On ( p , q ) -Analogues of Laplace-Typed Integral Transforms and Applications

: In this paper, we establish ( p , q ) -analogues of Laplace-type integral transforms by using the concept of ( p , q ) -calculus. Moreover, we study some properties of ( p , q ) -analogues of Laplace-type integral transforms and apply them to solve some ( p , q )-differential equations.


Introduction
Integral transform techniques are very important for solving many problems in applied mathematics, physics, astronomy, economics and engineering. The integral transform techniques have contributed largely to a variety of theories and applications, such as Laplace, Sumudu, σ-Integral Laplace, Mohand, Sawi, Kamal and Pourreza transforms. In the sequence of such integral transforms, in 2017, H. Kim [1] introduced the Laplace-typed integral transform or α G-transform, which is defined by where α ∈ R. The α G-transform can be applied directly to a suitable problem by choosing α appropriately. In Table 1, we list a few of them with their definitions and set u, α for converting α G-transform into appropriate transforms.   [6] s 2 ∞ 0 f (t)e −st dt u = 1/s and α = −2 Sawi [7] 1 s 2 ∞ 0 f (t)e −t/s dt u = s and α = −2 Kamal [8] ∞ 0 f (t)e −t/s dt u = s and α = 0 Pourreza [9] s ∞ 0 f (t)e s 2 t dt u = 1/s 2 and α = −1/2

Preliminaries
In this section, we give basic knowledge that will be used in our work. Throughout this paper, let 0 < q < p ≤ 1 be constants.
Let us introduce (p, q)-analogue or (p, q)-number for n ∈ N, which is defined by [n] p,q = p n − q n p − q .
If p = 1 in (1), then (1) is q-analogue of n or q-number; see [26] for more details.
Definition 1 ([35]). If f is an arbitrary function, then is the (p, q)-derivative of the function f .
If p = 1 in (4), then D p,q f (x) = D q f (x), which is the q-derivative of the function f ; in addition, if q → 1 in (4), then we get the classical derivative.
Proposition 1. The (p, q)-derivatives of the product and quotient rules of functions f and g are as follows: The proof of Proposition 1 is given in [35].
If p = 1 in (7), then (7) reduces to the q-integral of the function f ; also, if q → 1 in (7), then we get the classical integral.
Proposition 2. If f and g are arbitrary functions, then b a f (px)(D p,q g(x))d p, is the (p, q) integration by parts. Note that b = ∞ is allowed.
The proofs of Proposition 2 are given in [35].

Proposition 3 ([35]
). If n ∈ R, then the following identities hold: The proofs of the following Propositions are given in [48].

Definition 5 ([38]
). For s, t ∈ N, the (p, q)-beta function is defined by Theorem 1. For s, t ∈ N, the relation between the (p, q)-gamma function and the (p, q)-beta function is The proof of this Theorem is given in [38].

Properties of (p, q)-Analogues of Laplace-Typed Integral Transform
In this section, we introduce (p, q)-analogues of the Laplace-typed integral transform in the form 1 α G p,q and 2 α G p,q , which are called α G p,q -transform of type one and type two, respectively. Let and Now the definition of the α G p,q -transform of type one and type two is given by: Definition 6. The 1 α G p,q ( f (t); u) over the set A in (26) and the 2 α G p,q ( f (t); u) over the set B in (27) are defined as follows: If u = 1/s, p = 1 and α = 0, then (28) and (29) reduce to and respectively, which appeared in [28]. If u = s and α = −1, then (28) and (29) reduce to and respectively, which appeared in [49].

Theorem 2. (Linearity):
If f 1 , g 1 ∈ A and f 2 , g 2 ∈ B, then for constants c and d, we have Proof. The theorem follows immediately from Definition 6.

Theorem 3. (Scaling):
If f 1 ∈ A and g 1 ∈ B, then the following formulas hold for non-zero constants β and γ: Proof. Using (28) and Proposition 7, we have The proof of (33) is similar to (32), and therefore the proof is completed.
Theorem 4. Let α ∈ R; then the following formulas hold: Proof. Using (28) and (12) to prove (34), we get The proof of the part (35) utilizes a similar process as for (34). Therefore, the proof is completed.
Theorem 5. If n ∈ N, then the following identities hold: .
Proof. Using (8) and (28) to prove (i), we have We prove (iii) by mathematical induction: obviously, (iii) is true for n = 1. Assuming that (iii) is true and using the (p, q)-integration by parts, we obtain .
The proofs of (ii) and (iv) use (8) and (29); then we follow a similar process for (i) and (iii), respectively. Therefore, the proof is completed.
respectively, which appeared in [28]. If u = s and α = −1, then Theorem 5 (i) and (iii) reduce to S p,q (t; s) = s p and S p,q (t n ; s) = s n [n] p,q !

Theorem 9. (Transforms of integrals):
Let f ∈ A andf ∈ B, then the following identities hold: Proof. Using (8) and (28) to prove (i) − (iii), we have We give g(t) = E p,q − t u , h(t) = ∞ 0 f (x)d p,q x and apply the formula of (p, q)integration by parts, we obtain Next, we get Consequently, After continuing this process, we obtain the sequence The proofs of (iv) − (vi) utilize (8) and (29), and then follow the similar process for (i) − (iii). The proof is completed. Remark 6. If p = 1, then Theorem 9 (iii) reduces to Furthermore, if q → 1, then (36) reduces to the α G-transform of integrals, which appeared in [12].

Theorem 10. (Transforms of derivatives):
If f ∈ A and D n p,q has the 1 α G ( p, q)-transform of type one for each n ∈ N, then the transforms of the first, second and n-th derivatives of f can be written in the following forms: Proof. Using (8) and (28) to prove (i), we have Applying the equation above with n = 2 to prove (ii), we get 1 α G p,q (D (2) p,q f (t) In (iii), if n = 1, it is not difficult to see that 1 α G p,q (D (n) p,q f (t); u) = 1 α G p,q ( f (t); up n ) u n p nα p ( n+1 The proof of α G p,q -transform of the type two in (29) is similar to the one for Theorem 10, and is therefore is omitted.

Theorem 11. (Derivative of transforms):
For n ∈ N, the following formulas hold: Proof. Using (28) Taking (p, q)-derivative on both sides with respect to 1/u, we get From (37), taking the second (p, q)-derivative on both sides with respect to 1/u to prove (ii), we have Following the same process, we can prove (iii). Therefore, the proof is completed.
The proof of α G p,q -transform of the type two in (29) is similar to the one for Theorem 11 and therefore is omitted.
The proof of α G p,q -transform of the type two in (29) is similar to the one for Corollary 2, but changes e p,q (at) to E p,q (at), and therefore is omitted.
Theorem 13. (Transforms of the Dirac delta function): For a ≥ 0, let If δ(t − a) denotes the limit of f k as k → 0, then we have where δ is the Dirac delta function.
Proof. Using (28) to prove (44), we obtain If we take the limit of f k as k → 0, then The proof of (45) uses (29), and then follows the similar process for (44). Therefore, the proof is completed.
Therefore, the proof is completed.
The proof of α G p,q -transform of the type two in (29) is similar to one for Theorem 14 and therefore is omitted.

Examples
In this section, we solve the (p, q)-differential equations using the definition and properties of α G p,q -transform of type one. We consider the (p, q)-Cauchy problem and two second-order (p, q)-differential equations. where c is a constant.