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The new Laplace variational iterative method is used in this research for solving the (2+1)-D and (3+1)-D Burgers equations. This technique relies on the modified variational iteration method and the Laplace transform. To apply this approach, the differential problem is first transformed into an algebraic form using the Laplace transform, and then the algebraic equations are iteratively solved using the modified variational iterative approach. By utilizing this technique, the Burgers equations can be solved both numerically and analytically. The study demonstrates the effectiveness of the new Laplace variational iterative approach through three specific examples.
The J.M. Burgers equation, also known as Burgers’s equation, is a significant and commonly used non-linear PDE. It was first introduced by Bateman and later corrected by Burgers, and is sometimes referred to as the Bateman–Burgers equation. This equation is employed to simulate numerous physical phenomena, for example, acoustics, diffraction water waves, heat conduction, shock waves, and turbulence issues, among others.
This research focuses on the analytical solutions of the two-dimensional and three-dimensional Burgers equations. The new Laplace transform with the variational iteration method (LVIM) is utilized for solving these equations. Approximate results obtained using the LVIM approach are then compared with the analytical results of Burgers’s equation, the numerical approximations of the Burgers equation obtained via the Laplace Homotopy Perturbation method (LHPM) [1], and the numerical results of the Burgers equation obtained via the EHPM [2]. To demonstrate the effectiveness of the proposed method, a comparison study is given in Section 3.
In addition, the suggested strategy’s convergence is illustrated through graphs of both precise and approximate solutions. Various partial differential equations with linear and non-linear coefficients can be utilized to solve initial value and boundary value problems. To find approximate solutions to Burgers equations, several numerical schemes have been developed, including the spline FEM, ADM, Douglas FD scheme, exact explicit FDM, VIM, and others [3,4,5,6,7,8,9,10]. However, only a few analytical methods, such as the LHPM [11], Hopf–Cole Transformation [12], etc., have been developed to obtain the precise solution of certain PDEs. Laplace transform-based methods are extensively employed in mathematics to solve differential equations. Other techniques, such as the VIM and the HPM, can also be combined with it to make it hybrid. By combining these approaches with the Laplace Transform method, partial differential equations can be solved analytically. The VIM has been used to solve various differential equations [13]. It has been demonstrated that the VIM can also solve non-linear equations [14]. The Laplace Transformation and variational iteration approach have been used to solve Smoluchowski’s coagulation equations [15]. A new modified variational iterative approach has been proposed for the solution of boundary value problems of higher order [16]. The variational iteration approach and Laplace transformation have been combined in [17]. In [18], certain issues with the variational iterative approach and how the Laplace transform method fixes them are detailed. Modified fractional derivatives have a Laplace variational approach built into them [19]. A new Laplace Transformation and variational iterative approach can solve non-linear PDEs [20]. The new Laplace and variational iterative approach has been used to solve numerous equations [21,22,23,24].
Consider, the (2+1)-D non-linear Burgers’s equation can also be written as
with the initial conditions
where is the velocity component, is the kinematic viscosity, is any constant, and is the time.
Similarly, the (3+1)-D Non-linear Burgers equation is
with initial conditions
where is the velocity component, is the kinematic viscosity, is any constant, and is the time.
Non-linear partial differential equations find wide application in the fields of engineering, physics, and applied mathematics. Various approaches have been suggested in the literature to solve the two-dimensional Burgers equations as well as the two-dimensional and three-dimensional Burgers equations. The importance of discovering exact solutions to PDEs for developing novel techniques to obtain precise or approximate solutions remains a topic of great interest in mathematics, engineering, and physics, as evidenced by recent publications [25,26,27,28,29,30].
2. Materials and Methods
2.1. New LVIM for Solving (2+1)-D Burgers’s Equation
Consider the following (2+1)-D Burgers equation:
with given conditions as
Rewriting Equation (2), we have
By applying Laplace transformation on (3), we have
By using inverse Laplace transformation on (8), we obtain
Now, by modifying VIM from Equation (9), we obtain
Equation (10) represents the modified iteration formula of LVIM; the solution is given by
2.2. The Convergence of LVIM for (2+1)-D Partial Differential Equations
Consider the two-dimensional differential equation
with the initial conditions
where l, n, and g are a linear operator of the first order, a non-linear operator, and a non-homogeneous term, respectively.
The iteration formula of the new LVIM is
Now, define the operator as
define the components as
Hence,
For the analysis of convergence of new LVIM, let us discuss the following theorem.
Theorem1.
