Deformable Laplace transform and its applications

Priyanka Ahuja; Amit Ujlayan; Dinkar Sharma; Hari Pratap

doi:10.1515/nleng-2022-0278

Open Access Published by De Gruyter March 17, 2023

Deformable Laplace transform and its applications

Priyanka Ahuja , Amit Ujlayan , Dinkar Sharma and Hari Pratap

From the journal Nonlinear Engineering

https://doi.org/10.1515/nleng-2022-0278

Abstract

Recently, the deformable derivative and its properties have been introduced. In this work, we have investigated the concept of deformable Laplace transform (DLT) in more detail. Furthermore, some classical properties of the DLT are also included. The Heaviside expansion formula and convolution theorem for deformable inverse Laplace transform are also discussed. Furthermore, some illustrative numerical examples are also discussed to validate the applicability of the proposed DLT and finally conclude the theory.

Keywords: deformable derivative; deformable exponential function; deformable Laplace transform

MSC 2010: 26A33; 34G20; 34A08

1 Introduction

As we are aware with the fact that the Laplace transform is a particular case of the integral transform, and it is proposed by the famous French mathematician (1749–1827) Pierre Simon De Laplace [1,2]. It plays a very crucial role to convert one signal into another using a set of laws or equations. A very natural question arises in the mind, why should one learn Laplace transform technique, when the other method is available. The answer is quite simple. The Laplace transform method reduces the ordinary differential equation (ODE) into the algebraic equation, and after that, we can easily solve the problem of ODE [3]. More generally, the theory of Laplace transform has become an important part of modern physics. It also plays a significant role in various fields. We can see the applications of the Laplace transformation in electronics engineering, signal and system, digital signal processing, and control system. Moreover, in studying the dynamic control system [4–7].

This article is organized as follows. In Section 2, we give brief definition of deformable derivative and its basic properties [8–10]. Section 3 studies the definition of deformable integral corresponding to deformable derivative and its properties. Section 4 deals with the proof of the main definition of deformable Laplace transform (DLT) and its existence as well as basic properties of DLT. Section 5 contains the deformable inverse Laplace transform (DILT), and its basic properties. Finally, Section 6, shows the proficiency of the DLT for computing the solution of the IVP.

2 Basic definition of deformable derivative and its properties

Definition 2.1

Given a real-valued function h ( t ) defined on interval ( a , b ) , then for arbitrary order α deformable derivative of a function h is defined as:

(1) lim ε → 0 ( 1 + ε β ) h ( t + ε α ) − h ( t ) ε ,

where α + β = 1 . If the aforementioned limit exists for all, 0 ⩽ α ⩽ 1 , then, deformable derivative of the real-valued function h ( t ) is denoted by the symbol D α h ( t ) .

Remark 2.2

It is clear that the aforementioned definition is suitable with α = 0 and 1. When α = 0 , D 0 h ( t ) = h ( t ) , that is, the function itself and when α = 1 , D h ( t ) = h ′ ( t ) , that is, ordinary derivative of h . The deformable derivative can be considered as α -derivative as well.

The following theorems are useful results of the aforementioned definition.

Theorem 2.3

Let h be α -differentiable at a point t 0 for some α . Then h is continuous at t 0 .

It can be easily shown that operator D α has the following properties:

Theorem 2.4

Let h 1 and h 2 be α -differentiable at t . Then

Linearity: D α ( a h 1 + b h 2 ) = a D α h 1 + b D α h 2 , for all a , b ∈ R .
Commutativity: D α 1 ⋅ D α 2 = D α 2 ⋅ D α 1 .
D α ( s ) = β s , for all constant functions h ( t ) = s .
D α ( h 1 ⋅ h 2 ) = ( D α h 1 ) ⋅ h 2 + α h 1 ⋅ D h 2 , hence D α does not follow the Leibniz rule.
D α h ( t ) = β h ( t ) + α D h ( t ) .

