Solutions of certain initial-boundary value problems via a new extended Laplace transform

Yahya Almalki; Mohamed Akel; Mohamed Abdalla

doi:10.1515/nleng-2022-0353

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Solutions of certain initial-boundary value problems via a new extended Laplace transform

Yahya Almalki , Mohamed Akel and Mohamed Abdalla

Published/Copyright: March 2, 2024

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From the journal Nonlinear Engineering Volume 13 Issue 1

Abstract

In this article, we present a novel extended exponential kernel Laplace-type integral transform. The Laplace, natural, and Sumudu transforms are all included in the suggested transform. The existence theorem, Parseval-type identity, inversion formula, and other fundamental aspects of the new integral transform are examined in this article. Integral identities define the connections between the new transforms and the established transforms. In order to solve specific initial-boundary value problems, the new transforms are used.

Keywords: Laplace transform; Sumudu transform; natural transform; H-function; H-transform; inversion formula; Parseval-type identity; Borel–Džrbashjan transform

MSC 2010: 44A10; 44A20; 33C20; 33C60

1 Introduction

During the last two decades, integral transforms have become useful for dealing with differential and integral equations. The suitable choice of an integral transform helps convert differential and integral operators on a certain domain into the multiplication of operators on another domain where problems can be easily solved. Moreover, the inverse transforms help invert the manipulated solution back to the required solution of the original problem in the original domain (see, to exemplify, [1–10]). Nowadays, many authors have established new transforms such as the Laplace–Carson transform, which is used in the railway engineering [11,12], the z -transform, which is applied in signal processing [13], the Sumudu transform, which is used in engineering and many real-life problems [14,16], and the Hankel’s and Weierstrass transform, which is applied in heat and diffusion equations [16,17]. In addition, we have the natural transform [18–20] and the Yang transform [21,22], which are used in many fields of physical sciences and engineering.

The most well-known and often employed sort of integral transform is the Laplace transform, which is utilized in many fields including astronomy, physical applications, probability distributions, and engineering and mathematical modeling (see, e.g., [23–27]). The Laplace transform has been broadly generalized in the past, and the reader is directed to a number of recent works by Bosch et al. [28], Futcher and Rodrigo [29], Ortigueira and Machado [30], Kim [31], Ganie and Jain [32], Almalki et al. [33], and Jarad and Abdeljawad [34].

Recently, the factors exp ( − b τ ) and exp [ − b τ ( 1 − τ ) ] have been used to extend the domains of gamma and beta functions to the entire complex plane, respectively. These functions are called the extended gamma and the extended beta functions and are defined in (cf. [35]) as follows:

(1.1) Γ b ( ξ ) = ∫ 0 ∞ τ ξ − 1 exp − τ − b τ d τ , Re ( b ) > 0 ,

and

(1.2) ℬ b ( θ , ϑ ) = ∫ 0 1 τ θ − 1 ( 1 − τ ) ϑ − 1 exp − b τ ( 1 − τ ) d τ , Re ( b ) > 0 .

Furthermore, these extensions yield an interesting relationship with many special functions, such as the modified Bessel function [35], the error and Whittaker functions [36], the hypergeometric functions [37,38], and other special polynomials and functions [35]. Furthermore, these extensions have been applied in deriving certain probability distributions, which were proposed by Good [39], and in other various applications in physics and engineering [35,40–42].

However, the generalizations of known integral transforms by the factors exp ( − b τ ) and exp [ − b τ ( 1 − τ ) ] have not been defined and discussed. It is, therefore, of great interest to define the generalization of the extended Laplace-type integrals by the exponential factor exp ( − b ∕ t ) and study their properties and potential applications.

The present work is organized six sections. First, we introduce the extended Laplace integral transform or briefly ℒ b -transform. in Section 2. In Section 3, we discuss the basic properties of the ℒ b -transform, namely, the existence of the ℒ b -transform will be established. Furthermore, the linearity, scaling, and elimination properties are stated. The inversion of the ℒ b -transform is also given. In Section 4, integral identities involving the ℒ b -transform are obtained. Relationships of the ℒ b -transform with the Borel-Džrbashjan transform, the Mellin transform, the H -transform, and the Srivastava-Luo-Raina integral transform are established. In Section 5, illustrative examples are presented to show the applications of the ℒ b -transform for solving initial-boundary value problems. Finally, we summarize the obtained results in Section 6.

2 Definitions and special cases

In this section, we define a new extended Laplace integral transform or briefly ℒ b -transform as follows:

(2.1) ℒ b [ f ( x ) ] ( u , ω ) = ∫ 0 ∞ e − u x − b ∕ x f ( ω x ) d x ,

where u , b ∈ C satisfy Re ( u ) > 0 , Re ( b ) > 0 , and ω ∈ R + such that the domain of the transform (2.1) is

A = { f ( x ) : ∃ K , β > 0 , ∣ f ( x ) ∣ < K e β x , x ∈ [ 0 , ∞ ) } ,

the set of all functions that have an exponential order on [ 0 , ∞ ) .

