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Hahn Laplace transform and its applications

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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

Like q -calculus, Hahn calculus (or q , ω -calculus) is constructed by defining a difference derivative operator and an integral operator. The q , ω -analogs of the integral representations of the Laplace transform and related special functions, such as gamma and beta, are proposed in this article. Then, some basic properties similar to classical and q -analogs are investigated. Finally, a few examples are given to solve q , ω -initial value problems via the newly introduced q , ω -Laplace transform.

Keywords: q, ω-gamma function; q, ω-beta function; q, ω-Laplace transform; q, ω-differential equations; Hahn calculus; q-calculus
MSC 2010: 33D05; 05A30; 44A10; 39A70

1 Introduction

Quantum calculus ( q -calculus) is based on the derivative defined by the difference ratio without limit. It has essential applications in mathematics, engineering and mathematical physics such as quantum chromodynamics, quantum theory, the theory of relativity, basic hypergeometric functions, string theory, etc. (see [1–4]). One type of q -calculus is Hahn calculus ( q , ω -calculus). Let q ∈ ( 0 , 1 ) , ω > 0 , ω 0 ≔ ω 1 − q , and I be an interval of R containing ω 0 . The Hahn difference operator (see [5,6]) of f , which is defined on I , is as follows:

(1) D q , ω f ( x ) ≔ f ( q x + ω ) − f ( x ) ( q x + ω ) − x , x ≠ ω 0 , f ′ ( ω 0 ) , x = ω 0 .

After the definition of q , ω -integral in the study by Annaby et al. [7], interest in studies on the Hahn calculus has increased. Many physical and mathematical problems in classical and q -calculus are extended to Hahn calculus. The theory and applications of q , ω -differential equations were examined by Hamza and Ahmed [8,9]. The q , ω -Sturm-Liouville problem was presented by Annaby et al. [10], and the sampling theory for the same problem was constructed by Annaby and Hassan [11]. Fractional Hahn calculus was investigated in [12–14], and Taylor theory based on Hahn’s operator was established by Oraby and Hamza [15]. Hıra [16] presented the q , ω -Dirac system.

Different definitions and applications of the q -Laplace transform defined using the q -exponential functions e q ( x ) and E q ( x ) can be seen in [5,17–20]. The answer to the question of how to apply the q -Laplace transform to the q -differential equations was given in [6,21,22]. A few studies on the q -Sumudu transform associated with the q -Laplace transform can be cited as [23–25] (also cited therein). Researchers can find studies on fractional calculus on the timescale and its q -analog in [26] and [27], respectively. We also refer to the study by Sheng and Zhang [28] for solving some fractional q -differential equations, by Hajiseyedazizi et al. [29] q-integrodifferential equations, by Samei et al. [30] for singular fractional q -differential equations, and by Sheng and Zhang [31] for numerical solutions of the q -fractional boundary value problems. As it is known, there are some relations between the beta and the gamma functions, and some properties of the Laplace transform are obtained from the gamma function. Similarly, q -analogs of these functions were investigated in [2,3,32].

This study deals with the q , ω -analogs of the Laplace transform, gamma, and beta functions. First, we define integral representations of q , ω -gamma function Γ q , ω ( t ) (Definition 1) and q , ω -beta function B q , ω ( t , s ) (Definition 2) and obtain some of their properties and relations. Then, we define q , ω -Laplace transform F q , ω ( s ) (Definition 3) for suitable function f . As an alternative method, we give q , ω -Laplace transform of f function with the help of q , ω -Taylor series expansion (Theorem 2), then obtain the q , ω -analogs of the basic properties of the Laplace transform, such as linearity, scaling, shifting, and convolution. Finally, we give some examples for solving q , ω -initial value problems via the q , ω -Laplace transform.

Basic definitions and features in q -calculus and q , ω -calculus mentioned throughout this article can be found in [3,7,33,34], so we will not repeat them here. Since the Hahn difference operator is a combination of the q -difference operator D q and the forward difference operator Δ ω (see [1,3,4,35,36]), the definitions and properties given in the following sections for q , ω -calculus are reduced to their q -analogs as ω → 0 , and their classical equivalents as ω → 0 and q → 1 .

Shehata et al. [37] investigated the Laplace transform associated with the general quantum difference operator D β (see resources there for details). From the difference operator D β , the difference operator D q , ω is obtained by a special selection of some functions. Therefore, the study may appear to include our study. However, in both studies, different integral definitions are made, and the proof methods are also different. The limits of the integral representations for the q , ω -gamma function and the q , ω -Laplace transform in our study are consistent with those in the q -calculus for ω → 0 and the classical forms for ω → 0 , q → 1 .

2 q , w -gamma and q , w -beta functions

In [2,3, 36,38], the q -analogs of beta and gamma functions were introduced as the infinite product and then the integral representations of them were given in [38–40]. One of them is given as follows (see [39]):

(2) Γ q ( t ) ≔ ∫ 0 [ ∞ ] q x t − 1 E q ( − q x ) d q x ,

and

(3) B q ( t , s ) ≔ ∫ 0 1 x t − 1 ( 1 − q x ) q s − 1 d q x ,

where [ α ] q = 1 − q α 1 − q , [ n ] q = 1 + q + q 2 + ⋯ + q n − 1 for α = n ∈ N , and [ ∞ ] q = 1 1 − q for q ∈ ( 0 , 1 ) .

Now, we define q , ω -gamma and q , ω -beta functions based on the q , ω -exponential function E q , ω ( x ) . Then, we obtain q , ω -analogs of some properties.

