Solving Some Partial Differential Equations by Using Double Laplace Transform in the Sense of Nonconformable Fractional Calculus

Injrou, Sami; Hatem, Iyad

doi:https://doi.org/10.1155/2022/5326132

Mathematical Problems in Engineering

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Abstract Introduction Examples and Applications Discussion Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 5326132 | https://doi.org/10.1155/2022/5326132

Solving Some Partial Differential Equations by Using Double Laplace Transform in the Sense of Nonconformable Fractional Calculus

Sami Injrou¹and Iyad Hatem²

Academic Editor: D. L. Suthar

Received19 Sept 2022

Revised24 Nov 2022

Accepted03 Dec 2022

Published30 Dec 2022

Abstract

In this paper, we introduce the non-conformable double Laplace transform. Its properties are studied, and it is applied to solve some fractional PDEs involving the nonconformable fractional derivative. Graphical representations of the obtained solutions are shown in figures. The study shows that this transform is effective and easy to apply to create an exact solution for types of fractional PDEs.

1. Introduction

Throughout the decades, mathematics has played an influential role in the developed civilization. Through mathematical modelling, it has allowed to describe and predict phenomena in real world. From this viewpoint, it is essential to assure the importance of calculus to study many of the laws of nature.

On September 30^th, 1695, a generalization of calculus was born in a letter between L’Hopltal and Leibniz in which a question got raised about the meaning of taking a fractional derivative such as [1]. This generalization is called fractional calculus. In the last and present centuries, the importance of the fractional calculus has been growing because of its deep applications in all related fields of science and engineering [2–5].

Actually, many definitions of fractional derivative and fractional integral have been proposed. The most popular ones are Riemann–Liouville, Caputo, Grunwald–Letnikov, Hadamard, Erdely, Kober, Marchaud, and Riesz [6–9]. Most of these fractional derivatives do not satisfy the classical formulas of derivatives such as product, quotient, and chain rules except Caputo [10, 11].

Recently, Khalil et al. [12] introduced an extension of the ordinary limit definition for the derivative of a function, namely, the conformable derivative. More recently, Martinez proposed a new nonconformable local derivative in [13]. One can see that the last two definitions satisfy the classical properties mentioned above. Many researchers solved nonlinear fractional partial differential equations in the sense of conformable derivative and Caputo derivative [14–31]. There is an important study [32], where the authors treat the price adjustment equation in many senses of fractional derivatives, such as truncated M-derivative including the Mittag–Leffer function, beta-derivative, and conformable derivative defined in the form of limit for -differentiable functions.

In [13], Martinez introduced the nonconformable Laplace transform local fractional derivative, and they proved its existence beside main properties. Ozarslan et al. in [33] presented Atangana–Baleanu fractional derivative in the Caputo sense with the ^th order and its Laplace transform, and they studied that the wind influenced projectile motion equations involving this sense of derivative. In [34], Ozkan and Kurt introduced the conformable double Laplace transform and used it to solve some fractional partial differential equations that represent many physical and engineering models.

In this manuscript, we establish the nonconformable double Laplace transform and apply this new definition to solve a nonconformable fractional heat, wave, and telegraph equations with sense of nonconformable fractional derivative.

2. Preliminary

In this section, we present the definition of nonconformable fractional derivative and some important definitions.

Definition 1. (see [13]). Given a function , then the nonconformable fractional derivative of order of at is defined byIn addition, if the nonconformable fractional derivative of of order exists, then is N-differentiable.

Definition 2. (see [13]). Let and be a real number, then the fractional exponential is defined in the following way:

Definition 3. (see [13]). Let and , then we can say that a function is fractional integrable on , if the integralexists and is finite.
In the next section, we present the definitions of the nonconformable double and single Laplace transform of a function with tow variables. We prove some properties. We claim that the results presented here are new in its domain.

3. Nonconformable Double Laplace Transform

Definition 4. Let be a piecewise continuous function on of generalized exponential order; that is, there exist constants and such that for sufficiently large . The nonconformable double Laplace transform of is defined bywhere , , and the integrals are in sense of nonconformable fractional integral with respect to and .
Now, we define the single nonconformable Laplace transform of a function with two variables.

Definition 5. Let be a piecewise continuous function on of the generalized exponential order. The nonconformable Laplace transform with respect to of is defined byand the nonconformable Laplace transform with respect to of is defined bywhere the integrals are in nonconformable sense.

3.1. Some Properties of Nonconformable Double Laplace Transform

Here, we consider some of the properties and theorems of the nonconformable double Laplace Transform with their verification.

If we assume that the function provides the sufficient conditions [35] and the order of transformation can be changed, then

Therefore, we get

Theorem 1. Let and be two functions which have the nonconformable double Laplace transform. Then, we get(1), where and are real constants(2), where and are any real constants, and (3), where and are two nonzero real numbers(4)For ,

Proof. (1)We obtain the proof from the definition directly.(2)By using nonconformable Laplace transform definition, one can find but Therefore,(3)Taking into account the convergence properties of improper integral, one can change the order of the operation of differentiation and integration. Therefore, one can differentiate with respect to , under the sign of integral. Thus, where we repeated differentiation with respect to , , m and n times, respectively.

