A new integral transform and its applications

Acta Mathematica Scientia 2015,35B(6):1386–1400 http://actams.wipm.ac.cn

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 : [email protected]

ÛÞ#)

Minjie LUO (

†

Department of Mathematics, East China Normal University, Shanghai 200241, China E-mail : [email protected]

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 : rkraina [email protected] Abstract 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 the H-transform are characterized by means of some integral identities. The introduced transform is also used to find solution to a certain differential equation. Some illustrative examples are also given. Key words

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

2010 MR Subject Classification

1

33C60; 44A10; 44A20

Introduction In this paper we consider the following new integral transform defined by Z ∞ −ut e f (vt) dt Mρ,m [f (t)] (u, v) = (tm + v m )ρ 0 (ρ ∈ C; ℜ (ρ) ≧ 0; m ∈ Z+ = {1, 2, 3, · · · }) ∗ Received

August 12, 2014; revised April 1, 2015. author: Minjie LUO.

† Corresponding

(1.1)

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1387

in which both u ∈ C and v ∈ R+ are the transform variables. As we will show in the subsequent sections, the first 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) defined by Z ∞ N+ [f (t)] = R (s, u) = e−st f (ut) dt (s > 0; u > 0), (1.2) 0

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 defined (in the usual manner) by Z ∞
F (s) = L [f (t)] (s) = e−st f (t) dt ℜ (s) > 0 (1.3) 0

and the Sumudu transform over the set A of functions given by n t
o A = f : |f (t)| < M e τj t ∈ [0, ∞); M, τ1 , τ2 > 0 is defined by ([6]; see also [5] and [19]) Z G (u) = S [f (t)] =

0

∞

f (ut) e−t dt


u ∈ (−τ1 , τ2 ) .

(1.4)

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)]):     1 1 G = sF (s) and F = uG (u) , (1.5) s u 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)]) 1 s 1 u N+ [f (t)] = F and N+ [f (t)] = G , (1.6) u u s s respectively. In view of the above details, the M-transform defined by (1.1) appears to be sufficiently 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 the H-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 defined in

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terms 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, Definition E.1], [17, p. 58, Eq. (1.12.1)] and [23, p.2, Definition 1.1]):     (a , α ) (a , α ) , · · · , (a , α ) i 1,p 1 p p m,n  i m,n  1  = Hp,q  Hp,q z z (bj , βj )1,q (b1 , β1 ) , · · · , (bq , βq ) Z 1 = Hm,n (s) z −s ds, (1.7) 2πi L p,q where m,n Hp,q

(s) =

m Y

Γ (bj + βj s)

j=1 p Y

Γ (ai + αi s)

i=n+1

n Y

Γ (1 − ai − αi s)

i=1 q Y

,

(1.8)

Γ (1 − bj − βj s)

j=m+1

the contour L is suitably chosen, and an empty product, if it occurs, is taken to be 1. The H-transform is an integral transform involving the H-function (1.7) in its kernel and is defined by ([29, p. 42, Eq. (4.2.2)] and [16, p. 71, Eq. (3.1.1)]; see also [22, p. 299, Definition 5.6.1])   Z ∞ (a , α ) i i 1,p  m,n  (Hf ) (x) = Hp,q xt f (t) dt. (1.9) 0 (bj , βj )1,q

We will find later in Section 3 that there exist some close relations between the H-transform and the M-transform. The present paper is organized as follows. In Section 2, we first give the existence theorem for the M-transform. Some examples are then considered and the relation between the Mtransform 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 find some useful identities involving the Mtransform and H-transform. In the concluding section, we first prove an inversion formula for the M-transform and then use it to solve a certain differential equation.

2

Basic Properties of the M-transform We first begin by establishing the existence of the M-transform defined above by (1.1).

