The Solution of Coupled Burgers’ Equation by G-Laplace Transform

: The coupled Burgers’ equation is a fundamental partial differential equation with applications in various scientiﬁc ﬁelds. Finding accurate solutions to this equation is crucial for understanding physical phenomena and mathematical models. While different methods have been explored, this work highlights the importance of the G-Laplace transform. The G-transform is effective in solving a wide range of non-constant coefﬁcient differential equations, setting it apart from the Laplace, Sumudu, and Elzaki transforms. Consequently, it stands as a powerful tool for addressing differential equations characterized by variable coefﬁcients. By applying this transformative approach, the study provides reliable and exact solutions for both homogeneous and non-homogeneous coupled Burgers’ equations. This innovative technique offers a valuable tool for gaining deeper insights into this equation’s behavior and signiﬁcance in diverse disciplines.


Introduction
The Burgers' equation is a fundamental partial differential equation and convectiondiffusion equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow.The solution to this equation is quite important for mathematical models and physical phenomena.Several scientists have suggested analytical solutions to the one-dimensional coupled Burgers' equation.Many analytical methods have been produced to obtain the solution to Burgers' equation, see [1][2][3].In their work [4], the authors employed the space-time Sinc-collocation method to address the fourth-order nonlocal heat model arising in viscoelasticity.Nonlinear terms were handled using the MacCormack method, while the Riemann-Liouville (R-L) fractional integral term was managed through the second-order convolution quadrature formula, as outlined in [5].Furthermore, they discussed the modified Burgers' model with nonlocal dynamic properties and utilized an implicit robust difference method with graded meshes, as elaborated in [6].Additionally, in their study [7], the authors established the theory that the L1 scheme is effective in solving time-fractional partial differential equations with non-smooth data.In recent years, significant effort has been devoted to applying the Laplace decomposition method (LDM) and its modifications to study physical model equations [8].The exact solution of the Burgers' equation has been given in [9], employing the Adomian Decomposition method, and the authors of [10,11] offered a modified, expanded tanh-function method to receive its exact solution.The researchers in [12] suggested the homotopy perturbation method to achieve the exact solution of the nonlinear Burgers' equation.Ref. [13] introduced a groundbreaking approach that combined the Laplace transform and new homotopy perturbation method (NHPM) for obtaining closed-form solutions of coupled viscous Burgers' equations.This study holds significant importance as it sheds light on the understanding of polydispersity and its connection to gravity effects [14].The study of coupled Burgers' equations is paramount because it explains the precipitation of polydispersity commentary down to the effect of gravity [14].The authors of [15,16] discussed the solution of the time-fractional two-mode coupled Burgers' equation.G-transform was first proposed in [17] and later applied to solve certain nonlinear dynamical models with a non-integer order in [18].
The main goal of this work is to apply the G-Laplace transform in order to secure exact solutions with high reliability for homogeneous and nonhomogeneous coupled Burgers' equations.

Some Basic Idea of the G α -Laplace Transforms
In this work, we transact with G α Transform and Laplace Transform to support us with solving some partial differential equations.Definition 1.Let f (ν) be an integrable function, for all ν ≥ 0. The generalized integral transform G α of the function f (ν) is given by for, s ∈ C and α ∈ Z (see [19]).
As an illustration, if we set α = 0 and p = 1 s in Equation ( 1), we can derive the Laplace transform as follows: This demonstrates that the G α -transform serves as a generalized version of both the Laplace and other transforms.It encompasses a broader and more fundamental range than existing transforms.For a more in-depth understanding, we recommend referring to [19].Definition 2. The Laplace transform of the function f (µ) is determined by given by the following expression: where L µ G α indicates the G α -Laplace Transform, and consequently, , where n is a non-negative integer.If θ(> −1) and β(> −1) ∈ R, then can be derived from the definition of G α -Laplace Transform, so we have and by using the definition of Laplace transform for the integral inside the first bracket of Equation (3), we obtain In Equation ( 4), put r = ν s , ν = rs to obtain where, Gamma functions of β + 1 are defined by the convergent integral: Example 3. G α -Laplace Transform for the the following function where the H(µ, ν) = H(ν) ⊗ H(µ) is the dimensional Heaviside function and ⊗ is a tensor product (see [20]).Let ζ = pµ and η = 1 s ν, then the integral becomes Existence Condition for the G α -Laplace Transform In the following theorem, we establish the conditions for the existence of the G α -Laplace Transform of f (µ, ν).Let f (µ, ν) be considered of exponential order a (> 0) and b (> 0) on 0 ≤ µ, ν < ∞ if there exists a non-negative constant M, such that for all µ > X and ν > Y, the inequality holds.In this case, we can express f (µ, ν) as as µ → ∞ and ν → ∞.Equivalently, we have for p > a and s > 1 b .The function f (µ, ν) is simply referred to as having an exponential order as µ → ∞ and ν → ∞.Clearly, it does not grow faster than Me aµ+bν as µ → ∞ and ν → ∞.Theorem 1.If f (µ, ν) is a continuous function in every bounded interval , (0, µ) and (0, ν) and of exponential order e aµ+bν , then the L µ G α transform of f (µ, ν) exists for all p and s, provided that p > a and 1 s > b.
Proof.By using the definition of G α -Laplace Transform and Equations ( 6) and ( 7), one can obtain and has an exponential order at infinity with Me aµ+bν for µ ≥ µ and ν ≥ ν,where µ and ν are constant, then for any real number ρ ≥ 0 and σ ≥ 0, we have Proof.By using the definition of let z = µ − ρ and r = ν − σ, Equation ( 8) becomes Theorem 2. Suppose f (µ, ν) is a periodic function with periods λ and µ.For this periodicity condition to hold, we require that: Then G α -Laplace Transform of f (µ, ν) is given by Proof.Utilizing the definition of the G α -Laplace Transform given by we can apply the property of improper integrals to Equation (10), resulting in By setting µ = λ + γ and ν = µ + δ in the second part of the integral in Equation ( 11), we have Equation ( 12) can then be rewritten as follows: By the second integral in Equation ( 13), given the definition of G α -Laplace Transform, we obtain and hence, where and the symbol * * indicates the double convolution with respect to x and t.
Proof.On using the definition of G α -Laplace Transform, we obtain Set ρ = µ − ζ and σ = ν − η, and by applying the adequate expansion of the upper bound of integrals to µ → ∞ and ν → ∞, Equation ( 14) can be written as where the functions φ(µ, ν) and ϕ(µ, ν) equal zero at µ < 0, ν < 0; therefore, it follows with respect to the lower limit of integrations that It is thus easy to see that ∂ν n and and Proof.Now, by substituting n = 1 in Equation ( 16): first, we calculate the integral inside bracket to obtain and by substituting in Equation ( 18), we have at n = 2.In a similar way, one can easily see that and let us suppose that n = m is valid for some m.Thus, hold for ∂ 0 ∂ = 1, and now, we show that and by the notion of n = 1, we have of which, the formula inside the bracket is Therefore, Hence, Equation ( 16) can be written as follows: For Equation (17), by substituting n = 1 in Equation ( 16), we calculate the integral inside bracket Therefore, Now, assume that n = m, Equation ( 17) is then correct for some m.Thus, and by the notion of n = 1, we have Thus, the theorem is correct at an arbitrary natural number k.Hence, Equation ( 17) is correct.