Definition 1

For any t > 0 ,

(4) Γ q , ω ( t ) ≔ ∫ ω 0 ω 0 + [ ∞ ] q ( x − ω 0 ) t − 1 E q , − ω ( − ( q x + ω ) ) d q , ω x

is called the q , ω -gamma function.

Remark 1

If u = x − ω 0 is changed in equation (4), then Γ q , ω ( t ) ≡ Γ q ( t ) is obtained. This is equivalent to the reduction of equation (4) to the q -analog in equation (2) for ω → 0 (in this case, u = x ).

From equation (4) and using the q , ω -integration by parts, we have

Γ q , ω ( t + 1 ) = ∫ ω 0 ω 0 + [ ∞ ] q ( x − ω 0 ) t E q , − ω ( − ( q x + ω ) ) d q , ω x = − ∫ ω 0 ω 0 + [ ∞ ] q ( x − ω 0 ) t D q , ω E q , − ω ( − x ) d q , ω x = − ( ( x − ω 0 ) t E q , − ω ( − x ) ) ∣ ω 0 ω 0 + [ ∞ ] q + ∫ ω 0 ω 0 + [ ∞ ] q E q , − ω ( − ( q x + ω ) ) D q , ω ( x − ω 0 ) t d q , ω x = [ t ] q ∫ ω 0 ω 0 + [ ∞ ] q ( x − ω 0 ) t − 1 E q , − ω ( − ( q x + ω ) ) d q , ω x ,

where ( 1 − a ) q n ≔ ( a ; q ) n ≔ ∏ i = 0 n − 1 ( 1 − a q i ) and ( 1 − a ) q 0 ≔ ( a ; q ) 0 = 1 ; E q , ω ( x ) ≔ ∑ n = 0 ∞ q n ( n − 1 ) 2 ( x − ω 0 ) n [ n ] q ! = ( 1 + ( x ( 1 − q ) − ω ) ) q ∞ ; E q , − ω ( − ω 0 ) = 1 , E q , − ω ( − ( ω 0 + [ ∞ ] q ) ) = 0 , D q , ω E q , ω ( x ) = E q , ω ( q x + ω ) , [ n ] q ! = [ 1 ] q [ 2 ] q … [ n ] q , and D q , ω ( x − ω 0 ) n = [ n ] q ( x − ω 0 ) n − 1 . Hence, for any t > 0 , we obtain

(5) Γ q , ω ( t + 1 ) = [ t ] q Γ q , ω ( t ) ,

and for t = n ∈ N , we have

(6) Γ q , ω ( n + 1 ) = [ n ] q ! , Γ q , ω ( 1 ) = 1 .

Definition 2

For any t , s > 0 ,

(7) B q , ω ( t , s ) ≔ ∫ ω 0 ω 0 + 1 ( x − ω 0 ) t − 1 ( 1 − q ( x − ω 0 ) ) q s − 1 d q , ω x

is called the q , ω -beta function.

Theorem 1

The two relations between the q , ω -gamma and the q , ω -beta functions are given as follows:

(8) Γ q , ω ( t ) = B q , ω ( t , ∞ ) ( 1 − q ) t ,

(9) B q , ω ( t , s ) = Γ q , ω ( t ) Γ q , ω ( s ) Γ q , ω ( t + s ) .

Proof

If we put s = ∞ in Definition 2, use the definition E q , ω ( x ) = ( 1 + ( x ( 1 − q ) − ω ) ) q ∞ and change of variable x − ω 0 = ( 1 − q ) ( u − ω 0 ) , we obtain

B q , ω ( t , ∞ ) = ∫ ω 0 ω 0 + 1 ( x − ω 0 ) t − 1 ( 1 − q ( x − ω 0 ) ) q ∞ d q , ω x = ( 1 − q ) t ∫ ω 0 ω 0 + [ ∞ ] q ( u − ω 0 ) t − 1 E q , − ω ( − ( q u + ω ) ) d q , ω u = ( 1 − q ) t Γ q , ω ( t ) ,

which proves equation (8).

From equation (7), we obtain the following recurrence relations for B q , ω ( t , s ) :

(10) B q , ω ( t + 1 , s ) = ∫ ω 0 ω 0 + 1 ( x − ω 0 ) t ( 1 − q ( x − ω 0 ) ) q s − 1 d q , ω x = − 1 [ s ] q ∫ ω 0 ω 0 + 1 ( x − ω 0 ) t D q , ω ( 1 − ( x − ω 0 ) ) q s d q , ω x = − 1 [ s ] q ( ( x − ω 0 ) t ( 1 − ( x − ω 0 ) ) q s ) ω 0 ω 0 + 1 + 1 [ s ] q ∫ ω 0 ω 0 + 1 ( 1 − ( q x + ω − ω 0 ) ) q s D q , ω ( x − ω 0 ) t d q , ω x = [ t ] q [ s ] q ∫ ω 0 ω 0 + 1 ( x − ω 0 ) t − 1 ( 1 − q ( x − ω 0 ) ) q s d q , ω x = [ t ] q [ s ] q B q , ω ( t , s + 1 ) ,

and

(11) B q , ω ( t , s + 1 ) = ∫ ω 0 ω 0 + 1 ( x − ω 0 ) t − 1 ( 1 − q ( x − ω 0 ) ) q s d q , ω x = ∫ ω 0 ω 0 + 1 ( x − ω 0 ) t − 1 ( 1 − q ( x − ω 0 ) ) q s − 1 ( 1 − q s ( x − ω 0 ) ) d q , ω x = ∫ ω 0 ω 0 + 1 ( x − ω 0 ) t − 1 ( 1 − q ( x − ω 0 ) ) q s − 1 d q , ω x − q s ∫ ω 0 ω 0 + 1 ( x − ω 0 ) t ( 1 − q ( x − ω 0 ) ) q s − 1 d q , ω x = B q , ω ( t , s ) − q s B q , ω ( t + 1 , s ) .

