox, Applications of Mellin's Transformation to the integral equations, (1933)
C.Fox Application of Mellin Transformation to Integral Equation, 3rd March, 1934, 495-502.
Derek Naylor, On a Mellin Type Integral Transforms, Journal of Mathematics and Mechanics, 12(2) (1963).
Ian N. Sneddon, The use of Integral Transforms, TMH edition 1974.
I. S. Reed, The Mellin Type Double Integral, Cambridge ,London.
Jean M. Tchuenche and Nyimvua S. Mhare, An applications of double Sumudu Transform, Applied Mathematical Sciences, 1 (2007), 31-39.
J. M. Mendez and J. R.Negrin , On the finite Hankel-Schwartz Transformation ofDistributions, Ganita, 39(1) (1988).
Rugra Pratap, Getting Started With Matlab, Oxford University Press, NewYork, 2003.
Stephane Derrode and Faouzi Ghorbel, Robust and Efficient Fourier-Mellin Transform Appoximations for Gray-Level Image Reconsrruction and Complete Invarient Description, Computer Vision and Image Understandind 83 (2001), 57-78.
S. M. Khairnar, R.M. Pise and J. N. Salunke , Applications of the Mellin type integral transform in the range (1/a, ), IJMSA (Accepted).
S. M. Khairnar, R.M. Pise and J. N. Salunke , Study Of The Sumudu Mellin Integral Transform and Its Applications, Int. J. Mat. Sci. and Engg. Appl., 4(IV) (2010), 307-325.
S. M. Khairnar, R.M. Pise and J. N. Salunke , Bilateral Laplace-Mellin Integral Transform and its Applications, International Journal of Pure and Applied Sciences and Technology (Accepted).
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