Now, we present some special cases of the ℒ b -transform as follows:

When b = 0 and ω = 1 in Eq. (2.1), the transform ℒ b reduces to the classical Laplace transform
(2.2) ℒ [ f ( x ) ] ( u ) = ∫ 0 ∞ e − u x f ( x ) d x , Re u > β ,
where β is a bounding exponent for the function f . Specifically, when Re u > 0 , we have
ℒ [ f ( x ) ] ( u ) = ℒ 0 [ f ( x ) ] ( u , 1 ) .
Moreover, when b , u , ω > 0 , we observe that
(2.3) ℒ b [ f ( x ) ] ( u , ω ) = ℒ [ e − b ∕ x f ( ω x ) ] ( u ) ,

(2.4) ℒ b [ f ( x ) ] ( u , ω ) = 1 ω ℒ [ e − b ω ∕ x f ( x ) ] u ω ,
and
(2.5) ℒ b [ e b ω ∕ x f ( x ) ] ( u , ω ) = 1 ω ℒ [ f ( x ) ] u ω .
Setting b = 0 in Eq. (2.1), the ℒ b -transform reduces to the natural transform [18–20]
(2.6) N [ f ( x ) ] ( u , ω ) = ∫ 0 ∞ e − u x f ( ω x ) d x .
In particular, we note that
(2.7) N [ f ( x ) ] ( u , ω ) = ℒ 0 [ f ( x ) ] ( u , ω ) , u , ω > 0 .
Moreover, for b , u , ω > 0 , we observe that
(2.8) ℒ b [ f ( x ) ] ( u , ω ) = N [ e − b ω ∕ x f ( x ) ] ( u , ω ) ,
and
(2.9) ℒ b [ e b ω ∕ x f ( x ) ] ( u , ω ) = N [ f ( x ) ] ( u , ω ) .
The Sumudu transform defined by [14,16]
(2.10) S [ f ( x ) ] ( ω ) = ∫ 0 ∞ e − x f ( ω x ) d x , ω > 0 ,
can be viewed as a special case of the ℒ b -transform as follows:
S [ f ( x ) ] ( ω ) = ℒ 0 [ f ( x ) ] ( 1 , ω ) , ω > 0 .
Furthermore, when b , u , ω > 0 , we have
ℒ b [ f ( x ) ] ( u , ω ) = 1 u S [ e − b ω ∕ x f ( x ) ] ω u .

3 Properties of the ℒ b -transform

In this section, we establish the existence of the ℒ b -transform and demonstrate some of its properties.

Theorem 1

(Existence theorem) Let f ( x ) be a continuous function (or piecewise continuous) on ( 0 , ∞ ) satisfying

(3.1) ∣ f ( x ) ∣ < K e x β for a l l x > T ,

where K , T , and β are positive constants. Then, the ℒ b [ f ( x ) ] ( u , ω ) given by Eq. (2.1) exists for all u , b , and ω such that

ω ∈ ( 0 , μ ) , Re u > μ β ,

is satisfied for some positive constant μ . Furthermore, the integral (2.1) converges uniformly whenever Re u ⩾ a > μ β , holds.

Proof

From Eq. (3.1), it follows immediately

∣ ℒ b [ f ( x ) ] ( u , ω ) ∣ ⩽ ∫ 0 ∞ e − x Re u ∣ f ( ω x ) ∣ d x ⩽ K ∫ 0 ∞ e − Re u − ω β x d x .

Since

∣ e − u x − b ∕ x ∣ ⩽ e − x Re u , x ⩾ 0 ,

the last integral and hence the integral (2.1) exist provided that Re u > μ β and 0 < ω < μ .

The uniform convergence of the integral (2.1) follows directly from the Weierstrass’s test.

Proposition 1

Assume that f and g satisfy condition (3.1). Then, we have the following properties:

Linearity property:

(3.2) ℒ b [ α f ( x ) + β g ( x ) ] = α ℒ b [ f ( x ) ] + β ℒ b [ g ( x ) ] , α , β ∈ C .

Scaling property:

(3.3) ℒ b [ f ( α x ) ] ( u , ω ) = 1 α ℒ α b [ f ( x ) ] u α , ω , α > 0 .

Elimination property:

(3.4) ℒ b [ e a ω ∕ x f ( x ) ] ( u , ω ) = ℒ b − a [ f ( x ) ] ( u , ω ) , Re ( b − a ) > 0 .

Proof

These characteristics are plainly evident from the definition (2.1) of ℒ b -transform.

Remark 1

From Eq. (1.1), we see that the extended gamma function is defined in Eq. (1.1) and simple computations yield

(3.5) ∫ 0 ∞ x z − 1 e − σ x − b x d x = Γ σ b ( z ) σ z , Re z > 0 , Re σ > 0 , Re b > 0 .

In particular, for b = 0 , we obtain the Euler integral

(3.6) ∫ 0 ∞ x z − 1 e − σ x d x = Γ ( z ) σ z , Re z > 0 , Re σ > 0 .

This justifies naming the integral (3.5) as the extended Euler integral.

Table 1 illustrates the ℒ b -transform of some functions.

Table 1

The ℒ b -transform of some functions

Function	New integral transform
f ( x )	ℒ b [ f ( x ) ] ( u , ω )
1	1 u Γ b u ( 1 )
x z ‒ 1	ω z ‒ 1 u z Γ b u ( z ) , Re z > 0
e ‒ a x	1 u + a ω Γ b ( u + a ω ) ( 1 ) , a ⩾ 0
x z ‒ 1 e ‒ a x	ω z ‒ 1 ( u + a ω ) z Γ b ( u + a ω ) ( z ) , Re z > 0 , a ⩾ 0
x n f ( x )	( ‒ 1 ) n ω n ∂ n ∂ u n ℒ b [ f ( x ) ] ( u , ω )
x n f ( x )	ω n ∫ b ∞ ∫ s n ∞ ⋯ ∫ s 2 ∞ ℒ s 1 [ f ( x ) ] ( u , ω ) d s 1 ⋯ d s n ‒ 1 d s n
f ( x ) x n	ω ‒ n ∫ u ∞ ∫ s n ∞ ⋯ ∫ s 2 ∞ ℒ b [ f ( x ) ] ( s 1 , ω ) d s 1 ⋯ d s n ‒ 1 d s n
f ( x ) x n	( ‒ 1 ) n ω ‒ n ∂ n ∂ b n ℒ b [ f ( x ) ] ( u , ω )
e ‒ a ∕ x f ( x )	ℒ b + a ∕ ω [ f ( x ) ] ( u , ω ) , a ⩾ 0
e ‒ a x f ( x )	ℒ b [ f ( x ) ] ( u + a ω , ω ) , a ⩾ 0

Theorem 2

( ℒ b -transform of derivatives) If the n th derivative of a function f ( x ) satisfies the hypothesis of Theorem 1, then

(3.7) ℒ b [ f ( n ) ( x ) ] ( u , ω ) = u n ω n ℒ b [ f ( x ) ] ( u , ω ) − δ b , 0 ∑ k = 0 n − 1 u k ω k + 1 f ( n − k − 1 ) ( 0 ) − b ω ∑ k = 0 n − 1 u k ω k ℒ b [ x − 2 f ( n − k − 1 ) ( x ) ] ( u , ω ) ,

where δ b , 0 is the delta function and is defined as follows:

δ b , 0 = 1 , b = 0 , 0 , b ≠ 0 .