From equations (10) and (11), we obtain

(12) B q , ω ( t , s + 1 ) = [ s ] q [ t + s ] q B q , ω ( t , s ) .

By substituting equation (11) in equation (10), we arrive at

(13) B q , ω ( t + 1 , s ) = [ t ] q [ t + s ] q B q , ω ( t , s ) .

By comparing equations (5) and (13), we have that equation (9) holds for any t , s > 0 .□

3 q , w -Laplace transform

In the literature, several different definitions have been made for the q -Laplace transform using e q ( x ) and E q ( x ) (see [6,17–21]). In the study by Kobachi [18], the q -Laplace transform of f is defined as follows:

(14) L q ( f ) ( s ) = 1 s ∫ 0 [ ∞ ] q E q ( − q x ) f x s d q x ,

and if the function f has a series expansion of f ( x ) = ∑ n = 0 ∞ a n x n [ n ] q ! , then its q -Laplace transform is given by L q ( f ) ( s ) = ∑ n = 0 ∞ a n s n + 1 .

Now we define q , ω -analog of the Laplace transform and investigate some of its basic properties. Let V ≔ V ( [ ω 0 , ω 0 + [ ∞ ] q ) , C ) be the set of q , ω -integrable functions compact on the subintervals of [ ω 0 , ω 0 + [ ∞ ] q ) .

Definition 3

Let f ∈ V . Then,

(15) L q , ω ( f ) ( s ) ≔ F q , ω ( s ) ≔ 1 s − ω 0 ∫ ω 0 ω 0 + [ ∞ ] q E q , − ω ( − ( q x + ω ) ) f x s − ω 0 d q , ω x

is called the q , ω -Laplace transform of f .

Example 1

For f ( x ) = ( x − ω 0 ) n , n ∈ N , we have

L q , ω ( ( x − ω 0 ) n ) = 1 s − ω 0 ∫ ω 0 ω 0 + [ ∞ ] q E q , − ω ( − ( q x + ω ) ) x − ω 0 s − ω 0 n d q , ω x = 1 ( s − ω 0 ) n + 1 ∫ ω 0 ω 0 + [ ∞ ] q E q , − ω ( − ( q x + ω ) ) ( x − ω 0 ) n d q , ω x .

From equations (4) and (6), we obtain

(16) L q , ω ( ( x − ω 0 ) n ) = Γ q , ω ( n + 1 ) ( s − ω 0 ) n + 1 = [ n ] q ! ( s − ω 0 ) n + 1 .

The q , ω -Taylor expansion of a function f defined on I was established in the study by Oraby and Hamza [15]. Therefore, let f ( x ) = ∑ n = 0 ∞ a n ( x − ω 0 ) n [ n ] q ! . If lim n → ∞ a n + 1 a n = r ( < ∞ ) , then f ( x ) is convergent in ∣ x − ω 0 ∣ < [ ∞ ] q r . That is, if s − ω 0 > r , then f x s − ω 0 is convergent in ∣ x − ω 0 ∣ < ω 0 + [ ∞ ] q r . Inspired by the work of Kobachi [18], we can give the following theorem, which is more useful than Definition 3 for calculating the Laplace transform of functions.

Theorem 2

Let f ( x ) = ∑ n = 0 ∞ a n ( x − ω 0 ) n [ n ] q ! , ∣ x − ω 0 ∣ < [ ∞ ] q r . Then, the q , ω -Laplace transform of f is given as follows:

(17) L q , ω ( f ) ( s ) ≔ F q , ω ( s ) ≔ ∑ n = 0 ∞ a n ( s − ω 0 ) n + 1 .

Proof

From equation (16), we have

L q , ω ( f ) ( s ) = ∑ n = 0 ∞ a n [ n ] q ! L q , ω ( ( x − ω 0 ) n ) = ∑ n = 0 ∞ a n [ n ] q ! [ n ] q ! ( s − ω 0 ) n + 1 = ∑ n = 0 ∞ a n ( s − ω 0 ) n + 1 .□

Example 2

For the q , ω -exponential function e q , ω ( x ) = ∑ n = 0 ∞ ( x − ω 0 ) n [ n ] q ! , ∣ x − ω 0 ∣ < [ ∞ ] q (see [7, 19]), using Definition 3, we have

L q , ω ( e q , ω ( x ) ) = 1 s − ω 0 ∫ ω 0 ω 0 + [ ∞ ] q E q , − ω ( − ( q x + ω ) ) ∑ n = 0 ∞ ( x − ω 0 ) n [ n ] q ! ( s − ω 0 ) n d q , ω x = ∑ n = 0 ∞ 1 [ n ] q ! ( s − ω 0 ) n + 1 ∫ ω 0 ω 0 + [ ∞ ] q E q , − ω ( − ( q x + ω ) ) ( x − ω 0 ) n d q , ω x ,

and from equations (4) and (6), we obtain

L q , ω ( e q , ω ( x ) ) = ∑ n = 0 ∞ Γ q , ω ( n + 1 ) [ n ] q ! ( s − ω 0 ) n + 1 = ∑ n = 0 ∞ [ n ] q ! [ n ] q ! ( s − ω 0 ) n + 1 = ∑ n = 0 ∞ 1 ( s − ω 0 ) n + 1 = 1 s − ω 0 ∑ n = 0 ∞ 1 s − ω 0 n = 1 s − ω 0 1 1 − 1 s − ω 0 = 1 s − ω 0 − 1 ,

provided that ∣ s − ω 0 ∣ > 1 .