Following are the arbitrary order deformable derivatives of well-known elementary functions:

D α ( t k ) = β t k + k α t k − 1 , k ∈ R .
D α ( e t ) = β e t + α e t .
D α ( sin ω t ) = β sin ω t + α ω cos ω t .

3 Deformable integral

This section introduced deformable integral, the inverse operator for deformable derivative. Besides discussing some basic properties of this deformable fractional integral, we list out deformable fractional integral of some elementary functions. Also, we introduced deformable form of exponential function [10].

Definition 3.1

Let h be a continuous function defined on [ a , b ] . We define α -fractional integral of h , denoted by I a α h , by the integral

(2) I a α h ( t ) = ∫ a t e − β α ( t − x ) h ( x ) d α x ,

where α + β = 1 , α ∈ ( 0 , 1 ] , and d α x = 1 α d x .

Next theorem explains some basic properties of this fractional integral.

Theorem 3.2

The operator I a α possesses the following properties:

Linearity: I a α ( b h 1 + c h 2 ) = b I a α h 1 + c I a α h 2 .
Commutativity: I a α 1 I a α 2 = I a α 2 I a α 1 , where α i + β i = 1 , i = 1 , 2 .

Following are the fractional integral of some well-known elementary functions:

Proposition 3.3

I a α sin t = 1 α 2 + β 2 β sin t − α cos t + e β α ( a − t ) ( α cos a − β sin a ) .
I a α e t = e t − e ( a − β t ) α .
I a α λ = λ β 1 − e β α ( a − t ) , where λ is a constant.
I 0 α t m = 1 β ∑ r = 0 m ( − 1 ) r ( m ) ! ( m − r ) ! α β r t m − r + ( − 1 ) m + 1 m ! α β m e − β α t .

Furhermore, the deformable exponential function is defined as follows:

Definition 3.4

(Deformable exponential function) For some point s , t ∈ R with s ≤ t , the exponential function with respect to D α in (1) is defined as:

(3) e α ( t , s ) = e − ∫ s t β d α u ,

where α + β = 1 , α ∈ ( 0 , 1 ] , and d α u = 1 α d u .

4 DLT

In this section, we introduce DLT in very natural way. We also discuss existence theorem of DLT.

Definition 4.1

Let f : [ 0 , ∞ ) ⟶ R . For α ∈ ( 0 , 1 ] , we define DLT of order α , denoted by L α f , by the integral:

(4) L α f ( t ) = F α ( p ) = ∫ 0 ∞ e − s t f ( t ) e α ( t , 0 ) d α t ,

where α + β = 1 , p = s + β α , and d α t = 1 α d t .

Theorem 4.2

A continuous function f is of exponential order a ∗ , there exists a positive real number C s.t. ∣ f ( t ) ∣ ≤ C e a ∗ t , where a ∗ , t ≥ 0 , then DLT of f exists for s > a ∗ .

Proof

∣ L α f ( t ) ∣ = ∫ 0 ∞ e − s t f ( t ) e α ( t , 0 ) d α t ≤ ∫ 0 ∞ ∣ e − s t f ( t ) e α ( t , 0 ) ∣ d α t ≤ ∫ 0 ∞ e − s t ∣ f ( t ) ∣ e α ( t , 0 ) d α t ≤ ∫ 0 ∞ e − s t C e a ∗ t e α ( t , 0 ) d α t , s > a ∗ = C α 1 p − a ∗ .

Remark 4.3

It should however be kept in mind that the aforementioned condition is sufficient but not necessary. For example, L α 1 t exists though 1 t is discontinuous at t = 0 .