Proof

For n = 1 , if f ( x ) and f ′ ( x ) satisfy the conditions of Theorem 1, then

ℒ b [ f ′ ( x ) ] ( u , ω ) = ∫ 0 ∞ e − u x − b x f ′ ( ω x ) d x = 1 ω ∫ 0 ∞ e − u x − b x d f ( ω x ) = 1 ω e − u x − b x f ( ω x ) 0 ∞ − ∫ 0 ∞ f ( ω x ) ∂ ∂ x e − u x − b x d x .

Using condition (3.1), if Re u > ω β and Re b > 0 when b ∈ C ⧹ { 0 } , we have

e − u x − b x f ( ω x ) ⩽ K e − ( Re u − ω ∕ β ) x ,

which shows that

lim x → ∞ e − u x − b x f ( ω x ) = 0 , lim x → 0 e − u x − b x f ( ω x ) = f ( 0 ) δ b , 0 .

Since

∂ ∂ x e − u x − b x = − u − b x 2 e − u x − b x ,

we arrive at

(3.8) ℒ b [ f ′ ( x ) ] ( u , ω ) = − f ( 0 ) ω δ b , 0 + u ω ∫ 0 ∞ e − u x − b x f ( ω x ) d x − b ω ∫ 0 ∞ e − u x − b x f ( ω x ) x 2 d x = u ω ℒ b [ f ( x ) ] ( u , ω ) − f ( 0 ) ω δ b , 0 − b ω ℒ b [ x − 2 f ( x ) ] ( u , ω ) .

For the induction, assume that Eq. (3.7) holds for some integer n . Then, for n + 1 , from Eq. (3.8), one obtains

(3.9) ℒ b [ f ( n + 1 ) ( x ) ] ( u , ω ) = u ω ℒ b [ f ( n ) ( x ) ] ( u , ω ) − f ( n ) ( 0 ) ω δ b , 0 − b ω ℒ b [ x − 2 f ( n ) ( x ) ] ( u , ω ) .

Plugging Eq. (3.7) into Eq. (3.9) leads to

ℒ b [ f ( n + 1 ) ( x ) ] ( u , ω ) = u n + 1 ω n + 1 ℒ b [ f ( x ) ] ( u , ω ) − δ b , 0 ∑ k = 0 n − 1 u k + 1 ω k + 2 f ( n − k − 1 ) ( 0 ) − b ω ∑ k = 0 n − 1 u k + 1 ω k + 2 ℒ b [ x − 2 f ( n − k − 1 ) ( x ) ] ( u , ω ) − f ( n ) ( 0 ) ω δ b , 0 − b ω ℒ b [ x − 2 f ( n ) ( x ) ] ( u , ω ) ,

which can be rewritten as follows:

ℒ b [ f ( n + 1 ) ( x ) ] ( u , ω ) = u n + 1 ω n + 1 ℒ b [ f ( x ) ] ( u , ω ) − δ b , 0 ∑ k = 0 n u k ω k + 1 f ( n − k ) ( 0 ) − b ω ∑ k = 0 n u k ω k + 1 ℒ b [ x − 2 f ( n − k ) ( x ) ] ( u , ω ) .

This completes the proof of Theorem 2.

From Theorem 2, we obtain the following corollaries.

Corollary 1

[19] For b = 0 and v = 0 in Eq. (3.7), we obtain

N [ f ( n ) ( x ) ] ( u , ω ) = u n ω n N [ f ( x ) ] ( u , ω ) − ∑ k = 0 n − 1 u k ω k + 1 f ( n − k − 1 ) ( 0 ) ,

which can be rewritten as follows:

N [ f ( n ) ( x ) ] ( u , ω ) = u n ω n N [ f ( x ) ] ( u , ω ) − ∑ k = 0 n − 1 u n − k − 1 ω n − k f ( k ) ( 0 ) ,

where N is the natural transform defined by Eq. (2.6).

Corollary 2

[16] For b = v = 0 and u = 1 in Eq. (3.7), we obtain

S [ f ( n ) ( x ) ] ( ω ) = 1 ω n S [ f ( x ) ] ( ω ) − ∑ k = 0 n − 1 1 ω n − k f ( k ) ( 0 ) ,

where S is the Sumudu transform (2.10).

Corollary 3

[9] For b = v = 0 and ω = 1 in Eq. (3.7), we have

ℒ [ f ( n ) ( x ) ] ( u , ω ) = u n ℒ [ f ( x ) ] ( u , ω ) − ∑ k = 0 n − 1 u n − k − 1 f ( k ) ( 0 ) ,

where ℒ is the Laplace transform defined by Eq. (2.2).

Corollary 4

If f ( n ) , the n th -derivative of a function f, satisfies the assumptions of Theorem 1, then for ν ∈ C with Re ν > 0 , we have

ℒ b [ e ν ∕ x f ( n ) ( x ) ] ( u , ω ) = u n ω n ℒ b − ν ∕ ω [ f ( x ) ] ( u , ω ) − δ b − ν ∕ ω , 0 ∑ k = 0 n − 1 u k ω k + 1 f ( n − k − 1 ) ( 0 ) − ( b ω − ν ) ∑ k = 0 n − 1 u k ω k ℒ b − ν ∕ ω [ x − 2 f ( n − k − 1 ) ( x ) ] ( u , ω ) .

