Department of Mathematics and Statistics,University of Victoria, Victoria,British Columbia V8W 3R4,Canada

E-mail:harimsri@math.uvic.ca

Minjie LUO(羅旻杰)?

Department of Mathematics,East China Normal University,Shanghai 200241,China

E-mail:mathwinnie@live.com

R.K.RAINA

Department of Mathematics,M.P.University of Agriculture and Technology, Udaipur 313001,Rajasthan,India

(Current address:10/11 Ganpati Vihar,Opposite Sector 5, Udaipur 313002,Rajasthan,India)

E-mail:rkraina7@hotmail.com

A NEW INTEGRAL TRANSFORM AND ITS APPLICATIONS?

H.M.SRIVASTAVA

Department of Mathematics and Statistics,University of Victoria, Victoria,British Columbia V8W 3R4,Canada

E-mail:harimsri@math.uvic.ca

Minjie LUO(羅旻杰)?

Department of Mathematics,East China Normal University,Shanghai 200241,China

E-mail:mathwinnie@live.com

R.K.RAINA

Department of Mathematics,M.P.University of Agriculture and Technology, Udaipur 313001,Rajasthan,India

(Current address:10/11 Ganpati Vihar,Opposite Sector 5, Udaipur 313002,Rajasthan,India)

E-mail:rkraina7@hotmail.com

In the present paper,the authors introduce a new integral transform which yields a number of potentially useful(known or new)integral transfoms as its special cases.Many fundamental results about this new integral transform,which are established in this paper,include(for example)existence theorem,Parseval-type relationship and inversion formula.The relationship between the new integral transform with the H-function and theH-transform are characterized by means of some integral identities.The introduced transform is also used to fnd solution to a certain diferential equation.Some illustrative examples are also given.

Laplace transform;Sumudu transform;natural transform;H-function;H-transform;inversion formula;Parseval-type relationship;existence theorem; Borel-Dˇzrbashjan transform;Fubini’s theorem;Weierstrass’s test;Heaviside generalized function

2010 MR Subject Classifcation33C60;44A10;44A20

1 Introduction

In this paper we consider the following new integral transform defned byin which both u∈C and v∈R+are the transform variables.As we will show in the subsequent sections,the frst transform variable u is more important.But,in many cases,the transform variable v acts simply as a parameter.A complete description of this integral transform(1.1) including its conditions of convergence will be given in a theorem in Section 2 below.

If we set ρ=0 in(1.1),it corresponds to the N+-transform(that is,the so-called natural transform)defned by

whose properties were fully studied in[2]and[7]and whose applications in solving Maxwell’s equations were considered in[8]and[24].Indeed,as indicated in[7],the natural transform N+is closely connected with the Laplace and Sumudu transforms.The Laplace transform of f(t) is defned(in the usual manner)by

and the Sumudu transform over the set A of functions given by

is defned by([6];see also[5]and[19])

Another special case of the integral transform(1.1)when u=0 corresponds essentially to a known generalization of the classical Stieltjes transform,which was investigated by(for example)Srivastava[27].

Recently,many results and applications concerning the Sumudu transform have appeared in[3,4],[9,10],[13-15],[18],[20],[25,26],[28]and[31].The relationships between the Laplace transform(1.3)and the Sumudu transform(1.4)are given by the following equations(see[6, p.5,Eq.(3.2)]):

which may be referred to as the Laplace-Sumudu duality.An analogue of this relation for double Sumudu and Laplace transforms is given in[21].Similarly,for natural transform N, the natural-Laplace duality and the natural-Sumudu duality are given by(see[7,p.108,Eqs. (2.10)and(2.11)])

respectively.

In view of the above details,the M-transform defned by(1.1)appears to be sufciently interesting to investigate because of the fact that it has useful connections with the natural transform(1.2),the Laplace transform(1.3),the Sumudu transform(1.4)and the aforementioned generalized Stieltjes transform.

There also exists another important connection of the M-transform with theH-transform. For integers m,n,p,q such that 0≦m≦q and 0≦n≦p,for the parameters ai,bj∈C and for the parameters αi,βj∈R+=(0,∞)(i=1,···,p;j=1,···,q),the H-function is defned interms of a Mellin-Barnes type integral in the following manner([29,p.10,Eq.(2.11)]and[16, pp.1-2];see also[22,p.343,Defnition E.1],[17,p.58,Eq.(1.12.1)]and[23,p.2,Defnition 1.1]):

the contour L is suitably chosen,and an empty product,if it occurs,is taken to be 1.