Now, we list out DLT of some certain functions in the following preposition:

Proposition 4.4

L α 1 = 1 α 1 p .
L α t m = 1 α m ! p m + 1 , p > 0 , m = 0 , 1 , 2 , …
L α e a ∗ t = 1 α 1 p − a ∗ , p > a ∗ .
L α sin a ∗ t = 1 α a ∗ p 2 + a ∗ 2 .
L α sinh a ∗ t = 1 α a ∗ p 2 − a ∗ 2 .
L α cos a ∗ t = 1 α p p 2 + a ∗ 2 .
L α cosh a ∗ t = 1 α p p 2 − a ∗ 2 .

4.1 Basic properties of DLT

Apart from discussing fundamental properties of DLT like linearity and change of scale property, the section deals with fundamental theorems: First shifting and second shifting property.

Theorem 4.5

The operator L α possesses the following properties:

Linearity: L α ( a ∗ f + b ∗ g ) = a ∗ L α f + b ∗ L α g .
First translation or shifting property: If L α g ( t ) = G α ( p ) , then
L α ( e a t g ( t ) ) = G α ( p − a ) .
Second translation or shifting property:
L α g ( t ) = G α ( p ) and F ( t ) = g ( t − a ) , t > a 0 , t < a
then, L α F ( t ) = e − a p G α ( p ) .
Change of scale property: If L α g ( t ) = G α ( p ) , then
L α g ( a t ) = 1 a G α p a .
DLT of successive derivatives: If L α g ( t ) = G α ( p ) , then
L α ( D α D α ⋯ D α ︸ ) n times g ( t ) = ( α p + β ) n G α ( p ) − ( α p + β ) n − 1 g ( 0 ) − ( α p + β ) n − 2 D α g ( 0 ) − ( α p + β ) n − 3 D α D α g ( 0 ) − ⋯ − ( D α D α ⋯ D α ︸ ) n − 1 times g ( 0 ) .
DLT of t g ( t ) :
If L α g ( t ) = G α ( p ) .
then L α ( t n g ( t ) ) = ( − 1 ) n d n d p n G α ( p ) .
DLT of g ( t ) t :
If L α ( g ( t ) ) = G α ( p ) ,
then L α g t = ∫ p ∞ G α ( p ) d p .

Proof

Linearity is obvious from definition, parts (ii), (iv), (vii) can be established easily. For part (iii), we have

L α F ( t ) = ∫ 0 ∞ e − s t F ( t ) e α ( t , 0 ) d α t = 1 α ∫ 0 a e − p t F ( t ) d t + ∫ a ∞ e − p t F ( t ) d t = 1 α 0 + ∫ a ∞ e − p t g ( t − a ) d t = 1 α ∫ a ∞ e − p t g ( t − a ) d t = e − a p 1 α ∫ 0 ∞ e − p u g ( u ) d u = e − a p G α ( p ) .

To prove part (v), we have

L α ( D α g ( t ) ) = ∫ 0 ∞ e − s t D α ( g ( t ) ) e α ( t , 0 ) d α t = 1 α ∫ 0 ∞ e − s t [ β g + α g ] e α ( t , 0 ) d t .

Therefore,

(5) L α ( D α g ( t ) ) = ( α p + β ) G α ( p ) − g ( 0 ) .

Now replacing g by D α g and D α g by D α D α g in (5), we obtain

(6) L α ( D α D α g ( t ) ) = ( α p + β ) 2 G α ( p ) − ( α p + β ) g ( 0 ) − D α g ( 0 ) .

With the same process we have

(7) L α ( D α D α ⋯ D α ︸ ) n times g ( t ) = ( α p + β ) n G α ( p ) − ( α p + β ) n − 1 g ( 0 ) − ( α p + β ) n − 2 D α g ( 0 ) − ( α p + β ) n − 3 D α D α g ( 0 ) − ⋯ − ( D α D α ⋯ D α ︸ ) n − 1 times g ( 0 ) .