Next, an inversion formula for the ℒ b -transform shown in Eq. (2.1) will be obtained using the relation (2.3). If F ( s ) is the Laplace transform of a function f ( x ) , then the inverse Laplace transform is (see, [9])

(3.10) ℒ − 1 [ F ( s ) ] ( x ) = 1 2 π i ∫ α − i ∞ α + i ∞ e s x F ( s ) d s , α > 0 .

Theorem 3

( ℒ b -inversion formula) The inversion of the ℒ b -transform (2.1) is given as follows:

(3.11) f ( x ) = e ω b ∕ x ℒ − 1 [ ℒ b [ f ( x ) ] ( u , ω ) ] x ω ,

provided that the involved integrals absolutely converge.

Proof

Consider

(3.12) F ( x ; ω ) = f ( ω x ) e − b ∕ x , x > 0 , ω > 0 .

It is clear that under the condition (3.1), the function F is well-defined. Substituting Eq. (3.12) in Eq. (2.1) provides

ℒ b [ f ( x ) ] ( u , ω ) = ℒ [ F ( x , ω ) ] ( u , ω ) ,

where the variables b and ω of the ℒ b -transform are considered as parameters.

Applying the Laplace-inversion formula yields

ℒ − 1 [ ℒ b [ f ( x ) ] ( u , ω ) ] = F ( x , v ) .

By replacing x with x ∕ ω and using Eq. (3.12), the required result (Eq. (3.11)) follows directly.

4 Integral identities involving the ℒ b -transform

In this section, we obtain some integral identities involving the ℒ b -transform (2.1) and well-known integral transforms and special functions. The identities established here include Parseval-type identity, Fubini-type identity, Mellin integral, H -transform, Borel-Džrbashjan transform and the Srivastava-Luo-Raina transform of the ℒ b transform.

For the sake of completeness, we recall the definitions of some famous integral transforms needed in this study as follows.

The Borel–Džrbashjan transform is defined [43] as follows:

(4.1) B ν , μ [ f ( x ) ] ( s ) = ν s ν μ − 1 ∫ 0 ∞ e − s ν x ν x ν μ − 1 f ( x ) d x , ν , μ > 0 .

The Mellin transform is defined [5] as follows:

(4.2) ℳ [ f ( x ) ] ( z ) = ∫ 0 ∞ x z − 1 f ( x ) d x , Re z > 0 .

The H -transform is defined [44] as follows:

(4.3) H [ f ( x ) ] ( t ) = ∫ 0 ∞ H r , s m , n x t ( a i , α i ) 1 , r ( b j , β j ) 1 , s f ( x ) d x ,

where m , n , r , and s are positive integers such that 0 ⩽ m ⩽ s , 0 ⩽ n ⩽ r ; a i , b j ∈ C , and α i , β j ∈ R + ( 1 ⩽ i ⩽ r ; 1 ⩽ j ⩽ s ), and

(4.4) H r , s m , n z ( a i , α i ) 1 , r ( b j , β j ) 1 , s = 1 2 π i ∫ C ℋ r , s m , n ( θ ) z − θ d θ

is the H -function defined in terms of a Mellin–Barnes-type integral over a suitable contour C , with

(4.5) ℋ r , s m , n ( θ ) = ∏ j = 1 m Γ ( b j + β j θ ) ∏ j = m + 1 s Γ ( 1 − b j − β j θ ) × ∏ i = 1 n Γ ( 1 − a j − α j θ ) ∏ i = n + 1 r Γ ( a i + α i θ )

where an empty product, if it exists, is taken to equal 1.

Furthermore, the Srivastava–Luo–Raina integral transform is defined [45] as follows:

(4.6) M ρ , m [ f ( x ) ] ( u , ν ) = ∫ 0 ∞ e − u x f ( ν x ) ( x m + ν m ) ρ d x ,

where u , ρ ∈ C with Re ρ > 0 and ω ∈ R + .

Theorem 4

(Parseval-type identity) Let f and g satisfy the condition (3.1). Then,

(4.7) ∫ 0 ∞ f ( ω τ ) e − b 1 ∕ τ ℒ b 2 [ g ( x ) ] ( τ , ω ) d τ = ∫ 0 ∞ g ( ω x ) e − b 2 ∕ x ℒ b 1 [ f ( τ ) ] ( x , ω ) d x ,

where b i ∈ C , Re b i > 0 , ( i = 1 , 2 ) , and ω ∈ R + .

Proof

Using Eq. (2.1) and applying the Fubini’s theorem, one obtains

∫ 0 ∞ f ( ω τ ) e − b 1 ∕ τ ℒ b 2 [ g ( x ) ] ( τ , ω ) d τ = ∫ 0 ∞ f ( ω τ ) e − b 1 ∕ τ ∫ 0 ∞ e − τ x − b 2 ∕ x g ( ω x ) d x d τ = ∫ 0 ∞ g ( ω x ) e − b 2 ∕ x ∫ 0 ∞ e − τ x − b 1 ∕ τ f ( ω τ ) d τ d x = ∫ 0 ∞ g ( ω x ) e − b 2 ∕ x ℒ b 1 [ f ( τ ) ] ( x , ω ) d x ,

which completes the proof.

Remark 2

Setting b 1 = b 2 = 0 in Eq. (4.7) leads to the Parseval-type identity for the natural transform (2.6) (see [9,19]) as follows:

(4.8) ∫ 0 ∞ f ( ω τ ) N [ g ( x ) ] ( τ , ω ) d τ = ∫ 0 ∞ g ( ω x ) N [ f ( τ ) ] ( x , ω ) d x .

