The Solution of Two-Dimensional Coupled Burgers’ Equation by -Double Laplace Transform

Abstract

The two-dimensional coupled Burgers’ equation, a foundational partial differential equation, boasts widespread relevance across numerous scientific domains. Attaining precise solutions to this equation stands as a pivotal endeavor, fostering a comprehensive understanding of both physical phenomena and mathematical models. In this article, we underscore the paramount significance of the -double Laplace transform, a transformative mathematical tool. Leveraging this innovative technique, we furnish dependable and exact solutions, addressing both homogeneous and nonhomogeneous variants of the coupled Burgers’ equations. This approach not only delivers reliability but also serves as an invaluable instrument for delving deeper into the equation’s intricate behavior and its profound implications across diverse disciplinary landscapes.

1. Introduction

Burgers’ equation is a fundamental partial differential equation and convection-diffusion equation that finds applications in various fields of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow. Solving this equation holds great significance in the context of mathematical models and understanding physical phenomena. Over the years, numerous scientists and mathematicians have proposed analytical solutions for the one-dimensional coupled Burgers’ equation. Various analytical methods have been developed to tackle Burger’s equation, as evidenced in [1–3].

In recent years, a substantial endeavor has been directed toward employing the Laplace decomposition method (LDM) and its modified variants to investigate physical model equations, as discussed in detail in [4]. Furthermore, the scientific community has witnessed remarkable advancements in analytical approaches for tackling differential equations. Researchers have explored a myriad of techniques in this field, including the modified Adomian-Rach decomposition method (MDM) [5], which has been effectively utilized to obtain solutions for scenarios such as the isentropic flow of an inviscid gas model (IFIG) through the modified decomposition method (MDM), elaborated in [6]. The Adomian decomposition method (ADM) [7] has proven valuable in addressing systems of conservation laws featuring a mixed hyperbolic-elliptic character. Additionally, the reduced differential transform method (RDTM) [8] has found practical applications.

In a separate study, as documented in [9], an author enhanced the Laplace decomposition method to derive approximate analytical solutions for both linear and nonlinear differential equations and systems. Furthermore, researchers have ventured into exploring the application of the double Laplace-Sumudu transform [10] to tackle a diverse array of partial differential equations.

To delve into specifics, the Adomian decomposition method has been effectively employed to obtain exact solutions for Burgers’ equation [11]. A modified expanded tanh-function method was introduced to achieve exact solutions, as highlighted in references [12, 13]. Additionally, the homotopy perturbation method has been suggested for addressing the nonlinear Burgers’ equation [14]. An innovative approach combining the Laplace transform and the new homotopy perturbation method (NHPM) was proposed to derive closed-form solutions for coupled viscous Burgers’ equations [15]. This approach holds significant implications in understanding polydispersity and its relationship with gravitational effects [16].

The study of coupled Burgers’ equations is of paramount importance as it elucidates the precipitation of polydispersity in the presence of gravity effects [16]. Furthermore, the solution of time-fractional two-mode coupled Burgers’ equations has been explored [17, 18].

In this context, the transform, initially introduced in [19] and later applied to solve nonlinear dynamical models with noninteger order [20], offers a promising avenue for tackling complex equations.

The core focus of this article revolves around harnessing the formidable -double Laplace transform to attain precise solutions of exceptional reliability. Our primary aim is to unravel exact solutions for both homogeneous and nonhomogeneous instances of the two-dimensional coupled Burgers’ equations. This transformative mathematical approach not only promises exactitude but also offers an unprecedented opportunity to gain profound insights into the intricate behavior of these equations. Furthermore, it opens up new horizons for their application across a multitude of scientific domains.

2. An Introduction to -Laplace Transforms: Fundamental Concepts

In this study, we employ the transform and Laplace transform to assist us in solving a set of partial differential equations.

Definition 1 (see [21]). Consider an integrable function defined for all . The generalized integral transform denoted as of the function is defined as follows: where belongs to the complex numbers and is an integer represented by .

To provide a practical example, if we substitute and into Eq. (1), we can express the result as

This showcases that the transform serves as a comprehensive extension of not only the Laplace transform but also various other transforms. Its scope encompasses a broader and more fundamental range compared to existing transforms. For a deeper insight into this concept, we recommend referring to [21].

Definition 2. The Laplace transform of the function is defined as where belongs to the complex numbers .

Definition 3. The -double Laplace transform of the function is a well-behaved and integrable function defined for all nonnegative values of , , and . This transform is denoted as with expressed as Here, the notation signifies the -double Laplace transform, and , , and belong to the complex number set .

Definition 4. The inverse -double Laplace transform of is expressed as

Example 1. transform of the is given by where indicate to -double Laplace transform; consequently,

Example 2. The transform of is determined by where is nonnegative integer. If and , then can be derived from the definition of -double Laplace transform; we have By applying the definition of the double Laplace transform to the integral within the first bracket of Eq. (10), we arrive at the following result: In Eq. (11) put we get where the gamma function of is defined through the convergent integral expression.

Example 3. -Laplace transform for the function is expressed as where represents the dimensional Heaviside function and denotes a tensor product (as described in [22]).
By introducing the substitutions and , the integral transforms into the following form: where is Euler’s constant.

2.1. Existence Criteria for the -Double Laplace Transform

In the forthcoming theorem, we establish the prerequisites for the existence of the -Laplace transform of . Consider which exhibits exponential behavior with positive orders , , and over the domain . We say that belongs to this class if there exists a nonnegative constant such that, for all , , and , the inequality holds. In this scenario, we can express as as , , and . Equivalently, we have for , , and . We refer to the function as having exponential order as , , and . It is evident that it does not grow at a rate exceeding .