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

|f (t)| ≦ Ktmℜ(ρ) e β for all t > T ,

(2.1)

then the M-transform of f (t) given by (1.1) exists for all v ∈ (0, µ) and u such that ℜ (u) > βµ . Further, the integral in (1.1) converges uniformly with respect to the variable u, provided that ℜ (u) ≧ a > µβ . Proof

By using (2.1), we have vt

|f (vt)| ≦ Kv mℜ(ρ) tmℜ(ρ) e β ,

(2.2)

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then, from (1.1) and (2.2), we get Z |Mρ,m [f (t)] (u, v)| ≦

Z ∞ e−ℜ(u)t e−ut f (vt) dt ≦ |f (vt)| dt ρ (tm + v m ) 0 0 (tm + v m )ℜ(ρ) Z ∞ vt tmℜ(ρ) e−ℜ(u)t+ β dt ≦ Kv mℜ(ρ) 0 (tm + v m )ℜ(ρ) Z ∞ v mℜ(ρ) e−[ℜ(u)− β ]t dt ≦ Kµ

1389

∞

(2.3)

0

for ℜ (u) > µβ and in view of the inequality that 0 < v < µ. We note that, if µ > β, then the half-plane ℜ (u) > µβ will be smaller when µ becomes bigger. So we can use index β to make refinements for different functions. When we let β → ∞, then the condition (2.1) reduces to |f (t)| ≦ Ktmℜ(ρ) , 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 definition of Mρ,m -transform and Theorem 2.1, we give below its basic properties. • Scaling Property u  
 Mρ,m f α2 t (u, v) = αmρ−1 Mρ,m [f (t)] , αv (α > 0). (2.4) α This property can be easily verified by noting that Z u 
 αmρ ∞ e− α αt f (vα (αt)) d (αt) Mρ,m f α2 t (u, v) = α 0 ((αt)m + (αv)m )ρ Z ∞ −uw e α f (vαw) dw = αmρ−1 m ρ (wm + (αv) ) 0 u  = αmρ−1 Mρ,m [f (t)] , αv . α t

• Elimination Property If ρ ≧ β ≧ 0 and |f (t) | ≦ Ktmℜ(ρ−η) e β , then  m η  t m Mρ,m +v f (t) (u, v) = Mρ−η,m [f (t)] (u, v) . vm

(2.5)

For η = ρ, we have Mρ,m



tm + vm vm

ρ

 f (t) (u, v) = N+ [f (t)] (u, v) .

(2.6)


ρ m In view of (2.6), we notice that the M-transform can be used to eliminate the factor vt m + v m and thus reduce it to a simpler form. The following result gives images of power function and exponential function under the M-transform. Theorem 2.2 Each of the following assertions holds true:  
1  λ−1  v λ−mρ−1 u−λ 2,1  1, m Mρ,m t (u, v) = H1,2 vu
 , 1 mΓ (ρ) (λ, 1) , ρ, m
ℜ (ρ) > 0; λ > 0; m ∈ Z+ ,

(2.7)

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  Mρ,m e−at (u, v) = and

 
1 1, v −mρ m H 2,1 v (u + av)
 1 m (u + av) Γ (ρ) 1,2 (1, 1) , ρ, m
ℜ (ρ) > 0; a > 0; m ∈ Z+


1 1, 2,1  m H1,2 v (u + av)
 λ 1 m (u + av) Γ (ρ) (λ, 1) , ρ, m
ℜ (ρ) > 0; a > 0; m ∈ Z+ ; λ > 0 .

we obtain



v λ−mρ−1

  Mρ,m tλ−1 e−at (u, v) = Proof

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(2.8)

(2.9)

By means of the well-known Eulerian integral Z ∞
Γ (η) tη−1 e−σt dt = min {ℜ (η) , ℜ (σ)} > 0 , η σ 0   Mρ,m tλ−1 (u, v) = v λ−1 =

v λ−1 Γ (ρ)

Z

∞

e−ut tλ−1 dt ρ (tm + v m ) 0 Z ∞ Z ∞ m m e−ut tλ−1 sρ−1 e−(t +v )s dsdt. 0

(2.10)

0

By noting that the integrals in (2.10) above are uniformly convergent, we can interchange the order of integration to get Z ∞  Z   m v λ−1 ∞ ρ−1 −vm s Mρ,m tλ−1 (u, v) = s e tλ−1 e−ut−t s dt ds. (2.11) Γ (ρ) 0 0

To evaluate the inner integral:

I (s) =

Z

∞

m

tλ−1 e−ut−t

s

dt, (s > 0) ,

(2.12)

0

we apply the Mellin transform on both the sides of (2.12), and we thus find that Z ∞  Z ∞ m M [I (s)] (z) = sz−1 tλ−1 e−ut−t s dt ds 0 0 Z ∞  Z ∞ m λ−1 −ut = t e sz−1 e−t s ds dt 0 0 Z ∞ Γ (λ − mz) λ−mz−1 −ut = Γ (z) t e dt = Γ (z) uλ−mz 0
ℜ (z) > 0, ℜ (λ − mz) > 0 .