Moreover, if we set ω = 1 in Eq. (4.8), then we obtain the Parseval-type identity for the Laplace transform

∫ 0 ∞ f ( τ ) ℒ [ g ( x ) ] ( τ ) d τ = ∫ 0 ∞ g ( x ) ℒ [ f ( τ ) ] ( x ) d x .

Next, we show the relationship between the ℒ b -transform Eq. (2.1) and some well-known integral transforms.

Theorem 5

Let f and g satisfy the condition (3.1). Then,

(4.9) ∫ 0 ∞ f ( ω τ ) e − u τ ℒ τ [ g ( x ) ] ( u , ω ) d τ = ∫ 0 ∞ g ( ω x ) e − u x N [ f ( τ ) ] u + 1 x , ω d x ,

where N is the natural transform defined by Eq. (2.6).

Proof

This result can be easily established by applying the argument used in proving Theorem 4.

Theorem 6

Let f and g satisfy the condition (3.1). Then,

(4.10) ∫ 0 ∞ f ( ω u ) ℒ b [ e − a x g ( x ) ] ( u ω , v ) d u = ℒ b [ g ( x ) N [ f ( u ) ] ( x , ω ) ] ( a ω , ω ) , a > 0 ,

where N is the natural transform defined by Eq. (2.6).

Proof

Plugging Eq. (2.1) into the left-hand side of Eq. (4.10), we have

∫ 0 ∞ f ( ω u ) ℒ b [ e − a x g ( x ) ] ( u ω , ω ) d u = ∫ 0 ∞ f ( ω u ) ∫ 0 ∞ e − u ω x − b ∕ x e − a ω x g ( ω x ) d x d u = ∫ 0 ∞ e − a ω x − b ∕ x g ( ω x ) ∫ 0 ∞ e − u ω x f ( ω u ) d u d x = ∫ 0 ∞ e − a ω x − b ∕ x g ( ω x ) N [ f ( u ) ] ( ω x , ω ) d x = ℒ b [ g ( x ) N [ f ( u ) ] ( x , ω ) ] ( a ω , ω ) ,

which is the targeted result (Eq. (4.10)).

Theorem 7

Let f satisfy the condition (3.1). Then

(4.11) ℒ [ ℒ b [ f ( x ) ] ( u , ω ) ; u → s ] = S t 1 [ e − b ∕ x f ( ω x ) ] ( s ) ,

where ℒ is the Laplace transform defined by Eq. (2.2) and S t 1 is the Stieltjes transform defined as follows:

(4.12) S t 1 [ f ( x ) ] ( s ) = ∫ 0 ∞ f ( x ) x + s d x , s > 0 .

Proof

Loading Eq. (2.1) into Eq. (2.2) and applying the Fubini’s theorem, we arrive at

ℒ [ ℒ b [ f ( x ) ] ( u , ω ) ; u → s ] = ∫ 0 ∞ e − s u ∫ 0 ∞ e − u x − b ∕ x f ( ω x ) d x d u = ∫ 0 ∞ f ( ω x ) e − b ∕ x ∫ 0 ∞ e − ( x + s ) u d u d x = ∫ 0 ∞ f ( ω x ) e − b ∕ x x + s d x .

This gives the required result (Eq. (4.11)).

Theorem 8

Let f satisfy the condition (3.1), then

(4.13) N [ ℒ b [ f ( x ) ] ( u , ω ) ; u → s ] ( s , v ) = S t 1 [ e − b ∕ x f ( ω x ) ] s v ,

where N is the natural transform defined by Eq. (2.6) and S t 1 is the Stieltjes transform defined by Eq. (4.12).

Proof

Loading Eq. (2.1) into Eq. (2.6) and applying the Fubini’s theorem, we obtain

N [ ℒ b [ f ( x ) ] ( u , ω ) ; u → s ] ( s , v ) = ∫ 0 ∞ e − s u ∫ 0 ∞ e − u v x − b ∕ x f ( ω x ) d x d u = ∫ 0 ∞ f ( ω x ) e − b ∕ x ∫ 0 ∞ e − ( v x + s ) u d u d x = ∫ 0 ∞ f ( ω x ) e − b ∕ x v x + s d x = 1 v ∫ 0 ∞ f ( ω x ) e − b ∕ x x + s v d x .

This gives the required result (Eq. (4.13)).

Theorem 9

Let f satisfy the condition (3.1), then

(4.14) ℳ [ ℒ b [ e − a x f ( x ) ] ( u , ω ) ; u → z ] = Γ ( z ) ω z ℒ b [ x − z f ( x ) ] ( a ω , ω ) , a > 0 ,

where ℳ is the Mellin transform defined by Eq. (4.2).

Proof

It is clear that

ℳ [ ℒ b [ e − a x f ( x ) ] ( u , ω ) ; u → z ] = ∫ 0 ∞ u z − 1 ℒ b [ e − a x f ( x ) ] ( u , ω ) d u = ∫ 0 ∞ u z − 1 ∫ 0 ∞ e − u x − b ∕ x e − a ω x f ( ω x ) d x d u = ∫ 0 ∞ e − a ω x − b ∕ x f ( ω x ) ∫ 0 ∞ u z − 1 e − u x d u d x = Γ ( z ) ∫ 0 ∞ e − a ω x − b ∕ x f ( ω x ) x − z d x = Γ ( z ) ω z ℒ b [ f ( x ) x − z ] ( a ω , ω ) .

This completes the proof.