If we use Theorem 2 for a n = 1 , similarly we obtain a geometric series with

L q , ω ( e q , ω ( x ) ) = ∑ n = 0 ∞ 1 ( s − ω 0 ) n + 1 = 1 s − ω 0 − 1 ,

which converges as ∣ s − ω 0 ∣ > 1 .

Let f ( x ) = ∑ n = 0 ∞ a n ( x − ω 0 ) n [ n ] q ! , ∣ x − ω 0 ∣ < [ ∞ ] q r 1 , and g ( x ) = ∑ n = 0 ∞ b n ( x − ω 0 ) n [ n ] q ! , ∣ x − ω 0 ∣ < [ ∞ ] q r 2 such that { a n } and { b n } satisfy lim n → ∞ a n + 1 a n = r 1 ( < ∞ ) and lim n → ∞ b n + 1 b n = r 2 ( < ∞ ) . The q , ω -Laplace transform of these functions be L q , ω ( f ) ( s ) ≔ F q , ω ( s ) and L q , ω ( g ) ( s ) ≔ G q , ω ( s ) . The following theorem shows that the q , ω -Laplace transform satisfies the linearity property.

Theorem 3

(Linearity property) Let f , g ∈ V and c 1 , c 2 be constant. Then,

(18) L q , ω ( c 1 f ( x ) + c 2 g ( x ) ) = c 1 F q , ω ( s ) + c 2 G q , ω ( s ) .

Proof

From c 1 f ( x ) + c 2 g ( x ) = ∑ n = 0 ∞ ( c 1 a n + c 2 b n ) ( x − ω 0 ) n [ n ] q ! , we have

L q , ω ( c 1 f ( x ) + c 2 g ( x ) ) = ∑ n = 0 ∞ ( c 1 a n + c 2 b n ) ( s − ω 0 ) n + 1 = c 1 ∑ n = 0 ∞ a n ( s − ω 0 ) n + 1 + c 2 ∑ n = 0 ∞ b n ( s − ω 0 ) n + 1 = c 1 F q , ω ( s ) + c 2 G q , ω ( s ) .□

Theorem 4

(Scaling property) Let f ∈ V . Then,

(19) L q , ω ( f ( λ x ) ) = 1 λ F q , ω s λ , ∣ s − ω 0 ∣ > ∣ λ ∣ .

Proof

From f ( λ x ) = ∑ n = 0 ∞ a n λ n ( x − ω 0 ) n [ n ] q ! , we have

□ L q , ω ( f ( λ x ) ) = ∑ n = 0 ∞ a n λ n ( s − ω 0 ) n + 1 = 1 λ ∑ n = 0 ∞ a n λ s − ω 0 n + 1 = 1 λ ∑ n = 0 ∞ a n s − ω 0 λ n + 1 = 1 λ F q , ω s λ .

Example 3

From e q , λ ω ( λ x ) = ∑ n = 0 ∞ λ n ( x − ω 0 ) n [ n ] q ! , cos q , λ ω ( λ x ) = ∑ n = 0 ∞ ( − 1 ) n λ 2 n ( x − ω 0 ) 2 n [ 2 n ] q ! , and sin q , λ ω ( λ x ) = ∑ n = 0 ∞ ( − 1 ) n λ 2 n + 1 ( x − ω 0 ) 2 n + 1 [ 2 n + 1 ] q ! , we have

L q , ω ( e q , λ ω ( λ x ) ) = 1 s − ω 0 − λ , ∣ s − ω 0 ∣ > ∣ λ ∣ , L q , ω ( cos q , λ ω ( λ x ) ) = s − ω 0 ( s − ω 0 ) 2 + λ 2 , s − ω 0 > λ , L q , ω ( sin q , λ ω ( λ x ) ) = λ ( s − ω 0 ) 2 + λ 2 , s − ω 0 > λ .

Theorem 5

(Shifting property) Let f ∈ V , { b } q = ∏ k = 0 n − 1 q k b , ( a + { b } q ) n = ∏ k = 0 n − 1 ( a + q k b ) = a n − b a ; q n , and R = max r , [ ∞ ] q r . If ∣ x − ω 0 ∣ < R , s − ω 0 > R , then

(20) L q , ω ( e q , λ ω ( λ x ) f ( x ) ) = L q , ω ( f ( x ) ) ∣ s = s − { λ } q = F q , ω ( s − { λ } q ) .

Proof

From

e q , λ ω ( λ x ) f ( x ) = ∑ n = 0 ∞ λ n ( x − ω 0 ) n [ n ] q ! ∑ k = 0 ∞ a k ( x − ω 0 ) k [ k ] q ! = ∑ n = 0 ∞ ( x − ω 0 ) n [ n ] q ! ∑ k = 0 n n k q λ n − k a k ,

where n k q = [ n ] q ! [ k ] q ! [ n − k ] q ! , we have

□ L q , ω ( e q , λ ω ( λ x ) f ( x ) ) = ∑ n = 0 ∞ 1 ( s − ω 0 ) n + 1 ∑ k = 0 n n k q λ n − k a k = ∑ n = 0 ∞ 1 ( s − ω 0 ) n + 1 ∑ k = 0 n n n − k q λ n − k a k = ∑ k = 0 ∞ a k ( s − ω 0 ) k + 1 ∑ n = 0 ∞ n + k n q λ s − ω 0 n = ∑ k = 0 ∞ a k ( s − ω 0 ) k + 1 ∑ n = 0 ∞ ( q k + 1 ; q ) n ( q ; q ) n λ s − ω 0 n = ∑ k = 0 ∞ a k ( s − ω 0 ) k + 1 λ s − ω 0 ; q k + 1 = ∑ k = 0 ∞ a k ( s − ω 0 − { λ } q ) k + 1 = F q , ω ( s − { λ } q ) .