Theorem 5. If is a continous function in every bounded interval , , and and of exponential order , then the transform of exists for all , , and provided , , and .

Proof. Combining Eq. (17) and Eq. (19) within Eq. (4) yields for , , and .

Lemma 6. If is a piecewise continuous function defined on and has an exponential order at infinity with for and , where , , and are constant, then for any real number , , and , we have where and is the Heaviside function, defined by

Proof. Utilizing the definition of the transform, Let , , and ; Eq. (23) becomes

Theorem 7. If is a periodic function with periods , , and to satisfy this periodicity condition, we must have In that case, the -double Laplace transform of can be expressed as follows:

Proof. By utilizing the definition of the -double Laplace transform, Applying the property of improper integrals to Eq. (27), Substituting , , and in the second part of the integral in Eq. (28), we obtain Equation (29) can be rewritten as follows: The second integral in Eq. (30), given the definition of the -double Laplace transform, leads to Hence,

Theorem 8 (convolution theorem). Let and exist and , and ; then, where where the symbol denotes the double convolution with respect to and .

Proof. When we apply the definition of the -double Laplace transform, we obtain By setting , , and and employing the appropriate expansion for the upper bounds of the integrals as , , and , Eq. (35) can be expressed as The functions and are both zero for , , and . Therefore, considering the lower limits of integration, we have It is evident that

Theorem 9. If the -double Laplace transform of the function is denoted as , then the -double Laplace transform of and is as follows:
For : For :

Proof. Substituting into Eq. (39), we get Now, let us calculate the integral inside the bracket: Substituting this result back into Eq. (96), we obtain Then, At , in a similar manner, one can readily observe that Let us assume that holds for some . Therefore, Hang on for , now we show that Using the concept of , we have The formula inside bracket Therefore, Hence, Eq. (39) can be written as follows: For Eq. (40), substituting in Eq. (39) we calculate the integral inside bracket Therefore, Now, assume that , Eq. (40) is correct for some . Thus, Let us indicate By the notion of , we have Thus, the theorem is correct at an arbitrary natural number . Hence, Eq. (40) is correct.

3. -Application of the Laplace Transform Decomposition Method to Coupled Burgers’ Equation

In this section, we investigate the solutions of two problems using the -double Laplace transform decomposition method: the first problem is the standard Burgers’ equation represented as follows: with the boundary condition: where and are given functions. Applying the -double Laplace transform to both sides of Eq. (58) and the double Laplace transform to Eq. (59), we obtain

Now, by utilizing the inverse -double Laplace transform for Eq. (60), we have

The -double Laplace transform decomposition method (GDLTDM) assumes that the solution can be expanded into an infinite series as

Adomian’s polynomials and are defined as follows:

The Adomian polynomials for the nonlinear terms and are given by By substituting Eq. (62) into Eq. (61), we obtain

Upon comparing both sides of Eq. (74) and Eq. (62), we derive the following iterative algorithm:

The remaining component, represented as , for , is determined by applying the following relationship:

To illustrate the application of this method to two-dimensional Burgers’ equations, we consider the following example.

Example 4. Let us consider the one-dimensional Burgers’ equation given by subject to the initial condition By applying the -double Laplace transform to both sides of Eq. (77) and the Laplace transform to Eq. (78), we obtain By using Eq. (74), we have where and are given by Eq. (64) and Eq. (69). By equating both sides of Eq. (74) and Eq. (80), we get Overall, the recursive link is given by where . At , Similarly, for , we have Similarly, when , we obtain Utilizing Equation (62), we can determine the specific convergent solutions as follows: Hence, the intricate solution transforms into

Problem 10. Consider the next two-dimensional Burgers’ equations subject to On using -double Laplace transform and characteristic of differentiation double Laplace transform, we have The next stage in the double Laplace transform decomposition method within the context of involves expressing the solution of the following series: The nonlinear operators are defined or characterized by where terms , , and are determined by where the definitions of , , and are analogous to those in Equations (96), (100), and (104). Apply the inverse -double Laplace transform to Equation (91), and then utilize Equations (89), (94), and (??), yielding

This yields the desired recursive relation as follows: and the remaining terms are represented by

To illustrate this method for the two-dimensional coupled Burgers’ equation, we examine the following examples.

Example 5 (see [23]). Consider the following two-dimensional homogeneous Burgers’ equations: subject to By employing the aforementioned method, we obtain The subsequent terms are presented as follows:
Where At Similarly, at , we obtain Similarly, we obtain the remaining terms as follows: We keep the same style to obtain the approximate solutions: Hence, the solutions become ideal.

In the following example, we apply our method to solve the inhomogeneous coupled system of Burgers’ equation.

Example 6. Examine the following set of two-dimensional Burgers’ equations: subject to Using the method mentioned above, we obtain Furthermore, the subsequent terms are represented by Continuing in a similar fashion, we have Hence, the solution series is given by

4. Conclusion

In the course of this study, we introduced the innovative -double Laplace transform, meticulously elaborating on its diverse definitions, theorems, existence conditions, partial derivatives, and the double convolution theorems. Leveraging these novel insights, we successfully uncovered the exact solutions for both Burgers’ equation and its coupled counterpart. To validate the efficacy of our technique, we presented three illustrative examples. As a result, we strongly recommend the adoption of this method for future endeavors in addressing equations commonly encountered in the realms of physics and engineering. The -double Laplace transform, with its profound potential, stands poised to make significant contributions to the field, promising enhanced problem-solving capabilities for complex mathematical models in these domains.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to extend their sincere appreciation to Researchers Supporting Project (number RSPD2023R802) King Saud University, Riyadh, Saudi Arabia.