For part (vi), we have

L α ( g ( t ) ) = G α ( p ) = 1 α ∫ 0 ∞ e − p t g ( t ) d t = d d p G α ( p ) = d d p 1 α ∫ 0 ∞ e − p t g ( t ) d t = 1 α ∫ 0 ∞ ∂ ∂ p ( e − p t ) g ( t ) d t = 1 α ∫ 0 ∞ e − p t ( − t g ( t ) ) d t = L α ( − t g ( t ) ) ⟶ L α ( t g ( t ) ) = ( − 1 ) 1 d d p G α ( p ) .

Similarly,

L α ( t 2 g ( t ) ) = ( − 1 ) 2 d 2 d p 2 ( G α ( p ) ) , L α ( t 3 g ( t ) ) = ( − 1 ) 3 d 3 d p 3 ( G α ( p ) ) , L α ( t n g ( t ) ) = ( − 1 ) n d n d p n ( G α ( p ) ) .

5 DILT

This section explains about DILT and further Heaviside expansion formula and convolution theorem for DILT are also discussed. DILT is a process for determining the function which generates given DLT. If G α ( p ) is the DLT of g ( t ) , then g ( t ) is called DILT of G α ( p ) . The operator for DILT is ( L α ) − 1 .

Theorem 5.1

(Heaviside expansion formula for DILT) If G 1 ( p ) and G 2 ( p ) are two polynomials and G 2 ( p ) has degree less than degree of G 1 ( p ) , then

( L α ) − 1 G 2 ( p ) G 1 ( p ) = α ∑ r = 1 m G 2 ( a r ) G 1 ′ ( a r ) e a r t ,

where G 1 ( p ) has distinct zeroes a 1 , a 2 , … , a m .

Proof

We have

G 2 ( p ) G 1 ( p ) = G 2 ( p ) ( p − a 1 ) . ( p − a 2 ) … ( p − a m ) = A 1 ( p − a 1 ) + A 2 ( p − a 2 ) + ⋯ + A m ( p − a m ) .

Multiplying both sides by ( p − a r ) and taking the limit as p ⟶ a r

A r = lim p ⟶ a r G 2 ( p ) ( p − a r ) G 1 ( S ) = G 2 ( a r ) G 1 ′ ( a r ) ,

i.e.,

G 2 ( p ) G 1 ( p ) = G 2 ( a 1 ) G 1 ′ ( a 1 ) 1 p − a 1 + G 2 ( a 2 ) G 1 ′ ( a 2 ) 1 p − a 2 + ⋯ + G 2 ( a m ) G 1 ′ ( a m ) 1 p − a m

Taking DILT on both sides, we have

( L α ) − 1 G 2 ( p ) G 1 ( p ) = α ∑ r = 1 m G 2 ( a r ) G 1 ′ ( a r ) e a r t .

Theorem 5.2

(Convolution theorem) If

( L α ) − 1 ( G 1 α ( p ) ) = g 1 ( t )

and

( L α ) − 1 ( G 2 α ( p ) ) = g 2 ( t ) ,

then

(8) ( L α ) − 1 ( G 1 α ( p ) G 2 α ( p ) ) = g 1 ∗ g 2 = ∫ 0 t g 1 ( x ) g 2 ( t − x ) d α x .

6 Applications of DLT to arbitrary order differential equations

In first example, we discuss method of solving homogeneous linear differential equation of arbitrary order, whereas in second and third examples non-homogeneous linear arbitrary order differential equations are solved.

Example 6.1

A homogeneous deformable fractional differential equation:

D α y ( t ) − λ y ( t ) = 0 ,

with y ( 0 ) = y 0 and λ is a constant.

Solution: Take the DLT on both the sides using equation (7),

( α p + β ) Y α ( p ) − y ( 0 ) − λ Y α ( p ) = 0 ⇒ Y α ( p ) = y 0 α p + β − λ .

Using DILT, general solution is given by:

y ( t ) = y 0 e λ − β α t .

Example 6.2

A non-homogeneous deformable fractional equation:

D α y ( t ) = sin t ,

with y ( 0 ) = 0 .