Lemma 1. Under the assumptions of the definition of the nonconformable double Laplace transform, we havewhere and are the -th and -th order nonconformable fractional partial derivatives, respectively, and are the mixed -th and -th order nonconformable fractional partial derivatives.

Proof. (1). Using the definition of non-comformable double Laplace transform and the Proposition 2.3 in [13], we get(2)In the same manner, one can prove(3)Setting and using (13), we havewhereandNow, substituting equations (18) and (19) into equation (17) yieldsAnalogously, one can prove the following theorem.

Theorem 2. Let and such that , where . Also, let and for are N-transformable; then, we havewhere and are m, n times nonconformable fractional derivative of function with order and respectively, and are the mixed -th and -th order nonconformable fractional partial derivatives of function .

4. Examples and Applications

In this section, we use the proposed transform to solve some fractional PDEs in sense of nonconformable fractional derivative arising from several physical and engineering problems.

4.1. Homogeneous Space-Time Nonconformable Fractional Heat Equation

Consider the following homogeneous space-time nonconformable fractional heat equation in one dimension:subject to the initial and boundary conditions

Applying nonconformable double Laplace transform (4) to equation (22), we get

Then, applying the nonconformable single Laplace transform to the conditions (23)–(25) by using Theorem 1.2 in [13], we get

Substituting (27) into (26), one can obtain

Therefore, the solution of problem (22)–(25) is

4.2. Homogeneous Space-Time Nonconformable Fractional Wave Equation

Consider the following homogeneous space-time nonconformable fractional wave equation in one dimension:subject to the initial and boundary conditions:

Applying nonconformable double Laplace transform (4) to equation (30), we get

Then, applying the nonconformable single Laplace transform to the conditions (31)–(34) by using Theorem 1.2 in [13].

Substituting (36) into (35), one can get

Hence, the solution of problem (30)–(34) is

4.3. Nonhomogeneous Space-Time Nonconformable Fractional Wave Equation

Consider the following nonhomogeneous space-time nonconformable fractional wave equation in one dimension:subject to the initial and boundary conditions

Applying nonconformable double Laplace transform (4) to equation (39), we get

Then, applying the nonconformable single Laplace transform to the conditions (40)–(43) by using Theorem 1.2 in [13], we get

Substituting (45) into (44), one can obtain

Thus, the solution of problem (39)–(43) is

4.4. Nonhomogeneous Space-Time Nonconformable Fractional Telegraph Equation

Consider the following nonhomogeneous space-time nonconformable fractional telegraph equation in one dimension:subject to the initial and boundary conditions

Applying nonconformable double Laplace transform (4) to equation (48), we get

Then, applying the nonconformable single Laplace transform to the conditions (49)–(52) by using Theorem 1.2 in [13], we get

Substituting (54) into (53), one can obtain

Therefore, the solution of problem (48)–(52) is

5. Discussion

In this section, we illustrate the obtained solutions in the four examples for different values of and in 2D and 3D graphs by using Maple 13 software.

In Figure 1(a), the solutions of problem (22)–(25) obtained from the nonconformable double Laplace transform method, are shown for different values of and when and , Figure 1(b) shows the solution of problem (22)–(25) in x − t plane when , and .

(a)

(b)

Figure 2(a) illustrates the nonconformable double Laplace transform solutions of problem (30)–(34) with different values of and when and . Figure 2(b) shows the 3D graph of the solution, where , and .

(a)

(b)

With different values of and , the nonconformable double Laplace transform solutions of problem (39)–(43) are shown in Figure 3(a) with and . In Figure 3(b), the solution is illustrated in the x − t plane, where , and .

(a)

(b)

The solutions of problem (48)–(52) are shown in Figure 4(a) for different values of and when and , and the 3D graph of the solution is shown in Figure 4(b) where , and .

(a)

(b)

In Figures 1–4, one can observe that the solutions in the case of nonconformable fractional derivative approach values of the classical case, i.e., , whenever and approach one.

In Figures 2 and 3, one can see that the wavelength increases as and approach zero. Thus, the number of crests and troughs increases as and approach one, the wave amplitude increases in Figure 1 and decreases in Figures 2 and 3 as and approach one.

6. Conclusion

In this work, we introduce the nonconformable double Laplace transform with proof of its main properties. Then, we use the transform to solve selected fractional PDEs in sense of nonconformable derivative, such as homogeneous space-time nonconformable fractional heat equation, homogeneous and nonhomogeneous space-time nonconformable fractional wave equations, and nonhomogeneous space-time nonconformable fractional telegraph equation. In addition, 3D graphical representations are offered for obtained solutions with illustrations for different values of and with fixed. The 2D graphs show that the obtained solutions, by using the nonconformable double Laplace transform, are close to that of classical derivatives as and with fixed. We observe that the wavelength increases and the amplitude decreases in the case of the homogeneous space-time nonconformable fractional wave equation as and with fixed. It increases in the last two cases as and approaches zero with fixed. We may conclude that this transform can be a very powerful technique to solve many fractional PDEs involving nonconformable derivative arising in physics, chemistry, and engineering.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by Tishreen University and Manara University.

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Copyright © 2022 Sami Injrou and Iyad Hatem. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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