Theorem 10

Let g satisfy the condition (3.1), then, for the positive real numbers a , ν , and μ , we have

(4.15) B ν , μ [ ℒ b [ e − a x g ( x ) ] ( u , ω ) ; u → s ] ( s , ω ) = s ν μ − 1 ω ν μ ℒ b g ( x ) x ν μ H 1 , 1 1 , 1 ω s x ( 1 − ν μ , 1 ) 0 , 1 ν ( a ω , ω ) ,

where B is the Borel–Džrbashjan transform defined by Eq. (4.1) and H 1 , 1 1 , 1 is an H-function defined by Eq. (4.4).

Proof

From Eqs (2.1) and (4.1), we have

(4.16) B ν , μ [ ℒ b [ e − a x g ( x ) ] ( u , ω ) ; u → s ] ( s , ω ) = ν s ν μ − 1 ∫ 0 ∞ e − s ν u ν u ν μ − 1 ℒ b [ e − a x g ( x ) ] ( u , ω ) d u .

Operating Eq. (2.1) on the function f ( x ) = e − a x g ( x ) , with the Fubini’s theorem, we have

ν s ν μ − 1 ∫ 0 ∞ e − s ν u ν u ν μ − 1 ℒ b [ e − a x g ( x ) ] ( u , ω ) d u = ν s ν μ − 1 ∫ 0 ∞ g ( ω x ) e − a ω x e − b ∕ x ∫ 0 ∞ e − s ν u ν u ν μ − 1 e − u x d u d x .

In view of Eq. (2.2), the interior integral is the Laplace transform of the function f ( u ) = e − s ν u ν u ν μ − 1 , then

ν s ν μ − 1 ∫ 0 ∞ e − s ν u ν u ν μ − 1 ℒ b [ e − a x g ( x ) ] ( u , ω ) d u = ν s ν μ − 1 ∫ 0 ∞ g ( ω x ) e − a ω x e − b ∕ x ℒ [ e − s ν u ν u ν μ − 1 ] ( x ) d x .

According to [45, p. 1396] we have

ℒ [ u ν μ − 1 e − u ν ω ν ] ( x ) = 1 x ν μ H 1 , 1 1 , 1 s ν x ν ( 1 − ν μ , ν ) ( 0 , 1 ) = 1 ν x ν μ H 1 , 1 1 , 1 s x ( 1 − ν μ , 1 ) 0 , 1 ν .

Thus, Eq. (4.16) reduces to

(4.17) B ν , μ [ ℒ b [ e − a x g ( x ) ] ( u , ω ) ; u → s ] ( s , ω ) = s ν μ − 1 ∫ 0 ∞ g ( ω x ) e − a ω x e − b ∕ x 1 x ν μ H 1 , 1 1 , 1 × s x ( 1 − ν μ , 1 ) 0 , 1 ν ( x ) d x ,

which can be written on the required form (Eq. (4.15)).

Theorem 11

Let f satisfy the condition (3.1). Then, the following identity holds true.

(4.18) H [ ℒ b [ e − a x f ( x ) ] ( u , ω ) ] ( t , ω ) = ω ℒ b f ( x ) x H r + 1 , s m , n + 1 ω t x ( 0 , 1 ) , ( a i , α i ) 1 , r ( b j , β j ) 1 , s ( t , ω ) ,

where H is the integral transform defined by Eq. (4.3).

Proof

From Eqs (2.1) and (4.3), we have

(4.19) H [ ℒ b [ e − a x f ( x ) ] ( u , ω ) ] ( t , ω ) = ∫ 0 ∞ H r , s m , n u t ( a i , α i ) 1 , r ( b j , β j ) 1 , s ∫ 0 ∞ e − u x − b ∕ x e − a ω x f ( ω x ) d x d u = ∫ 0 ∞ e − a ω x − b ∕ x f ( ω x ) ∫ 0 ∞ e − u x H r , s m , n u t ( a i , α i ) 1 , r ( b j , β j ) 1 , s d u d x = ℒ b f ( x ) ℒ H r , s m , n u t ( a i , α i ) 1 , r ( b j , β j ) 1 , s ( x ) ( t , ω ) .

According to [44,45], we have

ℒ H r , s m , n v ( a i , α i ) 1 , r ( b j , β j ) 1 , s ( x ) = 1 x H r + 1 , s m , n + 1 1 x ( o , 1 ) , ( a i , α i ) 1 , r ( b j , β j ) 1 , s .

Thus, Eq. (4.19) can be rewritten as the targeted form Eq. (4.18).

Theorem 12

Let f satisfy the condition (3.1). Then, the following identity holds true:

(4.20) M ρ , m [ ℒ b [ e − a x f ( x ) ] ( u , ω ) ] ( t , v ) = v − m ρ m Γ ( ρ ) ℒ b f ( x ) t + v x ω H 1 , 2 2 , 1 v t + v x ω 1 , 1 m ( 1 , 1 ) , ρ , 1 m ( a ω , ω ) ,

where M ρ , m is the Srivastava–Luo–Raina integral transform defined by Eq. (4.6).

Proof

From Eqs (2.1) and (4.6), we obtain

M ρ , m [ ℒ b [ e − a x f ( x ) ] ( u , ω ) ] ( t , v ) = ∫ 0 ∞ e − u t ( u m + v m ) ρ ∫ 0 ∞ e − u x − b ∕ x e − a ω x f ( ω x ) d x d u = ∫ 0 ∞ e − a ω x − b ∕ x f ( ω x ) ∫ 0 ∞ e − u t ( u m + v m ) ρ e − u x d u d x = ∫ 0 ∞ e − a ω x − b ∕ x f ( ω x ) M ρ , m [ e − u x ] ( t , v ) d x .

According to [45, (2.8), p. 1390], one has

M ρ , m [ e − u x ] ( t , v ) = v − m ρ m Γ ( ρ ) 1 t + v x H 1 , 2 2 , 1 × v ( t + v x ) 1 , 1 m ( 1 , 1 ) , ρ , 1 m .