Theorem 6

(Differentiation property) Let f ∈ V . If any n ∈ N , then

(21) L q , ω ( D q , ω n f ( x ) ) = ( s − ω 0 ) n F q , ω ( s ) − ∑ i = 1 n ( s − ω 0 ) n − i D q , ω i − 1 f ( ω 0 ) .

Proof

From D q , ω f ( x ) = ∑ n = 1 ∞ a n ( x − ω 0 ) n − 1 [ n − 1 ] q ! = ∑ n = 0 ∞ a n + 1 ( x − ω 0 ) n [ n ] q ! , we have

L q , ω ( D q , ω f ( x ) ) = ∑ n = 0 ∞ a n + 1 ( s − ω 0 ) n + 1 = ( s − ω 0 ) ∑ n = 0 ∞ a n ( s − ω 0 ) n + 1 − a 0 s − ω 0 = ( s − ω 0 ) F q , ω ( s ) − a 0 = ( s − ω 0 ) F q , ω ( s ) − f ( ω 0 ) .

Therefore, relation (21) is true for n = 1 . Assume that relation (21) is true for n = k ∈ N

L q , ω ( D q , ω k f ( x ) ) = ( s − ω 0 ) k F q , ω ( s ) − ∑ i = 1 k ( s − ω 0 ) k − i D q , ω i − 1 f ( ω 0 ) .

Then,

L q , ω ( D q , ω k + 1 f ( x ) ) = L q , ω ( D q , ω ( D q , ω k f ( x ) ) ) = ( s − ω 0 ) L q , ω ( D q , ω k f ( x ) ) − D q , ω k f ( ω 0 ) = ( s − ω 0 ) ( s − ω 0 ) k F q , ω ( s ) − ∑ i = 1 k ( s − ω 0 ) k − i D q , ω i − 1 f ( ω 0 ) − D q , ω k f ( ω 0 ) = ( s − ω 0 ) k + 1 F q , ω ( s ) − ∑ i = 1 k ( s − ω 0 ) k + 1 − i D q , ω i − 1 f ( ω 0 ) − D q , ω k f ( ω 0 ) = ( s − ω 0 ) k + 1 F q , ω ( s ) − ∑ i = 1 k + 1 ( s − ω 0 ) k + 1 − i D q , ω i − 1 f ( ω 0 ) .

Therefore, relation (21) is true for n = k + 1 , and by induction, it is true for any n ∈ N .□

Theorem 7

Let f ∈ V . If any n ∈ N , then

(22) L q , ω ( ( x − ω 0 ) n f ( x ) ) = ( − 1 ) n q n D q − 1 , − ω q − 1 n F q , ω ( s ) .

Proof

For n = 1 , from ( x − ω 0 ) f ( x ) = ∑ n = 0 ∞ a n ( x − ω 0 ) n + 1 [ n ] q ! = ∑ n = 0 ∞ [ n + 1 ] q a n ( x − ω 0 ) n + 1 [ n + 1 ] q ! , we have

(23) L q , ω ( ( x − ω 0 ) f ( x ) ) = ∑ n = 0 ∞ [ n + 1 ] q a n ( s − ω 0 ) n + 2 .

On the other hand, using equation (1), we have

(24) D q − 1 , − ω q − 1 F q , ω ( s ) = F q , ω ( q − 1 s − ω q − 1 ) − F q , ω ( s ) ( q − 1 s − ω q − 1 ) − s = ∑ n = 0 ∞ a n ( q − 1 s − ω q − 1 − ω 0 ) n + 1 − ∑ n = 0 ∞ a n ( s − ω 0 ) n + 1 q − 1 ( 1 − q ) ( s − ω 0 ) = ∑ n = 0 ∞ a n ( s − ω 0 ) n + 1 q n + 1 − 1 1 − q q ( s − ω 0 ) = − q ∑ n = 0 ∞ [ n + 1 ] q a n ( s − ω 0 ) n + 2 .

By comparing equations (23) and (24), equation (22) is true for n = 1 . Similarly, for n = 2 , from ( x − ω 0 ) 2 f ( x ) = ∑ n = 0 ∞ a n ( x − ω 0 ) n + 2 [ n ] q ! = ∑ n = 0 ∞ [ n + 1 ] q [ n + 2 ] q a n ( x − ω 0 ) n + 2 [ n + 2 ] q ! , we obtain

(25) L q , ω ( ( x − ω 0 ) 2 f ( x ) ) = ∑ n = 0 ∞ [ n + 1 ] q [ n + 2 ] q a n ( s − ω 0 ) n + 3 .

Using equations (1) and (24) and comparing the results,

(26) D q − 1 , − ω q − 1 2 F q , ω ( s ) = D q − 1 , − ω q − 1 − q ∑ n = 0 ∞ [ n + 1 ] q a n ( s − ω 0 ) n + 2 = ∑ n = 0 ∞ − q [ n + 1 ] q a n ( s q − 1 − ω q − 1 − ω 0 ) n + 2 − ∑ n = 0 ∞ − q [ n + 1 ] q a n ( s − ω 0 ) n + 2 s q − 1 − ω q − 1 − s = ∑ n = 0 ∞ q [ n + 1 ] q a n ( s − ω 0 ) n + 2 1 − q n + 2 1 − q q ( s − ω 0 ) = q 2 ∑ n = 0 ∞ [ n + 1 ] q [ n + 2 ] q a n ( s − ω 0 ) n + 3 .