Let A, as defined in (14), be an operator from Hilbert space H to H; the solution, as defined in (16), converges if there existssuch that
for all .
Proof.
Define the sequence
as
Now, we will show that sequence
is a Cauchy sequence in the Hilbert space H.
Consider
For every , we have
Since , therefore,
Hence, is a Cauchy sequence in the Hilbert space H and it implies that the series solution (16) converges. □
2.3. New LVIM for Solving (3+1)-D Burgers’s Equation
Consider the following (3+1)-D Burgers equation:
with given conditions as
Rewriting Equation (18), we have
By applying Laplace transformation on (19), we have
By using inverse Laplace transformation on (24), we obtain
Now, by modifying VIM from Equation (25), we obtain
Equation (26) represents the modified iteration formula of LVIM; the solution is given by
2.4. The Convergence of LVIM for (3+1)-D Partial Differential Equations
Consider the three-dimensional differential equation
with the initial conditions
where l, n, and g are a linear operator of the first order, a non-linear operator, and a non-homogeneous term, respectively.
The iteration formula of the new LVIM is
Now, define the operator as
define the components as
Hence,
For the analysis of convergence of new LVIM, let us discuss the following theorem.
Theorem2.
Let A, as defined in (30), be an operator from Hilbert space H to H; the solution, as defined in (32), converges if there existssuch that
for all .
Proof.
Define the sequence as
Now, we will show that sequence is a Cauchy sequence in the Hilbert space H.
Consider
For every , we have
Since , therefore,
Hence, is a Cauchy sequence in the Hilbert space H and it implies that the series solution (32) converges. □
3. Numerical Examples
Examples are provided in this part to illustrate the effectiveness and precision of the suggested Laplace variational iterative method.
Example1.
Consider the following Two-Dimensional Burgers Equation
with conditions given as
By applying the LT on (35), we have
By applying the inverse Laplace transformation on (37), we get
Using the proposed variational method from (38), we obtain
From (39), we obtain
Similarly, we can find the fourth, fifth, and other iterations.
The solution can be found as
After simplification, we obtain
This implies
or
This series solution is valid only if
.
Table 1 shows the comparison study of solutions obtained by new Laplace variation iteration method (up to fourth term), variational homotopy perturbation method (up to fourth term (as discussed in [1])), and the exact solutions for and A = 2 of Example 1. Table 2 shows the comparison of absolute errors obtained by new Laplace variation iteration method (up to fourth term) and variational homotopy perturbation method (up to fourth term (as discussed in [1])) for and A = 2 of Example 1. Table 3 shows the comparison of absolute errors obtained by new Laplace variation iteration method (up to fourth term) for different value of . Figure 1 shows the physical behavior of solutions for at different domain of and .
Example2.
Consider the following (3+1)-D Burgers equation
with the initial conditions
By using the Laplace transformation on (41), we obtain
By the inverse Laplace transformation on (42), we obtain
Using the modified variational iteration method from Equation (43), we obtain
From (44), we obtain
The solution can be obtained by
Now, after simplification, we obtain
This implies
or
This series solution is valid only if
.
Table 4 shows the comparison study of solutions obtained by new Laplace variation iteration method (up to fourth term), variational homotopy perturbation method (up to fourth term (as discussed in [1])), and the exact solutions for and B = 3 of Example 2. Table 5 shows the comparison of absolute errors obtained by new Laplace variation iteration method (up to fourth term) and variational homotopy perturbation method (up to fourth term (as discussed in [1])) for particular values of variables and B = 3 of Example 3. Table 6 shows the comparison of absolute errors obtained by new Laplace variation iteration method (up to fourth term) for different value of . Figure 2 shows the physical behavior of solutions for at different domain of and .
4. Conclusions
Based on the preceding discussion and experiments, the combination of the Laplace transforms, and the variational iteration technique presents an effective approach to solve the (2+1)-D and (3+1)-D Burgers equations. Compared to the variational homotopy perturbation technique (VHPM), the new Laplace variational iteration method (LVIM) is more effective in obtaining an approximate solution that closely approximates the actual one. It is possible that this technique may be utilized in the future to solve the three-dimensional Burgers equation system.
Author Contributions
Conceptualization, G.S.; methodology, G.S.; software, G.S.; validation, I.S., A.M.A. (Afrah M. AlDerea) and A.M.A. (Agaeb Mahal Alanzi); formal analysis, G.S.; investigation, G.S.; resources, G.S.; data curation, G.S.; writing—original draft preparation, G.S.; writing—review and editing, G.S.; visualization, H.A.E.-W.K.; supervision, I.S.; This research work is related to the Ph.D research work of Gurpreet Singh (G.S.), no other person can use this research work for getting any degree from any institute/University. All authors have read and agreed to the published version of the manuscript.