G α -Laplace Transform Decomposition Method Applied to Coupled Burgers' Equation
In this section, we discuss the solutions of two problems by applying G α -Laplace Transform decomposition method: The first problem: Regular Burgers' equation is given by for ν, µ > 0. Here, f (µ, ν) and f 1 (µ) are given functions.By taking G α -Laplace Transform for both sides of Equation ( 22) and Laplace transform for Equation ( 23), we obtain By utilizing the inverse G α -Laplace Transform for Equation ( 22), we have The G α -Laplace Transform decomposion method (GLTDM) supposes the solution φ(µ, ν) can be expanded into infinite series as We can present Adomian's polynomials A n , respectively, as follows: where, the Adomian polynomials for the nonlinear term φφ µ are given by By substituting Equation (26) into Equation (25), we have Upon comparing both sides of the Equations ( 26) and (29), we obtain the following iterative algorithm: The value of the rest component φ n+1 , where n ≥ 0, is determined by utilizing the following relation: Here, we applied the inverse G α -Laplace Transform to each term on the right-hand side of the above equation, with respect to p and s, to obtain the corresponding expressions for each term.
To examine this method for one-dimensional Burgers' equations, we use the following example: Example 4. Consider that one-dimensional Burgers' equation is given by By taking G α -Laplace Transform for both sides of Equation ( 30) and Laplace transform for Equation (31), we obtain and by using Equation (29), we have where A n is given by Equation (28).On matching both sides of Equations ( 29) and (33), we have Overall, the recursive link is given by where n ≥ 0. At n = 0, In a similar manner, at n = 1, we have In a similar way, at n = 2, we obtain Upon applying Equation ( 26), the convergent solutions are thus specified by and therefore, the delicate solution becomes The second problem: Consider the following one-dimensional Burgers' equations On using G α -Laplace Transform and the characteristic of the differentiation of Laplace transform, we have The following step in the G α -Laplace Transform decomposition method illustrates the solution of the series given below: where the nonlinear operators are determined by and the terms ϕϕ µ , ϕφ µ , and φϕ µ are determined by And Implement the inverse G α -Laplace Transform into Equation (36) and, using Equations ( 27), (37), and (38), we have provides our desired recursive relation in this way with the rest terms presented by and To explain this method for the coupled Burgers' equation problem, we check the next examples Example 5 ([21]).Consider the following one-dimensional homogeneous Burgers' equations: subject to φ(µ, 0) = sin µ, ϕ(µ, 0) = sin µ. (48) By using the mentioned method, we receive φ 0 = sin µ, ϕ 0 = sin µ, (49) and The following terms are presented, wherein n ≥ 0. At n = 0, we have and In the same way at n = 1, we obtain and Similarly, we obtain the remaining terms as follows: We keep the same style to obtain the approximate solutions Therefore, the perfect solutions become φ(µ, ν) = e −ν sin µ, ϕ(µ, ν) = e −ν sin µ.
The result we have reached is similar to the one in [21].