Then,

M ρ , m [ ℒ b [ e − a x f ( x ) ] ( u , ω ) ] ( t , v ) = v − m ρ m Γ ( ρ ) ∫ 0 ∞ e − a ω x − b ∕ x f ( ω x ) t + v x H 1 , 2 2 , 1 × v ( t + v x ) 1 , 1 m ( 1 , 1 ) , ρ , 1 m d x ,

which leads directly to Eq. (4.20).

5 Applications

The rise of the exponential factor exp ( − b ∕ t ) in the ℒ b -transform not only helps in simplifying and tackling computational difficulties in the results of Srivastava et al. [45] and Haberman [46] but also generalizes their works. Therefore, it is evident that the ℒ b -transform can be a strong mathematical tool for solving such problems that surfaced in the literature (cf. [45–49]). We provide two illustrative examples of the key findings from Sections 2 and 3.

Example 1

First-order initial-boundary value problem.

We consider the following first-order initial-boundary value problem:

(5.1) w t + w x = p ( t , ω ) e − b ω ∕ t r ( t , x ) , t > 0 , x > 0 ,

(5.2) w ( 0 , x ) = ω ϕ ( ω ) , x ⩾ 0 ,

(5.3) w ( t , 0 ) = 0 , t ⩾ 0 ,

where

0 < p ( t , ω ) ≔ t m ω m + ω m − ρ , ω > 0 , t ⩾ 0 , m ∈ N 0 = { 0 , 1 , 2 , … } ,

and r ( t , x ) and ϕ ( ω ) are given functions, where v and ω are taken as parameters.

Eq. (5.1) can be rewritten as follows:

(5.4) e b ω ∕ t w t + e b ω ∕ t w x = p ( t , ω ) r ( t , x ) , t > 0 , x > 0 .

Applying the ℒ b -transform shown in Eq. (2.1) to Eq. (5.4), and using the elimination property (3.4) and the duality relation (2.7) of the ℒ b -transform with the natural transform, we obtain

(5.5) u ω w ˆ ( u , x , v ) − 1 ω w ( 0 , x ) + w ˆ x = ℒ b [ p ( t , ω ) r ( t , x ) ] ( u , x , ω ) ,

where w ˆ is the natural transform image of w .

Using the initial condition (5.2) yields

(5.6) w ˆ x + u ω w ˆ = F ( x ) ,

where

(5.7) F ( x ) = ℒ b [ r ( t , x ) ] ( u , x , v ) + ϕ ( ω ) .

Transforming the boundary condition Eq. (5.3) gives

(5.8) w ˆ ( u , 0 , v ) = 0 , u ⩾ 0 .

The solution to problem Eqs (5.6), (5.8) is

(5.9) w ˆ ( u , x , v ) = e − u ω x ∫ 0 x F ( y ) e u ω y d y .

Now, applying the inverse natural transform and using Eq. (5.7), we obtain

(5.10) w ( t , x , v ) = N − 1 e − u ω x ∫ 0 x e u ω y ℒ b [ p ( t , ω ) r ( t , x ) ] ( u , y , ω ) d y + N − 1 e − u ω x ϕ ( ω ) ∫ 0 x e u ω y d y = N − 1 e − u ω x ∫ 0 x e u ω y ℒ b [ p ( t , ω ) r ( t , x ) ] ( u , y , ω ) d y + ω ϕ ( ω ) N − 1 1 u 1 − e − u ω x = N − 1 e − u ω x ∫ 0 x e u ω y ℒ b [ p ( t , ω ) r ( t , x ) ] ( u , y , ω ) d y + ω ϕ ( ω ) [ θ ( t ) − θ ( t − x ) ] ,

where θ ( t ) is the Heaviside function defined as follows:

θ ( t ) = 1 ; t > 0 , 0 ; t < 0 .

Remark 3

When b = 0 , the formula (5.10) recovers the solution to problems (5.1)–(5.3) obtained via the Srivastava–Luo–Raina generalized M -transform in [45]. According to Eqs (2.1) and (4.6), we have

M [ r ( t , x ) ] ( u , x , ω ) = ℒ 0 [ p ( t , ω ) r ( t , x ) ] ( u , x , ω ) .

Example 2

Second-order initial-boundary value problem.

We consider the following second-order initial-boundary value problem:

(5.11) ∂ φ ∂ t = ∂ 2 φ ∂ x 2 + ( t m + 1 ) − ρ e − b ∕ t r ( t , x ) , x ∈ ( 0 , π ) , t > 0 φ ( x , 0 ) = f ( x ) , x ∈ [ 0 , π ] , φ ( 0 , t ) = 0 , φ ( π , t ) = 0 , t ⩾ 0 ,

where f ( x ) and r ( x , t ) are known functions.

We rewrite Eq. (5.11) as follows:

(5.12) e b ∕ t ∂ φ ∂ t − e b ∕ t ∂ 2 φ ∂ x 2 = ( t m + 1 ) − ρ r ( x , t ) , x ∈ ( 0 , π ) , t > 0 .

Applying the ℒ b -transform (2.1) (when ω = 1 ) to Eq. (5.12) and using the duality relation (2.5) with the Laplace transform yields

(5.13) φ ˆ x x − u φ ˆ ( x , u ) = − ℒ b [ ( t m + 1 ) − ρ r ( t , x ) ] ( u , x , 1 ) − f ( x ) , x ∈ [ 0 , π ] ,

where φ ˆ is the Laplace transform of φ . Transforming the boundary data in Eq. (5.11) yields

(5.14) φ ˆ ( 0 , u ) = φ ˆ ( π , u ) = 0 .