Funding
The researchers would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project.
Data Availability Statement
All data is available within the article.
Acknowledgments
The researchers would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Suleman, M.; Wu, Q.; Abbas, G. Approximate analytic solution of (2 + 1) dimensional coupled differential Burger’s equation using Elzaki Homotopy Perturbation Method. Alex. Eng. J.2016, 55, 1817–1826. [Google Scholar] [CrossRef] [Green Version]
Kutluay, S.; Bahadir, A.; Özdeş, A. Numerical solution of one dimensional Burgers’ equation by explicit and exact-explicit finite difference methods. J. Comput. Appl. Math.1999, 103, 251–261. [Google Scholar] [CrossRef] [Green Version]
Kutluay, S.; Esen, A.; Dag, I. Numerical solutions of the Burgers’ equations by the least squares quadratic B spline finite element method. J. Comput. Appl. Math.2004, 167, 21–33. [Google Scholar] [CrossRef] [Green Version]
Pandey, K.; Verma, L.; Verma, A.K. On a finite difference scheme for Burgers’ equations. Appl. Math. Comput.2009, 215, 2206–2214. [Google Scholar] [CrossRef]
Aksan, E.N. A numerical solution of Burgers’ equation by finite element method constructed on the method of discretization in time. Appl. Math. Comput.2005, 170, 895–904. [Google Scholar] [CrossRef]
Abdou, M.A.; Soliman, A.A. Variational iteration method for solving Burgers’ and coupled Burgers’ equations. J. Comput. Appl. Math.1996, 181, 245–251. [Google Scholar] [CrossRef] [Green Version]
Mittal, R.; Singhal, P. Numerical solution of Burgers’ equation. Commun. Num. Methods Eng.1993, 9, 397–406. [Google Scholar] [CrossRef]
Abbasbandy, S.; Darvishi, M.T. A numerical solution of Burgers’ equation by modified Adomian Decomposition method. Appl. Math. Comput.2005, 163, 1265–1272. [Google Scholar]
Öziş, T.; Aksan, E.N.; Özdeş, A. A finite element approach for solution of Burgers’ equation. Appl. Math. Comput.2003, 139, 417–428. [Google Scholar] [CrossRef]
Aminikhah, H. A new efficient method for solving two-dimensional Burgers’ equation. ISRN Comput. Math.2012, 2012, 603280. [Google Scholar] [CrossRef] [Green Version]
Hopf, E. The partial differential equation ut + uux = µuxx. Commun. Pure Appl. Math.1950, 3, 201–230. [Google Scholar] [CrossRef]
He, J.H. Variational iteration method-a kind of non-linear analytical technique: Some examples. Int. J. Nonlinear Mech.1999, 34, 699–708. [Google Scholar] [CrossRef]
Hammouch, Z.; Mekkaoui, T. A Laplace-variational iteration method for solving the homogeneous Smoluchowski coagulation equation. Appl. Math. Sci.2012, 6, 879–886. [Google Scholar]
Arife, A.S.; Yildirim, A. New modified variational iteration transform method (MVITM) for solving eighth-order boundary value problems in one step. World Appl. Sci. J.2011, 13, 2186–2190. [Google Scholar]
Hesameddini, E.; Latifizadeh, H. Reconstruction of variational iteration algorithms using the Laplace transform. Int. J. Nonlinear Sci. Numer. Simul.2009, 10, 1377–1382. [Google Scholar] [CrossRef]
Wu, G.C. Laplace transform overcoming principle drawbacks in application of the variational iteration method to fractional heat equations. Therm. Sci.2012, 16, 1257–1261. [Google Scholar] [CrossRef]
Martinez, H.Y.; Gomez-Aguilar, J.F. Laplace variational iteration method for modified fractional derivatives with non-singular kernel. J. Appl. Comput. Mech.2020, 6, 684–698. [Google Scholar]
Elzaki, T.M. Solution of Nonlinear Partial Differential Equations by New Laplace Variational Iteration Method. In Differential Equations: Theory and Current Research; IntechOpen: London, UK, 2018. [Google Scholar]
Singh, G.; Singh, I. Laplace variational iterative method for solving 3D Schrodinger equations. J. Math. Comput. Sci.2020, 10, 2015–2024. [Google Scholar]
Singh, G.; Singh, I. Laplace variational iterative method for solving Two-dimensional Telegraph equations. J. Math. Comput. Sci.2020, 10, 2943–2954. [Google Scholar]