(2.13)

The integral I (s) given by (2.12) can now be expressed in terms of the inverse Mellin transform, and we have Z u−λ +i∞ I (s) = Γ (z) Γ (λ − mz) umz s−z dz. (2.14) 2πi −i∞

Substituting (2.14) into (2.11), interchanging the order of integrations, and using the H-function definition given by (1.7), we obtain Z Z  λ−1  v λ−1 u−λ ∞ ρ−1 −vm s +i∞ Mρ,m t (u, v) = s e Γ (z) Γ (λ − mz) umz s−z dzds 2πiΓ (ρ) 0 −i∞ Z Z ∞ m v λ−1 u−λ +i∞ = Γ (z) Γ (λ − mz) umz sρ−z−1 e−v s dsdz 2πiΓ (ρ) −i∞ 0

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v λ−mρ−1 u−λ = 2πiΓ (ρ)

Z

1391

+i∞

Γ (−z) Γ (λ + mz) Γ (ρ + z) (v m um )−z dz   λ−mρ−1 −λ (1, 1) v u 2,1  m m . (2.15) H1,2 v u = Γ (ρ) (λ, m) , (ρ, 1) −i∞

Now, by using the following known relation (see [16, p. 31, Eq. (2.1.4)]):     (a , kα ) (a , α ) i i i i 1,p  1,p  m,n  m,n  k (k > 0) , Hp,q z = kHp,q z (bj , kβj )1,q (bj , βj )1,q

(2.15) can further be simplified to yield the desired result (2.7). For the second result, it easily follows that Z ∞  Z ∞  −at  1 −(u+av)t−tm s ρ−1 −v m s (u, v) = Mρ,m e e dt ds. s e Γ (ρ) 0 0

(2.16)

(2.17)

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: −1 Z +i∞ (u + av) mz I1 (s) = Γ (z) Γ (1 − mz) (u + av) s−z dz. (2.18) 2πi −i∞ After substituting (2.18) into (2.17) and using (1.7), we find that    −at  (1, 1) v −mρ m . Mρ,m e (u, v) = H 2,1 v m (u + av) (u + av) Γ (ρ) 1,2 (1, m) , (ρ, 1)

(2.19)

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 2.3 In the special case when λ = 1 in (2.7) or when a = 0 in (2.8), we obtain  
1 1 v −mρ 2,1  1, m Mρ,m [1] (u, v) = H vu (2.20)
 . 1 Γ (ρ) mu 1,2 (1, 1) , ρ, m

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)]):    
∞ 1/β1 k X (a , α ) (a , α ) 1 − λ i i i i 1,p  1,p m,n  m,n   , (2.21) λz = λb1 /β1 Hp,q z Hp,q k! (bi , βj )1,q (b + k, β ) , (b , β ) 1 1 j j 2,q k=0 1/β where λ ∈ C when m > 0, while λ 1 − 1 < 1 for m > 1. Thus, by using the series expansion of e−at and the result (2.7), we have ∞   X   (−a)i Mρ,m e−at = Mρ,m ti (u, v) i! i=0  
i ∞ (1, 1) v −mρ X − av 2,1  m m u . = H1,2 v u uΓ (ρ) i=0 i! (1 + i, m) , (ρ, 1)

(2.22)

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To apply (2.21), we set 1 av  av 1 1 av  − = 1 − λm ⇒ λm − 1 = with λ m − 1 = < 1 u u u and  av m . (2.23) λ= 1+ u We thus find that   
∞   av i   −mρ X  −at  (1, 1) − v av 2,1  m m u 
= Mρ,m e 1 + H v u 1,2  u i=0 i! 1 + av (1 + i, m) , (ρ, 1)  u uΓ (ρ)    (1, 1) v −mρ av m m m 2,1   (by using (2.21)) H1,2 v u = 1+ (u + av) Γ (ρ) u (1, m) , (ρ, 1)   (1, 1) v −mρ m , = H 2,1 (u + av) v m (2.24) (u + av) Γ (ρ) 1,2 (1, m) , (ρ, 1) which is equal to the right-hand side of (2.8).