By comparing equations (25) and (26), equation (22) is true for n = 2 . Continuing this process, we obtain that equation (22) is true for any n ∈ N .□

Remark 2

It can be easily seen from equation (1) that ( D q , ω f ) x − ω q = D q − 1 , − ω q − 1 f ( x ) . Let h ( x ) = q x + ω , h n ( x ) = q n x + ω [ n ] q and h − n ( x ) = x − ω [ n ] q q n (see [7] for further details). Using these connections, equation (22) can be written as follows:

(27) L q , ω ( ( x − ω 0 ) n f ( x ) ) = ( − 1 ) n q n ( D q , ω n F q , ω ) ( h − n ( s ) ) ,

which is more convenient when the q , ω -Laplace transform is applied to differential equations with variable coefficients.

Theorem 8

(Integration property) Let f ∈ V. Then,

  1. (28) L q , ω ∫ ω 0 x f ( t ) d q , ω t = 1 s − ω 0 F q , ω ( s ) .

  2. If f ( ω 0 ) = 0 , then

    (29) L q , ω f ( x ) x − ω 0 = ∫ s ∞ F q , ω ( t ) d q , ω t .

Proof

(i) From the fundamental theorems of q , ω -integral (see [7]) and the q , ω -Taylor expansion of f , we have

∫ ω 0 x f ( t ) d q , ω t = ∫ ω 0 x ∑ n = 0 ∞ a n ( t − ω 0 ) n [ n ] q ! d q , ω t = ∑ n = 0 ∞ a n [ n ] q ! ∫ ω 0 x ( t − ω 0 ) n d q , ω t = ∑ n = 0 ∞ a n ( x − ω 0 ) n + 1 [ n + 1 ] q ! .

Thus, we obtain

L q , ω ∫ ω 0 x f ( t ) d q , ω t = ∑ n = 0 ∞ a n ( s − ω 0 ) n + 2 = 1 s − ω 0 ∑ n = 0 ∞ a n ( s − ω 0 ) n + 1 = 1 s − ω 0 F q , ω ( s ) .

(ii) From f ( ω 0 ) = 0 , we have a 0 = 0 . Thus, we obtain

f ( x ) x − ω 0 = ∑ n = 0 ∞ a n + 1 ( x − ω 0 ) n [ n + 1 ] q ! .

Therefore, we obtain

(30) L q , ω f ( x ) x − ω 0 = ∑ n = 0 ∞ a n + 1 [ n + 1 ] q ( s − ω 0 ) n + 1 = ∑ n = 1 ∞ a n [ n ] q ( s − ω 0 ) n .

Since we need the improper q , ω -integral of f on [ ω 0 , ∞ ) , it is defined as follows (similar to q -analog; see [3], p. 70):

(31) ∫ ω 0 ∞ f ( x ) d q , ω x = ∑ k = − ∞ ∞ ∫ h k + 1 ( a ) h k ( a ) f ( x ) d q , ω x = ( a − h ( a ) ) ∑ k = − ∞ ∞ q k f ( h k ( a ) ) , a ∈ [ ω 0 , ∞ ) .

From q , ω -integral and equation (31), we can write

∫ s ∞ F q , ω ( t ) d q , ω t = ∫ ω 0 ∞ F q , ω ( t ) d q , ω t − ∫ ω 0 s F q , ω ( t ) d q , ω t = ( s − h ( s ) ) ∑ k = − ∞ 0 q k F q , ω ( h k ( s ) ) = ( s − h ( s ) ) ∑ k = 0 ∞ q − k F q , ω ( h − k ( s ) ) .

Using equation (17), we have

(32) ∫ s ∞ F q , ω ( t ) d q , ω t = ( s − h ( s ) ) ∑ k = 0 ∞ q − k ∑ n = 0 ∞ a n ( h − k ( s ) − ω 0 ) n + 1 = ( s − h ( s ) ) ∑ k = 0 ∞ q − k ∑ n = 0 ∞ a n ( q k ) n + 1 ( s − ω 0 ) n + 1 = ( 1 − q ) ( s − ω 0 ) ∑ k = 0 ∞ q − k ∑ n = 1 ∞ a n ( q k ) n + 1 ( s − ω 0 ) n + 1 = ( 1 − q ) ( s − ω 0 ) ∑ k = 0 ∞ ∑ n = 1 ∞ a n ( q k ) n ( s − ω 0 ) n + 1 = ( 1 − q ) ∑ n = 1 ∞ a n ( s − ω 0 ) n ∑ k = 0 ∞ ( q n ) k = ( 1 − q ) ∑ n = 1 ∞ a n ( s − ω 0 ) n 1 1 − q n = ∑ n = 1 ∞ a n [ n ] q ( s − ω 0 ) n

by comparing equations (30) and (32) and thus equation (29) is true.□

Definition 4

The q , ω -analog of convolution of two functions f and g is defined as follows:

(33) ( f ∗ g ) ( x ) = ∫ ω 0 x f ( t ) g ( x − ω 0 − q t ) d q , ω t ,

where g ( x − ω 0 − q t ) = ( x − ω 0 − { q ( t − ω 0 ) } q ) .