TheH-transform is an integral transform involving the H-function(1.7)in its kernel and is defned by([29,p.42,Eq.(4.2.2)]and[16,p.71,Eq.(3.1.1)];see also[22,p.299,Defnition 5.6.1])

We will fnd later in Section 3 that there exist some close relations between theH-transform and the M-transform.

The present paper is organized as follows.In Section 2,we frst give the existence theorem for the M-transform.Some examples are then considered and the relation between the M-transform and the H-function is further exhibited.A theorem on the M-transform of higher derivatives is also obtained.In Section 3,some integral identities including the Parseval-type theorem of M-transform are established.We also fnd some useful identities involving the M-transform andH-transform.In the concluding section,we frst prove an inversion formula for the M-transform and then use it to solve a certain diferential equation.

2 Basic Properties of the M-transform

We frst begin by establishing the existence of the M-transform defned above by(1.1).

Theorem 2.1If a function f(t)is continuous or piecewise continuous in[0,∞)satisfying the property that,for given K＞0,T＞0 and β＞0,

ProofBy using(2.1),we havethen,from(1.1)and(2.2),we get

and range of v is then(0,∞).

The second assertion of Theorem 2.1 follows immediately from the well-known Weierstrass’s test.?

From the defnition of Mρ,m-transform and Theorem 2.1,we give below its basic properties.

·Scaling Property

This property can be easily verifed by noting that

For η=ρ,we have

The following result gives images of power function and exponential function under the M-transform.

Theorem 2.2Each of the following assertions holds true:

ProofBy means of the well-known Eulerian integral

we obtain

By noting that the integrals in(2.10)above are uniformly convergent,we can interchange the order of integration to get

To evaluate the inner integral:

we apply the Mellin transform on both the sides of(2.12),and we thus fnd that

The integral I(s)given by(2.12)can now be expressed in terms of the inverse Mellin transform, and we have

Substituting(2.14)into(2.11),interchanging the order of integrations,and using the H-function defnition given by(1.7),we obtain

Now,by using the following known relation(see[16,p.31,Eq.(2.1.4)]):

(2.15)can further be simplifed to yield the desired result(2.7).

For the second result,it easily follows that

Let I1(s)denote the inner integral of(2.17).Then,by similar method as employed above in the derivation of(2.7),I1(s)can be evaluated as follows:

After substituting(2.18)into(2.17)and using(1.7),we fnd that

The proof of the third result(2.9)is similar to that of(2.7)and(2.8).We,therefore,omit its details involved.?

Remark 2.3In the special case when λ=1 in(2.7)or when a=0 in(2.8),we obtain

The evaluation of(2.8)and(2.9)make use of particular forms of the contour integral representations in(1.7).We below explore alternative way to establish(2.8)and(2.9).By using some basic properties of the H-function,the evaluation procedure would be more direct.To consider another evaluation of Mρ,m[e-at](u,v),let us recall the following expansion of the H-function (see[16,p.39,Eq.(2.3.12)]):

To apply(2.21),we set

We thus fnd that

which is equal to the right-hand side of(2.8).

Next,we fnd the M-transform of derivatives which is given in the following theorem.

Theorem 2.4(M-transform of derivatives)If f(n)(t)is the nth derivative of the function f(t)with respect to t and satisfes the assumptions(stated with Theorem 2.1)such that its M-transform exists,then

ProofSuppose that f(t)and f′(t)satisfy the conditions given in Theorem 2.1.Using the defnition of the M-transform(1.1)and integrating by parts,we have

which shows that

Also,we have

Combining(2.27)and(2.28)with(2.26),we get

Similarly,we obtain

Upon repeating this process,we are led to(2.25).?

Remark 2.5If we set ρ=0 in(2.25),then we get

By setting n-k-1=j in this last sum(2.31),we notice that 0≦j≦n-1 and k=n-(j+1). Finally,(2.31)can be rewritten as follows:

which is the corresponding result for the natural transform(see[7,p.109,Theorem 3.3]).

The following theorem shows that,under certain conditions,the M-transform of derivatives possesses a simpler form.