Next, we find 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 satisfies the assumptions (stated with Theorem 2.1) such that its M-transform exists, then n−1 h i X un uk (n) Mρ,m f (t) (u, v) = n Mρ,m [f (t)] (u, v) − f (n−k−1) (0) v v mρ+k+1 k=0

+

mρ vm

n−1 X k=0

k

h

i u Mρ+1,m tm−1 f (n−k−1) (t) (u, v) . k v

(2.25)

Proof Suppose that f (t) and f ′ (t) satisfy the conditions given in Theorem 2.1. Using the definition of the M-transform (1.1) and integrating by parts, we have Z ∞ −ut ′ Z 1 ∞ e−ut df (vt) e f (vt) dt Mρ,m [f ′ (t)] (u, v) = = ρ v 0 (tm + v m )ρ (tm + v m ) 0   Z ∞ 1 e−ut f (vt) 1 ∞ ∂ e−ut = − f (vt) dt. (2.26) v (tm + v m )ρ 0 v 0 ∂t (tm + v m )ρ

By considering the asymptotic property (2.1), we find for u > βv that −ut e f (vt) e−ut mℜ(ρ) −ut+ vt β → 0 as t → ∞, e (tm + v m )ρ ≦ (tm + v m )ρ |f (vt)| ≦ Kv which shows that

Also, we have

1 e−ut f (vt) lim m = 0. v t→∞ (t + v m )ρ

(2.27)

  ∂ e−ut e−ut e−ut tm−1 = −u − mρ . ρ ρ ∂t (tm + v m ) (tm + v m ) (tm + v m )ρ+1

(2.28)

Combining (2.27) and (2.28) with (2.26), we get Z Z m−1 f (0) mρ ∞ e−ut (vt) f (vt) u ∞ e−ut f (vt) ′ Mρ,m [f (t)] (u, v) = − mρ+1 + dt + dt ρ ρ+1 m m m m m v v 0 (t + v ) v (t + v ) 0

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= Similarly, we obtain

  f (0) mρ u Mρ,m [f (t)] (u, v) − mρ+1 + m Mρ+1,m tm−1 f (t) (u, v) . v v v

1393 (2.29)

  u f ′ (0) mρ Mρ,m [f ′ (t)] (u, v) − mρ+1 + m Mρ+1,m tm−1 f ′ (t) (u, v) v v v u2 uf (0) f ′ (0) = 2 Mρ,m [f (t)] (u, v) − mρ+2 − mρ+1 v v v  m−1  mρ u + m Mρ+1,m t f (t) (u, v) v v   mρ (2.30) + m Mρ+1,m tm−1 f ′ (t) (u, v) v

Mρ,m [f ′′ (t)] (u, v) =

and

h i u3 u2 f (0) uf ′ (0) f ′′ (0) Mρ,m f (3) (t) (u, v) = 3 Mρ,m [f (t)] (u, v) − mρ+3 − mρ+2 − mρ+1 v v v v  m−1  mρ u2 + m 2 Mρ+1,m t f (t) (u, v) v v   mρ u + m Mρ+1,m tm−1 f ′ (t) (u, v) v v h i mρ + m Mρ+1,m tm−1 f (2) (t) (u, v) . v Upon repeating this process, we are led to (2.25).



Remark 2.5 If we set ρ = 0 in (2.25), then we get n−1 h i X uk un f (n−k−1) (0) . N+ f (n) (t) (u, v) = n N+ [f (t)] (u, v) − v v k+1

(2.31)

k=0

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: n−1 h i X un−(j+1) un N+ f (n) (t) (u, v) = n N+ [f (t)] (u, v) − f (j) (0) , n−j v v j=0

(2.32)

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. Theorem 2.6 If f (n) (t) is the n-th derivative of the function f (t) with respect to t and t f (t) satisfies |f (t) | ≦ Ke β (K > 0; β > 0) , then  m ρ  n−1 X uk un + t m (n) + v f (t) (u, v) = N [f (t)] (u, v) − f (n−k−1) (0) . (2.33) Mρ,m m n v v v k+1 k=0

Using the definition (1.1) of the M-transform and (2.5), we have ρ   m Z 1 ∞ −ut t m ′ Mρ,m + v f (t) (u, v) = e df (vt) vm v 0 Z ∞ u ∞ −ut 1  −ut = e f (vt) 0 + e f (vt) dt. v v 0