Using the variation of parameters method, one can obtain the solution to the boundary value problem defined by Eqs (5.13) and (5.14) as follows:

(5.15) φ ˆ ( x , u ) = − 1 u ∫ 0 x sinh u x sinh u ( π − y ) − sinh u π sinh u ( x − y ) sinh u π G ( y ) d y − 1 u ∫ x π sinh u x sinh u ( π − y ) sinh u π G ( y ) d y ,

where

(5.16) G ( x ) = − ℒ b [ ( t m + 1 ) − ρ r ( t , x ) ] − f ( x ) = − ℒ e − b ∕ t r ( t , x ) ( t m + 1 ) ρ ( u ) − f ( x ) , x ∈ [ 0 , π ] .

Here, the duality relation (2.3) is used.

Since

Δ ( u , ω ) ≔ sinh u x sinh u ( π − y ) u sinh u π = O ( 1 ) as u → 0 ,

then Δ ( u , ω ) and φ ˆ ( x , u ) have simple poles at u k = − k 2 for all k = 1 , 2 , … . Therefore, we arrive at

(5.17) ℒ − 1 sinh u x sinh u ( π − y ) u sinh u π = Θ ( x , t ; y ) ,

with

(5.18) Θ ( x , t ; y ) = 2 π ∑ k = 1 ∞ ( − 1 ) k + 1 e − k 2 t sin k x sin k y .

In view of Eqs (5.16)–(5.18), the relation (5.15) can be rewritten as follows:

(5.19) φ ˆ ( x , u ) = − ∫ 0 π ℒ [ Θ ( x , t ; y ) ] ℒ e − b ∕ t r ( t , y ) ( t m + 1 ) ρ d y − ∫ 0 π ℒ [ Θ ( x , t ; y ) ] f ( y ) d y .

By the Laplace convolution theorem, the inverse Laplace transform of Eq. (5.19) is

(5.20) φ ( x , t ) = − ∫ 0 π ∫ 0 t Θ ( x , t − ζ ; y ) e − b ∕ ζ r ( ζ , y ) ( ζ m + 1 ) ρ d ζ d y − ∫ 0 π Θ ( x , t ; y ) f ( y ) d y .

Substituting from Eq. (5.18) into Eq. (5.20) gives

(5.21) φ ( x , t ) = 2 π ∑ k = 1 ∞ ( − 1 ) k ∫ 0 π ∫ 0 t e k 2 ζ − b ∕ ζ r ( ζ , y ) ( ζ m + 1 ) ρ sin k y d ζ d y e − k 2 t sin k x + 2 π ∑ k = 1 ∞ ( − 1 ) k ∫ 0 π sin k y f ( y ) d y e − k 2 t sin k x ,

the solution of the problem (5.11). In particular, if

r ( t , x ) = e − t sin 3 x , x ∈ [ 0 , π ] , t ⩾ 0 ,

we find

φ ( x , t ) = 2 π ∑ k = 1 ∞ ( − 1 ) k × ∫ 0 π ∫ 0 t e ( k 2 − 1 ) ζ − b ∕ ζ ( ζ m + 1 ) ρ sin 3 y sin k y d ζ d y e − k 2 t sin k x + 2 π ∑ k = 1 ∞ ( − 1 ) k ∫ 0 π sin k y f ( y ) d y e − k 2 t sin k x .

Since

∫ 0 π sin 3 y sin k y d y = π 2 , k = 3 0 , k ≠ 3 ,

then

(5.22) φ ( x , t ) = − ∫ 0 t e 8 ζ − b ∕ ζ ( ζ m + 1 ) ρ d ζ e − 9 t sin k x + 2 π ∑ k = 1 ∞ ( − 1 ) k ∫ 0 π sin k y f ( y ) d y e − k 2 t sin k x .

Remark 4

Formula (5.22) recovers the solution to problem (5.11), when b = 0 and r ( t , x ) = e − t sin 3 x , obtained via the Srivastava–Luo–Raina integral transform method in [1].

Remark 5

Formula (5.22) recovers the solution to problem (5.11), when ρ = 0 , b = 0 , and r ( t , x ) = e − t sin 3 x , obtained via the eigenfunction expansion method in [46].

6 Conclusion

In this work, we defined a new extended Laplace integral transform. With the right parameter selection, the suggested extended Laplace transform can, in its specific circumstances, generalize the well-known integral transforms such as the Laplace transform, the natural transform, and the Sumudu transform. The existence, linearity, scaling, and elimination properties of the suggested transform (Eq. (2.1)) are explored. The extended Laplace transform (Eq. (2.1)), its inversion formula, and a Parseval-type identity are used to visualize differential derivatives. Moreover, the transform (2.1) is usefully connected to several well-known integral transforms, including the Borel–Dzrbashjan transform, the H-transform, the Mellin transform, the natural transform, and the Srivastava-Luo-Raina transform. Finally, we provided illustrative examples to show the effectiveness of the extended Laplace transform as a powerful mathematical instrument for addressing a variety of problems. In addition, the study contributes to the field of mathematics by introducing a new integral transform that can be used to solve a wide range of partial differential equations and fractional differential equations as discussed in recent works [47–49].

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through a Large Group research project under grant number RGP2/366/44.

Author contributions: Methodology, Mohamed Abdalla.; investigation, Yahya Almalki., Mohamed Akel., and Mohamed Abdalla.; writing – original draft, Yahya Almalki., Mohamed Akel., and Mohamed Abdalla.; writing – review & editing, Mohamed Akel., and Mohamed Abdalla.; supervision, Mohamed Akel. All authors have read and agreed to the published version of the manuscript.
Conflict of interest: The authors declare that they have no conflict of interest.
Data availability statement: Not applicable.

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Received: 2023-09-05

Revised: 2023-10-22

Accepted: 2023-11-02

Published Online: 2024-03-02

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

Articles in the same Issue

https://doi.org/10.1515/nleng-2022-0353

Keywords for this article

Laplace transform; Sumudu transform; natural transform; H-function; H-transform; inversion formula; Parseval-type identity; Borel–Džrbashjan transform

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