Theorem 9

( q , ω -convolution) Let F q , ω ( s ) and G q , ω ( s ) be the q , ω -Laplace transform of f and g, respectively. Then,

(34) L q , ω ( ( f ∗ g ) ( x ) ) = F q , ω ( s ) G q , ω ( s ) .

Proof

From equation (33), we have

(35) ( f ∗ g ) ( x ) = ∫ ω 0 x f ( t ) g ( x − ω 0 − q t ) d q , ω t = ∑ n = 0 ∞ ∑ k = 0 ∞ a n b k [ n ] q ! [ k ] q ! ∫ ω 0 x ( t − ω 0 ) n ( x − ω 0 − { q ( t − ω 0 ) } q ) k d q , ω t .

In equation (35), change of variable t − ω 0 = ( x − ω 0 ) ( u − ω 0 ) , and using the equality ( x − ω 0 − { q ( t − ω 0 ) } q ) k = ( x − ω 0 ) k ( 1 − q ( u − ω 0 ) ) q k , we obtain

(36) ( f ∗ g ) ( x ) = ∑ n = 0 ∞ ∑ k = 0 ∞ a n b k ( x − ω 0 ) n + k + 1 [ n ] q ! [ k ] q ! ∫ ω 0 ω 0 + 1 ( u − ω 0 ) n ( 1 − q ( u − ω 0 ) ) q k d q , ω u .

From equations (7) and (9), we obtain

( f ∗ g ) ( x ) = ∑ n = 0 ∞ ∑ k = 0 ∞ a n b k ( x − ω 0 ) n + k + 1 [ n + k + 1 ] q !

and then obtain

□ L q , ω ( ( f ∗ g ) ( x ) ) = ∑ k = 0 ∞ ∑ l = 0 ∞ a k b l ( s − ω 0 ) k + l + 2 = ∑ k = 0 ∞ a k ( s − ω 0 ) k + 1 ∑ l = 0 ∞ b l ( s − ω 0 ) l + 1 = F q , ω ( s ) G q , ω ( s ) .

4 Applications to q , w -differential equations

Let L q , ω ( y ( x ) ) = Y q , ω ( s ) be the q , ω -Laplace transform of y function and L q , ω − 1 be the inverse q , ω -Laplace transform, i.e., L q , ω − 1 ( Y q , ω ( s ) ) = y ( x ) .

Example 4

Let

(37) D q , ω 2 y ( x ) + y ( x ) = 0 , y ( ω 0 ) = 0 , D q , ω y ( ω 0 ) = 1 .

By applying the q , ω -Laplace transform of equation (37), we obtain

( s − ω 0 ) 2 Y q , ω ( s ) − ( s − ω 0 ) y ( ω 0 ) − D q , ω y ( ω 0 ) + Y q , ω ( s ) = 0 ,

and then, we have

Y q , ω ( s ) = 1 ( s − ω 0 ) 2 + 1 .

Thus,

y ( x ) = L q , ω − 1 ( Y q , ω ( s ) ) = L q , ω − 1 1 ( s − ω 0 ) 2 + 1 = sin q , ω ( x )

is the solution of equation (37).

Example 5

Let

(38) D q , ω 2 y ( x ) − 3 [ 2 ] q D q , ω y ( x ) + 9 q y ( x ) = 0 , y ( ω 0 ) = 1 , D q , ω y ( ω 0 ) = − 2 .

Taking the q , ω -Laplace transform of equation (38), we obtain

Y q , ω ( s ) = s − ω 0 − 3 q − 5 ( s − ω 0 ) 2 − 3 ( 1 + q ) ( s − ω 0 ) + 9 q = 1 s − ω 0 − 3 − 5 ( s − ω 0 − { 3 } q ) 2 .

Thus,

y ( x ) = L q , ω − 1 ( Y q , ω ( s ) ) = L q , ω − 1 1 s − ω 0 − 3 − 5 L q , ω − 1 1 ( s − ω 0 − { 3 } q ) 2 .

Hence,

y ( x ) = ( 1 − 5 ( x − ω 0 ) ) e q , 3 ω ( 3 x ) ,

is the solution of equation (38).

Example 6

Consider the q , ω -diffusion equation

(39) ∂ q , ω ∂ q , ω t u ( x , t ) = ∂ q , ω 2 ∂ q , ω x 2 u ( x , t ) ,

with the initial condition

(40) u ( x , ω 0 ) = e q , ω ( x ) .

Let L q , ω ( u ( x , t ) ) = U ( x , s ) . Taking the q , ω -Laplace transform of both sides of equation (39) with respect to t , we obtain

( s − ω 0 ) U ( x , s ) − u ( x , ω 0 ) = d q , ω 2 d q , ω x 2 U ( x , s ) .

Rearranging and using equation (40), we write

(41) d q , ω 2 d q , ω x 2 U ( x , s ) − ( s − ω 0 ) U ( x , s ) = − e q , ω ( x ) .

Solving the q , ω -differential equation (41) with the initial condition (40) we have

U ( x , s ) = 1 s − ω 0 − 1 e q , ω ( x ) ,

and taking the inverse q , ω -Laplace transform, we obtain the following solution:

u ( x , t ) = e q , ω ( x ) e q , ω ( t ) .

  1. Funding information: This work was supported by the Ondokuz Mayıs University, with the project numbered PYO.FEN.1901.20.003.