Thus,it follows that

In general,with the help of(2.31),we can fnd that

as desired.

3 Integral Identities Involving the M-transform

In this section,we will present some integral identities involving the M-transform which yield identities involving the well-known integral transforms and special functions.

Theorem 3.1(Parseval-Type Theorem of M-Transform)Under the hypotheses of Theorem 2.1,the following assertion holds true

In particular,when ρ1=0,

where the integral transform N+is defned by(1.2).

ProofFrom the defnition(1.1)of the M-transform,we fnd(by using the well-known Fubini’s theorem)that

The integral relation(3.2)follows rather immediately from(3.1).This completes the proof of Theorem 3.1.?

If we set ρ1=ρ2=0 in(3.1),then we at once get the following Parseval-Type Theorem for the N+-transform(See[2,p.731]).

Corollary 3.2(Parseval-type theorem for N+-transform)The following assertion holds true

Theorem 3.3Under the hypotheses of the Theorem 2.1,the following assertion holds true

Z

ProofUsing the defnition(1.1)of the M-transform,we have

It may be noted that in the above proof,the transform variable v is actually treated as a parameter.?

As a corollary of(3.4),we have the following result.

Corollary 3.4The following assertion holds true

ProofWe frst set f(t)=tλ-1in(3.4).Then,by noting that

the result(3.6)follows immediately. ?

The following two corollaries provide the relationship between the M-transform and some important integral transforms.

Corollary 3.5Under the hypotheses of the Theorem 2.1,there exists the relation given by

is the Borel-Dˇzrbashjan transform(see[1]and[12]).

ProofLet us set f(t)=ωtμω-1e-tωin(3.4).Then

The N+-transform of uμω-1e-uωcan be evaluated by observing that

where we have applied the result giving the Laplace transform of the H-function[16,p.47, Eq.(2.5.25)](with a=vω＞0 and=μω＞0).

If we use(2.16),we obtain

Finally,by substituting(3.11)into(3.10)and interpreting the integral in the left-hand side as a Borel-Dˇzrbashjan transform(3.9),we arrive at the desired result(3.8).?

More generally,if we consider

in(3.4),then we get theH-transform of the M-transform which is given by the following corollary.

Corollary 3.6There exists the relationship given by

whereHis the integral transform defned by(1.9).

ProofFrom the natural-Laplace Duality(1.6),we have

Since the Laplace transform of the H-function is given by([16,p.45,Eq.(2.5.16)])

it follows that

Also,by applying(3.15)in Theorem 3.3,we get the relation(3.12).?

4 An Inversion Formula

The inversion formula for the Laplace transform is given by([11,p.134,Eq.(3.2.6)];see also[30])

By using the duality relations(1.5)and(1.6),one can fnd the corresponding inversion formulas for the Sumudu transform and the natural transform(see[7]).We now give an inversion formula for the M-transform by using its connection with the Laplace transform.

Theorem 4.1The inversion of the M-transform defned by(1.1)is given by

provided that the integral involved converges absolutely.

ProofLet us frst write

If the variable v in the transform M defned by(1.1)can be considered as a parameter of the function F(v;t),then this integral transform can be expressed as follows:

Taking the inverse Laplace transform of both the sides of(4.4),we can then formally obtain

which completes the proof. ?

Remark 4.2If we set ρ=0,then Theorem 4.1 will become the inversion formula for the natural transform.That is,if

To illustrate the application of the introduced transform(1.1),we consider the following example.

Example(First-Order Initial-Boundary Value Problem)

where function p(t,v)is given by

and φ(v)and r(t,s;v)are known functions.It may be pointed out here that v is always considered as a parameter instead of a constant.

The above equation(4.8)can now be rewritten as follows:

By applying the M-transform with respect to t to(4.11),we get

where the variables in(u,v)denote the transform variables.

Using the elimination property(2.6),we get

Also,by means of(2.31),we obtain

The use of the condition(4.10)yields

It is now easily verifed that

and,therefore,the equation(4.15)fnally becomes

with the initial-value condition that

It is convenient to use the following notations:

Thus,clearly,(4.16)is actually an initial-value problem:

Its solution is given byor,equivalently,by

Finally,on using(4.7),we get

where θ(x)is the Heaviside generalized function which is defned by

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?Received August 12,2014;revised April 1,2015.

?Corresponding author:Minjie LUO.