Proof

t

Since f (t) satisfies |f (t) | ≦ Ke β , therefore, we get from Theorem 2.1 that 1 lim e−ut f (vt) = 0. v t→∞

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Thus, it follows that Mρ,m



tm + vm vm

ρ

 u f (0) f ′ (t) (u, v) = N+ [f (t)] (u, v) − . v v

(2.34)

In general, with the help of (2.31), we can find that ρ   m t (n) m f (t) (u, v) +v Mρ,m vm h i u 1 = N+ f (n−1) (t) (u, v) − f (n−1) (0) v( v ) n−2 X uk u un−1 + 1 (n−k−2) = N [f (t)] (u, v) − f (0) − f (n−1) (0) v v n−1 v k+1 v k=0

n

=

u + N [f (t)] (u, v) − vn n

=

u + N [f (t)] (u, v) − vn n

= as desired.

3

u + N [f (t)] (u, v) − vn

n−2 X k=0

n−1 X k=1

n−1 X k=0

k+1

u 1 f (n−k−2) (0) − f (n−1) (0) v k+2 v uk (n−k−1) 1 f (0) − f (n−1) (0) k+1 v v uk (n−k−1) f (0) , v k+1

(2.35) 

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 Z ∞ Z ∞ g (vt) f (vu) Mρ1 ,m [f (u)] (t, v) dt. (3.1) ρ1 Mρ2 ,m [g (t)] (u, v) du = m m m (u + v ) (t + v m )ρ2 0 0 In particular, when ρ1 = 0, Z ∞ Z f (vu) Mρ2 ,m [g (t)] (u, v) du = 0

0

∞

g (vt) + ρ N [f (u)] (t, v) dt, (tm + v m ) 2

(3.2)

where the integral transform N+ is defined by (1.2). Proof From the definition (1.1) of the M-transform, we find (by using the well-known Fubini’s theorem) that Z ∞ −ut  Z ∞ Z ∞ f (vu) f (vu) e g (vt) dt du ρ Mρ2 ,m [g (t)] (u, v) du = ρ ρ (um + v m ) 1 (um + v m ) 1 0 (tm + v m ) 2 0 0 Z ∞ −ut  Z ∞ g (vt) e f (vu) = du dt (tm + v m )ρ2 0 (um + v m )ρ1 Z0 ∞ g (vt) = Mρ1 ,m [f (u)] (t, v) dt. m + v m )ρ2 (t 0

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

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1395

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

Z

∞

+

f (vu) N [g (t)] (u, v) du =

0

Z

∞

g (vt) N+ [f (u)] (t, v) dt.

(3.3)

0

Theorem 3.3 Under the hypotheses of the Theorem 2.1, the following assertion holds true Z ∞ 0

   f (vu) Mρ,m e−at g (t) (uv, v) du = Mρ,m g (t) N+ [f (u)] (t, v) (av, v) (a > 0) . (3.4)

Proof Using the definition (1.1) of the M-transform, we have Z ∞ −uvt −avt  Z ∞ Z ∞   e e g (vt) dt f (vu) Mρ,m e−at g (t) (uv, v) du = f (vu) du (tm + v m )ρ 0 0 0 Z ∞  Z ∞ −avt e g (vt) −uvt = e f (vu) du dt ρ (tm + v m ) 0 0 Z ∞ −avt e g (vt) N [f (u)] (tv, v) dt = m + v m )ρ (t 0 = Mρ,m {g (t) N [f (u)] (t, v)} (av, v) .

(3.5)

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 3.4 The following assertion holds true   Z ∞  −at  g (t) λ−1 u Mρ,m e g (t) (uv, v) du = Γ (λ) Mρ,m (av, v) tλ 0
a > 0; ℜ (λ) > 0 . Proof

We first set f (t) = tλ−1 in (3.4). Then, by noting that Z ∞ Γ (λ) , v λ−1 uλ−1 e−tvu du = v λ−1 λ (tv) 0

(3.6)

(3.7)

the result (3.6) follows immediately.