Singh, G.; Singh, I. New Hybrid Technique for solving Three-dimensional Telegraph equations. Adv. Differ. Equ. Control. Process.2021, 24, 153–165. [Google Scholar] [CrossRef]
Singh, G.; Singh, I. The exact solution of 3D Diffusion and wave equations using Laplace variational iterative method. Int. J. Adv. Res. Eng. Technol.2020, 11, 36–43. [Google Scholar]
Aksan, E.N. Quadratic B-spline finite element method for numerical solution of the Burgers’ equation. Appl. Math. Comput.2006, 174, 884–896. [Google Scholar] [CrossRef]
Kutluay, S.; Esen, A. A lumped galerkin method for solving the burgers equation. Int. J. Comput. Math.2004, 81, 1433–1444. [Google Scholar] [CrossRef]
Sirendaoreji. Exact solutions of the two-dimensional Burgers equation. J. Phys. A1999, 32, 6897–6900. [Google Scholar] [CrossRef]
Sharma, K.D.; Kumar, R.; Kakkar, M.; Ghangas, S. Three dimensional waves propagation in thermos-viscoelastic medium with two temperature and void. IOP Conf. Ser. Mater. Sci. Eng.2021, 1033, 012059. [Google Scholar] [CrossRef]
Singh, V.; Saluja, N.; Singh, C.; Malhotra, R. Computational and Experimental study of microwave processing of susceptor with multiple topologies of launcher waveguide. AIP Conf. Proc.2022, 2357, 040019. [Google Scholar]
Khan, M. A novel solution technique for two-dimensional Burgers equation. Alex. Eng. J.2014, 53, 485–490. [Google Scholar] [CrossRef] [Green Version]
Figure 1.
Description of solutions of Example 1 for τ = 0.2.
Figure 1.
Description of solutions of Example 1 for τ = 0.2.
Figure 2.
Description of solutions of Example 2 for .
Figure 2.
Description of solutions of Example 2 for .
Table 1.
The comparison study of new LVIM (up to fourth term), VHPM (up to fourth term (as mentioned in [1])), and the exact solution for (, ) = (0.1, 0.1) and A = 2.
Table 1.
The comparison study of new LVIM (up to fourth term), VHPM (up to fourth term (as mentioned in [1])), and the exact solution for (, ) = (0.1, 0.1) and A = 2.
Table 2.
The comparison of absolute errors obtained by new LVIM (up to fourth term) and VHPM (up to fourth term (as mentioned in [1])) for (, ) = (0.1, 0.1) and A = 2.
Table 2.
The comparison of absolute errors obtained by new LVIM (up to fourth term) and VHPM (up to fourth term (as mentioned in [1])) for (, ) = (0.1, 0.1) and A = 2.
Table 3.
The comparison of absolute errors obtained by new LVIM (up to fourth term) and VHPM (up to fourth term (as mentioned in [1])) for (, ) = (0.1, 0.1) and A = 2 at different .
Table 3.
The comparison of absolute errors obtained by new LVIM (up to fourth term) and VHPM (up to fourth term (as mentioned in [1])) for (, ) = (0.1, 0.1) and A = 2 at different .
Table 4.
The comparison of new LVIM (up to fourth term), VHPM (up to fourth term (as mentioned in [1])), and exact solution for (, , z) = (0.1, 0.1, 0.1) and B = 3.
Table 4.
The comparison of new LVIM (up to fourth term), VHPM (up to fourth term (as mentioned in [1])), and exact solution for (, , z) = (0.1, 0.1, 0.1) and B = 3.
Table 5.
The comparison of absolute errors obtained by new LVIM (up to fourth term) and VHPM (up to fourth term (as mentioned in [1])) for (, , z) = (0.1, 0.1, 0.1) and B = 3.
Table 5.
The comparison of absolute errors obtained by new LVIM (up to fourth term) and VHPM (up to fourth term (as mentioned in [1])) for (, , z) = (0.1, 0.1, 0.1) and B = 3.
Table 6.
The comparison of absolute errors obtained by new LVIM (up to fourth term) and VHPM (up to fourth term (as mentioned in [1])) for (, , z) = (0.1, 0.1, 0.1) and B = 3 at different .
Table 6.
The comparison of absolute errors obtained by new LVIM (up to fourth term) and VHPM (up to fourth term (as mentioned in [1])) for (, , z) = (0.1, 0.1, 0.1) and B = 3 at different .
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