Solution: Take the DLT on both the sides, then we have the expression

( α p + β ) Y α ( p ) = 1 α ( p 2 + 1 ) ⇒ Y α ( p ) = 1 α ( p 2 + 1 ) ( α p + β ) ,

and the general solution is given by:

y ( t ) = β α 2 + β 2 sin t − α α 2 + β 2 cos t + α α 2 + β 2 e − β α t .

Example 6.3

Consider the problem

( D 1 4 D 1 4 ) u ( t ) = e 3 t ; u ( 0 ) = 1 , D α u ( 0 ) = 0 .

Taking DLT both sides by using Eq. (6) we obtain

( α p + β ) L α ( u ) − ( α p + β ) u ( 0 ) − D α u ( 0 ) = 4 p − 3

L α ( u ) = 4 ( p − 3 ) ( α p + β ) 2 + 1 α p + β ,

which implies

u = ( L α ) − 1 4 ( p − 3 ) ( α p + β ) 2 + 1 α p + β .

Here α = 1 4 and β = 3 4 , therefore

(9) u = ( L 1 4 ) − 1 64 ( p − 3 ) ( p + 3 ) 2 + 4 p + 3 .

Using convolution theorem for DILT

( L 1 4 ) − 1 64 ( p − 3 ) ( p + 3 ) 2 = 4 ∫ 0 t x e 3 t − 6 x d α x = 16 e 3 t ∫ 0 t x e − 6 x d x = − 8 3 t e − 3 t − 4 9 e − 3 t + 4 9 e 3 t

and

( L 1 4 ) − 1 1 ( p − 3 ) = 1 4 e − 3 t .

Therefore, complete solution is

u ( t ) = 5 9 e − 3 t − 8 3 t e − 3 t + 4 9 e 3 t .

Remark

It should be noted that all the aforementioned solutions coincide with α = 1 .

7 Conclusion

This work is motivated by the concept of classical Laplace transform. We tried to produce Laplace transform via the deformable integral approach with its corresponding properties and observed that these all coincide with classical Laplace transform for α = 1 . Moreover, we end up the article with some important questions that are yet to be answered. What is geometric interpretation and physical significance of the DLT? Is there any similarity between the classical fractional Laplace transform corresponding to various fractional derivative and DLT. The authors hope that the work would be meaningful to other researchers in the future.

Acknowledgments

The Authors sincerely thanks to the Editor and anonymous reviewers for their careful reviews and useful suggestions on improving the presentation of the article.

Funding information: There is no funding to declare for this research article.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission. Priyanka Ahuja: formal analysis and validation of the concept and initial draft. Amit Ujlayan: actualization, validation, methodology, analysis, and investigation. Dinkar Sharma: actualization, methodology, formal analysis, validation and supervision of the original draft and editing. Hari Pratap: complete investigation of analysis and initial draft. All authors read and approved the final version.
Conflict of interest: The authors declare that they have no known conflict of interest that could appear to influence the work presented in this article.
Data availability statement: No data were used to support this study.

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Received: 2022-09-04

Revised: 2023-01-11

Accepted: 2023-01-16

Published Online: 2023-03-17

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

Deformable Laplace transform and its applications

Abstract

1 Introduction

2 Basic definition of deformable derivative and its properties

Definition 2.1

Remark 2.2

Theorem 2.3

Theorem 2.4

3 Deformable integral

Definition 3.1

Theorem 3.2

Proposition 3.3

Definition 3.4

4 DLT

Definition 4.1

Theorem 4.2

Proof

Remark 4.3

Proposition 4.4

4.1 Basic properties of DLT

Theorem 4.5

Proof

5 DILT

Theorem 5.1

Proof

Theorem 5.2

6 Applications of DLT to arbitrary order differential equations

Example 6.1

Example 6.2

Example 6.3

Remark

7 Conclusion

Acknowledgments

References

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