The following two corollaries provide the relationship between the M-transform and some important integral transforms. Corollary 3.5 Under the hypotheses of the Theorem 2.1, there exists the relation given by    g (t)   −at  v µω−1 1,1  v Bω,µ Mρ,m e g (t) (uv, v) (v) = Mρ,m H1,1 µω t ω t

where

Bω,µ [f (z)] (s) = ωsµω−1

Z

∞

ω ω

e−s

z

z µω−1 f (z) dz

0

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

 (1 − µω, 1) 
 (av, v) ,  0, ω1 (3.8)

(ω > 0; µ > 0)

(3.9)

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ω

Let us set f (t) = ωtµω−1 e−t in (3.4). Then Z ∞   ω ω e−v u uµω−1 Mρ,m e−at g (t) (uv, v) du ωv µω−1 i o h n 0 ω = Mρ,m g (t) N+ uµω−1 e−u (t, v) (av, v) .

(3.10)

ω

The N+ -transform of uµω−1 e−u can be evaluated by observing that Z ∞ i h ω ω ω e−tu−u v uµω−1 du N+ uµω−1 e−u (t, v) = v µω−1 0

 i h v µω−1 1,1  a µω−1 ̺−1 −auω (t) = =v L u e H1,1 ω t̺ t

 (1 − ̺, ω) , (0, 1)

where we have applied the result giving the Laplace transform of the H-function [16, p. 47, Eq. (2.5.25)] (with a = v ω > 0 and ̺ = µω > 0). If we use (2.16), we obtain   h i µω−1 (1 − µω, 1) ω v v N+ uµω−1 e−u (t, v) = H 1,1  (3.11)
. ωtµω 1,1 t 0, 1 ω

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   (a , αi )1,p m,n  i  f (t) = Hp,q t (bj , βj )1,q

in (3.4), then we get the H-transform of the M-transform which is given by the following corollary. Corollary 3.6 There exists the relationship given by    H Mρ,m e−at g (t) (uv, v) (v)      g (t) m,n+1  v (0, 1) , (ai , αi )1,p  (av, v) Hp+1,q = Mρ,m   t t (bj , βj )

(a > 0) ,

(3.12)

1,q

where H is the integral transform defined by (1.9).

From the natural-Laplace Duality (1.6), we have          t (a , α ) (a , α ) i 1,p i 1,p m,n  i m,n  i  (t, v) = 1 L Hp,q  N+ Hp,q u u .  v  (bj , βj )1,q  (bj , βj )1,q  v

Proof

Since the Laplace transform of the H-function is given by ([16, p. 45, Eq. (2.5.16)])        (a , α ) (0, 1) , (a , α ) 1 1 i i i i m,n+1  1,p  1,p  m,n  L Hp,q u (t) = Hp+1,q ,   t t (bj , βj ) (bj , βj ) 1,q

(3.13)

(3.14)

1,q

it follows that

 

    (a , α ) 1 i i m,n+1  v 1,p  m,n  N+ Hp,q u (t, v) = Hp+1,q   t t (bj , βj )1,q

 (0, 1) , (ai , αi )1,p . (bj , βj )1,q

(3.15)

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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]) Z c+i∞ 1 L−1 [F (s)] (t) = est F (s) ds (c > 0). (4.1) 2πi c−i∞ By using the duality relations (1.5) and (1.6), one can find 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 4.1 The inversion of the M-transform defined by (1.1) is given by ρ    m t t −1 m L {Mρ,m [f (t)] (u, v)} f (t) = +v , v ∈ (0, µ) , m v v

(4.2)

provided that the integral involved converges absolutely. Proof

Let us first write F (v; t) :=

f (vt) ρ + vm )


v ∈ (0, µ) .

(tm

(4.3)

If the variable v in the transform M defined by (1.1) can be considered as a parameter of the function F (v; t), then this integral transform can be expressed as follows: Mρ,m [f (t)] (u, v) = L [F (v; t)] (u) .

(4.4)

Taking the inverse Laplace transform of both the sides of (4.4), we can then formally obtain L−1 {Mρ,m [f (t)] (u, v)} (t) = F (v; t) . Upon changing t →

(4.5)

t v

in (4.5) and using (4.3), we find that  m ρ   t t m −1 f (t) = + v , L {M [f (t)] (u, v)} ρ,m vm v

which completes the proof.