  2. Conflict of interest: The author declares that there is no conflict of interest.

  3. Data availability statement: Not applicable.

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Received: 2022-11-22
Revised: 2023-05-03
Accepted: 2023-06-11
Published Online: 2023-10-04

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/dema-2023-0259
Keywords for this article
q, ω-gamma function; q, ω-beta function; q, ω-Laplace transform; q, ω-differential equations; Hahn calculus; q-calculus
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  23. Some aspects of the n-ary orthogonal and b(αn,βn)-best approximations of b(αn,βn)-hypermetric spaces over Banach algebras
  24. Numerical solution of a malignant invasion model using some finite difference methods
  25. Increasing property and logarithmic convexity of functions involving Dirichlet lambda function
  26. Feature fusion-based text information mining method for natural scenes
  27. Global optimum solutions for a system of (k, ψ)-Hilfer fractional differential equations: Best proximity point approach
  28. The study of solutions for several systems of PDDEs with two complex variables
  29. Regularity criteria via horizontal component of velocity for the Boussinesq equations in anisotropic Lorentz spaces
  30. Generalized Stević-Sharma operators from the minimal Möbius invariant space into Bloch-type spaces
  31. On initial value problem for elliptic equation on the plane under Caputo derivative
  32. A dimension expanded preconditioning technique for block two-by-two linear equations
  33. Asymptotic behavior of Fréchet functional equation and some characterizations of inner product spaces
  34. Small perturbations of critical nonlocal equations with variable exponents
  35. Dynamical property of hyperspace on uniform space
  36. Some notes on graded weakly 1-absorbing primary ideals
  37. On the problem of detecting source points acting on a fluid
  38. Integral transforms involving a generalized k-Bessel function
  39. Ruled real hypersurfaces in the complex hyperbolic quadric
  40. On the monotonic properties and oscillatory behavior of solutions of neutral differential equations
  41. Approximate multi-variable bi-Jensen-type mappings
  42. Mixed-type SP-iteration for asymptotically nonexpansive mappings in hyperbolic spaces
  43. On the equation fn + (f″)m ≡ 1
  44. Results on the modified degenerate Laplace-type integral associated with applications involving fractional kinetic equations
  45. Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂn#
  46. Commentary
  47. On I. Meghea and C. S. Stamin review article “Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,” Demonstratio Mathematica 2022; 55: 354–379
  48. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part II
  49. On Cauchy problem for pseudo-parabolic equation with Caputo-Fabrizio operator
  50. Fixed-point results for convex orbital operators
  51. Asymptotic stability of equilibria for difference equations via fixed points of enriched Prešić operators
  52. Asymptotic behavior of resolvents of equilibrium problems on complete geodesic spaces
  53. A system of additive functional equations in complex Banach algebras
  54. New inertial forward–backward algorithm for convex minimization with applications
  55. Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem with the p-Laplacian operator
  56. Analysis of Cauchy problem with fractal-fractional differential operators
  57. Common best proximity points for a pair of mappings with certain dominating property
  58. Investigation of hybrid fractional q-integro-difference equations supplemented with nonlocal q-integral boundary conditions
  59. The structure of fuzzy fractals generated by an orbital fuzzy iterated function system
  60. On the structure of self-affine Jordan arcs in ℝ2
  61. Solvability for a system of Hadamard-type hybrid fractional differential inclusions
  62. Three solutions for discrete anisotropic Kirchhoff-type problems
  63. On split generalized equilibrium problem with multiple output sets and common fixed points problem
  64. Special Issue on Computational and Numerical Methods for Special Functions - Part II
  65. Sandwich-type results regarding Riemann-Liouville fractional integral of q-hypergeometric function
  66. Certain aspects of Nörlund ℐ-statistical convergence of sequences in neutrosophic normed spaces
  67. On completeness of weak eigenfunctions for multi-interval Sturm-Liouville equations with boundary-interface conditions
  68. Some identities on generalized harmonic numbers and generalized harmonic functions
  69. Study of degenerate derangement polynomials by λ-umbral calculus
  70. Normal ordering associated with λ-Stirling numbers in λ-shift algebra
  71. Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators
  72. Compositions of positive integers with 2s and 3s
  73. Kinematic-geometry of a line trajectory and the invariants of the axodes
  74. Hahn Laplace transform and its applications
  75. Discrete complementary exponential and sine integral functions
  76. Special Issue on Recent Methods in Approximation Theory - Part II
  77. On the order of approximation by modified summation-integral-type operators based on two parameters
  78. Bernstein-type operators on elliptic domain and their interpolation properties
  79. A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities
  80. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part II
  81. Application of fractional quantum calculus on coupled hybrid differential systems within the sequential Caputo fractional q-derivatives
  82. On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative
  83. A certain class of fractional difference equations with damping: Oscillatory properties
  84. Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals
  85. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems - Part II
  86. The behavior of hidden bifurcation in 2D scroll via saturated function series controlled by a coefficient harmonic linearization method
  87. Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves
  88. Petri net analysis of a queueing inventory system with orbital search by the server
  89. Asymptotic stability of an epidemiological fractional reaction-diffusion model
  90. On the stability of a strongly stabilizing control for degenerate systems in Hilbert spaces
  91. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part I
  92. New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization
  93. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation
  94. Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations
  95. Some results on fractional Hahn difference boundary value problems
  96. Properties of a subclass of analytic functions defined by Riemann-Liouville fractional integral applied to convolution product of multiplier transformation and Ruscheweyh derivative
  97. Special Issue on Development of Fuzzy Sets and Their Extensions - Part I
  98. The cross-border e-commerce platform selection based on the probabilistic dual hesitant fuzzy generalized dice similarity measures
  99. Comparison of fuzzy and crisp decision matrices: An evaluation on PROBID and sPROBID multi-criteria decision-making methods
  100. Rejection and symmetric difference of bipolar picture fuzzy graph
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