Some Significant Novel identities of Fractional Laplace Transform

Fractionalization of transform has been an attention grabbing topic for the scientists since last four decades, because of its better performance as compare to classical transform. Infact fractionalization of a transform makes it much competent and powerful tool in various domains of science and engineering. In this context one of the most effective methods for handling several physical issues in mathematical physics, applied mathematics, and engineering is the Laplace transform. Because of its amazing applications in the field of dynamic and delayed differential equations, it attracted significant interest from researchers [1-10]. In 2009 Jummarie introduced the fractional Laplace transform and provided a convolution theorem of fractional Laplace transform along with some applications [11].

Even after the rigrous work by Jummarie in 2009, only a few investigators paid attention on fractionalization of Laplace transform. Some researchers mainly K.K. Sharma, Huang, Tan, Lepag, Schiff and Rudolf presented applications on fractional Laplace transforms [12, 13-17]. Recently Teekam et. al. applied the fractional Laplace transform given by Jummarie to solve fractional wave and heat equtions [18]. Although Jummarie presented consolidated definition of fractional Laplace transform but over the period of 15 years there is  still a scope of developing its analytical theory produce.

Purpose of present paper is to derive Product theorem and Parseval’s identity of fractional Laplace transform given by Jummarie using Mittage-Leffler function in enviornment of fractional calculus.

Jummarie introduced a novel definition of fractional Laplace transform and produced some properties of this newly introduced transform [11]. But many identities of the transform are still missing. In the present paper we have attempted to derive product theorem and parseval’s identity of this novel Laplace transform of fractional order.

2. PRELIMINARIES

[I] Laplace Transform

The Laplace transform of the function ft  is denoted by Fs and is defined as [2]

Fs= 0+∞e-stft dt , s > 0                                                           (1.1)

[II] Inverse Laplace Transform [2]

If  Fs  is the Laplace transform of a function ft,  i.e. Lft=Fs

Then ft  is called the inverse Laplace transform of the function Fs  and  is written as [2]

ft=L-1{F(s)}

L-1  is called the inverse Laplace transform operator.

It is inverse complex formula

If  ft  has a continuous derivative and is exponential order and if

Lft = Fs , then L-1{F(s)}  is given by

ft= 12πiγ-i∞γ-i∞estF(s)ds,     t ≥ 0,  γ > real part of all poles of F                                                                     (1.2)

and  f t = 0   for t < 0.

This result is called the complex inversion integral or formula. It is also known as Bromwich’s integral formula.

[III] Fractional Laplace Transform

Let fx  denote a function which vanishes for negative values of x, then  its Laplace’s transform Lα {fx}  of order α  (or its α -th fractional Laplace’s transform) is defined by the following expression [11].

Lα fx=Fαs=0∞Eα(-sαxα) fx(dx)α                                                     (1.3)

Provided the integral converges. It can also be expressed as

            =lim M↑∞  0MEα(-sαxα) fx(dx)α                                                 (1.4)

where s∈C,  and Eαu is the Mittage-Leffler function , for α > 0 is given by [11]

         Eα (u)=∞ k=0 ukΓ(αk+1),                                                                         (1.5)

where α ∈ C, u∈ C and Re(α ) > 0. The Mittag-Leffler function Eα (u ) is an entire function of type 1, which reduces to exponential function eu  = E1 (u ) for α =1.

 We will use following identity of Mittage-Leffler function [11]

Eα(λ(x+y)α=EαλxαEαλyα.  

[IV] Inverse Fractional  Laplace Transform

 Given the Laplace’s transform (1.3) that we recall here for convenience [11]:

Fαs=0+∞Eα-sαxαfx(dx)α,   0<α<1,                                      (1.6)

one has the inverse formula

fx=1(Mα)α-i∞+i∞EαsαxαFαs(ds)α                                                        (1.7)

3. SOME NOVEL IDENTITIES OF FRLT

In this section we have attempted to derive some identities of FRLT.

Following definition of fractional convolution was given by Jummarie  [11]

Fractional Convolution

Convolution of order α  of the two functions fx  and g(x ) is defined by [11]

(fx*g(x))α=0xfx-ugu(du)α.                                                    (1.8)

He presented following convolution theorem using the definition of fractional convolution (1.7) for fractional Laplace transform [11].

Let  Lαfx=Fαs ,

Lαgx=Gαs

Lαfx.gxα=0∞Eα(-sαxα) fxgx(dx)α .                 using (1.3)

Inverse transform of  g (x)

g (x) = 1(2π)α0∞Eα(σαxα)Gασ(dσ)α

Lαf(x)*gxα=1(2π)α0∞Eα(-sαxα) fx0∞Eα(σαxα)Gασ(dσ)α(dx)α .

Interchange the integrals

1(2π)α0∞Gα(σ)0∞Eα(-sαxα)Eα(σαxα)fx(dx)α(dσ)α

Use the Mittage-Leffler property [11]

EαaαxαEαbαxα=Eα(a+b)αxα

We get

Eα-sαxαEασαxα=Eα-(s-σ)αxα

= 1(2π)α0∞Gα(σ)0∞Eα[-(s-σ)αxαfx(dx)α ] (dσ)α

= 1(2π)α0∞Fαs-σGα(σ) (dσ)α

By definition of fractional convolution

(Fα * gα)αs=0∞Fαs-σGα(σ) (dσ)α

Hence,

Lαfx.gxα=1(2π)α(Fα * gα)αs

Let  Lαfx=Fαs , Lαgx=Gαs ,

Using the inverse fractional Laplace transform.

g (x)= 1(2π)α0∞Eα(sαxα)Gαs (d s)α

Multiplying both sides by fx

fxg (x) = 1(2π)αf (x) 0∞Eα(sαxα)Gαs (d s)α

0∞fxgx(dx)α=1(2π)α0∞0∞f(x)Eα(sαxα)(dx)αGαs (d s)α

Changing the order of integration and using the transform definition.

0∞fxEα(sαxα) (dx)α=Fα-s

Therefore,

0∞fxgx(dx)α=1(2π)α0∞Fα-sGαs(ds)α

0∞fxgx(dx)α=1(2π)α0∞Fαsgα-s(ds)α
 Hence,

0∞fxgx(dx)α=1(2π)α0∞Fαsgα-s(ds)α

CONCLUSION

Fractional Laplace transform is also an area of interest due to its applicability and modifiability in diverse disciplines of engineering and sciences but theoretical development of fractional Laplace transform still needs attention of investigators. Parseval identity of transforms proved to be significant phenomena in signal processing area. Similarly, Product theorem is also very important for specific applications. We have attempted these two significant theorems for fractional Laplace transform. Digital computation of our results can engendre variety of applications in multiple sectors of science and engineering.

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