(4.6) 

Remark 4.2 If we set ρ = 0, then Theorem 4.1 will become the inversion formula for the natural transform. That is, if R (u, v) = N+ [f (t)] (u, v) , then N

−1

−1

[R (u, v)] = f (t) = L

  t {R (u, v)} . v

(4.7)

To illustrate the application of the introduced transform (1.1), we consider the following example. Example (First-Order Initial-Boundary Value Problem)  U + Us = p (t, v) r (t, s; v) , (4.8)   t U (t, 0; v) = 0 for t > 0, (4.9)   U (0, s; v) = vϕ (v) for s > 0, (4.10)

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where function p (t, v) is given by 1 = p (t, v)



tm + vm vm

ρ

(v > 0) ,

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: Ut (t, s; v) Us (t, s; v) + = r (t, s; v) . p (t, v) p (t, v) By applying the M-transform with respect to t to (4.11), we get     Ut (t, s; v) Us (t, s; v) Mρ,m (u, v) + Mρ,m (u, v) = Mρ,m [r (t, s; v)] (u, v) , p (t, v) p (t, v)

(4.11)

(4.12)

where the variables in (u, v) denote the transform variables. Using the elimination property (2.6), we get N+ [Ut (t, s; v)] (u, v) + N+ [Us (t, s; v)] (u, v) = Mρ,m [r (t, s; v)] (u, v) .

(4.13)

Also, by means of (2.31), we obtain u + U (0, s; v) N [U (t, s; v)] (u, v) − + N+ [Us (t, s; v)] (u, v) = Mρ,m [r (t, s; v)] (u, v) . v v The use of the condition (4.10) yields u + N [U (t, s; v)] (u, v) − ϕ (v) + N+ [Us (t, s; v)] (u, v) = Mρ,m [r (t, s; v)] (u, v) . v It is now easily verified that   ∂ + ∂ N+ f (t, s) (u, v) = N [f (t, s)] (u, v) ∂s ∂s

(4.14)

(4.15)

and, therefore, the equation (4.15) finally becomes ∂ + u N [U (t, s; v)] (u, v) + N+ [U (t, s; v)] (u, v) = Mρ,m [r (t, s; v)] (u, v) + ϕ (v) , ∂s v with the initial-value condition that

(4.16)

N+ [U (t, 0; v)] (u, v) = 0. It is convenient to use the following notations: R (s) ≡ R (t, s, u; v) := N+ [U (t, s; v)] (u, v) and F (s) ≡ F (t, s, u; v) := Mρ,m [r (t, s; v)] (u, v) + ϕ (v) . Thus, clearly, (4.16) is actually an initial-value problem:   ∂R + u R = F (s) . ∂s v  R (0) = 0.

Its solution is given by

u

R (s) = e− v s

Z

0

s

u

e v w F (w) dw,

(4.17) (4.18) (4.19)

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or, equivalently, by u

N+ [U (t, s; v)] (u, v) = e− v s

Z

s

u

u

e v w Mρ,m [r (t, w; v)] (u, v) dw + e− v s ϕ (v)

0

Z

s

u

e v w dw. (4.20)

0

Finally, on using (4.7), we get     Z s Z s u u −u −u s w −1 s w −1 v v v v e Mρ,m [r (t, w; v)] (u, v) dw + N e e ϕ (v) e dw U (t, s; v) = N 0 0   Z s u u e v w Mρ,m [r (t, w; v)] (u, v) dw = N−1 e− v s    0   −1 1 − u s −1 1 v −N e + vϕ (v) N u u   Z s u −u −1 s w v v e =N e Mρ,m [r (t, w; v)] (u, v) dw + vϕ (v) {θ (t) − θ (t − s)} , (4.21) 0

where θ (x) is the Heaviside generalized function which is defined by  1, x > 0 θ (x) = 0, x < 0. References

[1] Al-Musallam F, Kiryakova V, Tuan V K. A multi-index Borel-Dˇ zrbashjan transform. Rocky Mountain J Math, 2002, 32: 409–428 [2] Al-Omari S K Q. On the application of natural transforms. Internat J Pure Appl Math, 2013, 84: 729–744 [3] Atangana A, Kılı¸cman A. The use of Sumudu transform for solving certain nonlinear fractional heat-like equations. Abstr Appl Anal, 2013, 2013: Article ID 737481 [4] Belgacem F B M. Sumudu applications to Maxwell’s equations. The Plan Index Enquiry and Retrieval System Online, 2009, 5: 355–360 [5] Belgacem F B M, Karaballi A A, Kalla S L. Analytical investigations of the Sumudu transform and applications to integral production equations. Math Prob Engrg, 2003, 3: 103–118 [6] Belgacem F B M, Karaballi A A. Sumudu transform fundamental properties investigations and applications. J Appl Math Stochast Anal, 2005, 2005: Article ID 91083 [7] Belgacem F B M, Silambarasan R. Theory of natural transform. Math Engrg Sci Aerospace, 2012, 3(1): 105–135 [8] Belgacem F B M, Silambarasan R. Advances in the natural transform. AIP Conf Proc, 2012, 1493: Article ID 106 [9] Bulut H, Baskonus H M, Belgacem F B M. The Analytical solution of some fractional ordinary differential equations by the Sumudu transform method. Abstr Appl Anal, 2013, 2013: Article ID 203875 [10] Chaurasia V B L, Dubey R S, Belgacem F B M. Fractional radial diffusion equation analytical solution via Hankel and Sumudu transforms. Math Engrg Sci Aerospace, 2012, 3(2): 1–10 [11] Debnath L, Bhatta D. Integral Transforms and Their Applications. New York: Chapman and Hall (CRC Press Company), 2007 [12] Dˇ zrbashjan M M. Integral Transforms and Representations of Functions in the Complex Domain. Moscow: Nauka, 1966 [13] Eltayeb H, Kılı¸cman A. A note on the Sumudu transforms and differential equations. Appld Math Sci, 2010, 4 (22): 1089–1098 [14] Hussain M G M, Belgacem F B M. Transient solutions of Maxwell’s equations based on Sumudu transformation. J Progr Electromag Res, 2007, 74: 273–289 [15] Katatbeh Q D, Belgacem F B M. Applications of the Sumudu transform to fractional differential equations. Nonlinear Stud, 2011, 18: 99–112 [16] Kilbas A A, Saigo M. H-Transforms: Theory and Applications. Boca Raton, London, New York and Washington, D C: Chapman and Hall (CRC Press Company), 2004

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[17] Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier, 2006 [18] Kılı¸cman A, Gadain H E. On the applications of Laplace and Sumudu transforms. J Franklin Inst, 2010, 247: 848–862 [19] Kılı¸cman A, Eltayeb H. On a new integral transform and differential equations. Math Prob Engrg, 2010, 2010: Article ID 463579 [20] Kılı¸cman A, Eltayeb H, Agarwal R P, On Sumudu transform and system of differential equations. Abstr Appl Anal, 2010, 2010: Article ID 598702 [21] Kılı¸cman A, Eltayeb H, Atan K A M. A note on the comparison between Laplace and Sumudu transforms. Bull Iranian Math Soc, 2011, 37: 131–141 [22] Kiryakova V. Generalized Fractional Calculus and Applications. Harlow, New York: Longman & John, 1994 [23] Mathai A M, Saxena R K, Haubold H J. The H-Function: Theory and Applications. New York, Dordrecht, Heidelberg and London: Springer, 2010 [24] Silambarasn R, Belgacem F B M. Applications of the natural transform to Maxwell’s equations//Progress Electromagnetics Research Symposium Proceedings. Suzhou, People’s Republic of China; September 12–16, 2011 [25] Singh J, Kumar D, Kılı¸cman A. Homotopy perturbation method for fractional gas dynamics equation using Sumudu transform. Abstr Appl Anal, 2013, 2013: Article ID 934060 [26] Singh J, Kumar D, Kumar S. New treatment of fractional Fornberg-Whitham equation via Laplace transform. Ain Shams Eng J, 2013, 4 (3): 557–562 [27] Srivastava H M. Some remarks on a generalization of the Stieltjes transform. Publ Math Debrecen, 1976, 23: 119–122 [28] Srivastava H M, Golmankhaneh A K, Baleanu D, Yang X J. Local fractional Sumudu transform with applications to IVPs on Cantor sets. Abstr Appl Anal, 2014, 2014: Article ID 620529 [29] Srivastava H M, Gupta K C, Goyal S P. The H-Functions of One and Two Variables with Applications. New Delhi and Madras: South Asian Publishers, 1982 [30] Su Feng, Peng Zhigang. The inverse Laplace transform theorem of Henstock-Kurzweil integrable function. Acta Mathematica Scientia, 2007, 27A(6): 1155–1163 [31] Sushila, Singh J, Shishodia Y S. A reliable approach for two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability using Sumudu transform. Ain Shams Eng J, 2014, 5(1): 237–242