# Solutions of (2+1)-D & (3+1)-D Burgers Equations by New Laplace Variational Iteration Technique

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. New LVIM for Solving (2+1)-D Burgers’s Equation

#### 2.2. The Convergence of LVIM for (2+1)-D Partial Differential Equations

**l**,

**n**, and

**g**are a linear operator of the first order, a non-linear operator, and a non-homogeneous term, respectively.

**Theorem**

**1.**

**Proof.**

#### 2.3. New LVIM for Solving (3+1)-D Burgers’s Equation

#### 2.4. The Convergence of LVIM for (3+1)-D Partial Differential Equations

**l**,

**n**, and

**g**are a linear operator of the first order, a non-linear operator, and a non-homogeneous term, respectively.

**Theorem**

**2.**

**Proof.**

## 3. Numerical Examples

**Example**

**1.**

**Example**

**2.**

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Hendi, F.A.; Kashkari, B.S.; Alderremy, A.A. The variational Homotopy Perturbation method for solving ((n × n) + 1). Dimensional Burgers’ equations. J. Appl. Math.
**2016**, 2016, 4146323. [Google Scholar] [CrossRef] [Green Version] - 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 u
_{t}+ uu_{x}= µu_{xx}. Commun. Pure Appl. Math.**1950**, 3, 201–230. [Google Scholar] [CrossRef] - He, J.H. Variational iteration method for delay differential equations. Commun. Nonlinear Sci. Numer. Simul.
**1997**, 2, 235–236. [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. A
**1999**, 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]

**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 ($\alpha $, $\beta $) = (0.1, 0.1) and A = 2.

$\mathit{\tau}$ | Exact | LVIM | VHPM [1] |
---|---|---|---|

0.01 | 0.40404040 | 0.40404040 | 0.40404040 |

0.02 | 0.40816326 | 0.40816324 | 0.40816320 |

0.03 | 0.41237113 | 0.41237101 | 0.41237080 |

0.04 | 0.41666666 | 0.41666629 | 0.41666560 |

0.05 | 0.42105263 | 0.42105170 | 0.42105000 |

0.06 | 0.42553191 | 0.42552996 | 0.42552640 |

0.07 | 0.43010752 | 0.43010383 | 0.43009720 |

0.08 | 0.43478260 | 0.43477617 | 0.43476480 |

0.09 | 0.43956043 | 0.43954990 | 0.43953160 |

0.10 | 0.44444444 | 0.44442804 | 0.44440000 |

**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 ($\alpha $, $\beta $) = (0.1, 0.1) and A = 2.

$\mathit{\tau}$ | $\left|{\mathit{\phi}}_{\mathit{e}\mathit{x}\mathit{a}\mathit{c}\mathit{t}}-{\mathit{\phi}}_{\mathit{L}\mathit{V}\mathit{I}\mathit{M}}\right|$ | $\left|{\mathit{\phi}}_{\mathit{e}\mathit{x}\mathit{a}\mathit{c}\mathit{t}}-{\mathit{\phi}}_{\mathit{V}\mathit{H}\mathit{P}\mathit{M}}\right|$ [1] |
---|---|---|

0.01 | 1.3604 × 10^{−9} | 4.0404 × 10^{−9} |

0.02 | 2.2210 × 10^{−8} | 6.5306 × 10^{−8} |

0.03 | 1.1475 × 10^{−7} | 3.3402 × 10^{−7} |

0.04 | 3.7016 × 10^{−7} | 1.0667 × 10^{−6} |

0.05 | 9.2255 × 10^{−7} | 2.6316 × 10^{−6} |

0.06 | 1.9531 × 10^{−6} | 5.5149 × 10^{−6} |

0.07 | 3.6948 × 10^{−6} | 1.0327 × 10^{−5} |

0.08 | 6.4373 × 10^{−6} | 1.7809 × 10^{−5} |

0.09 | 1.0532 × 10^{−5} | 2.8840 × 10^{−5} |

0.10 | 1.6399 × 10^{−5} | 4.4444 × 10^{−5} |

**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 ($\alpha $, $\beta $) = (0.1, 0.1) and A = 2 at different $\tau $.

$\mathit{\tau}$ | Exact Solutions | $\left|{\mathit{\phi}}_{\mathit{e}\mathit{x}\mathit{a}\mathit{c}\mathit{t}}-{\mathit{\phi}}_{\mathit{L}\mathit{V}\mathit{I}\mathit{M}}\right|$ | $\left|{\mathit{\phi}}_{\mathit{e}\mathit{x}\mathit{a}\mathit{c}\mathit{t}}-{\mathit{\phi}}_{\mathit{V}\mathit{H}\mathit{P}\mathit{M}}\right|$ [1] |
---|---|---|---|

0.2 | 0.50000000 | 3.2774 × 10^{−4} | 8.0000 × 10^{−4} |

0.3 | 0.57142857 | 2.1108 × 10^{−3} | 4.6286 × 10^{−3} |

0.4 | 0.66666666 | 8.6822 × 10^{−3} | 1.7067 × 10^{−2} |

0.5 | 0.80000000 | 2.8423 × 10^{−2} | 5.0000 × 10^{−2} |

0.6 | 1.00000000 | 8.2421 × 10^{−2} | 1.2960 × 10^{−1} |

**Table 4.**The comparison of new LVIM (up to fourth term), VHPM (up to fourth term (as mentioned in [1])), and exact solution for ($\alpha $, $\beta $, z) = (0.1, 0.1, 0.1) and B = 3.

$\mathit{\tau}$ | Exact | NLVIM | VHPM [1] |
---|---|---|---|

0.01 | 0.90909090 | 0.90909090 | 0.90909090 |

0.02 | 0.91836734 | 0.91836729 | 0.91836720 |

0.03 | 0.92783505 | 0.92783479 | 0.92783430 |

0.04 | 0.93750000 | 0.93749916 | 0.93749760 |

0.05 | 0.94736842 | 0.94736634 | 0.94736250 |

0.06 | 0.95744680 | 0.95744241 | 0.95743440 |

0.07 | 0.96774193 | 0.96773362 | 0.96771870 |

0.08 | 0.97826086 | 0.97824638 | 0.97822080 |

0.09 | 0.98901098 | 0.98898729 | 0.98894610 |

0.10 | 1.00000000 | 0.99996310 | 0.99990000 |

**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 ($\alpha $, $\beta $, z) = (0.1, 0.1, 0.1) and B = 3.

$\mathit{\tau}$ | $\left|{\mathit{\phi}}_{\mathit{e}\mathit{x}\mathit{a}\mathit{c}\mathit{t}}-{\mathit{\phi}}_{\mathit{L}\mathit{V}\mathit{I}\mathit{M}}\right|$ | $\left|{\mathit{\phi}}_{\mathit{e}\mathit{x}\mathit{a}\mathit{c}\mathit{t}}-{\mathit{\phi}}_{\mathit{V}\mathit{H}\mathit{P}\mathit{M}}\right|$ [1] |
---|---|---|

0.01 | 3.0608 × 10^{−9} | 9.0909 × 10^{−9} |

0.02 | 4.9972 × 10^{−8} | 1.4694 × 10^{−7} |

0.03 | 2.5818 × 10^{−7} | 7.5155 × 10^{−7} |

0.04 | 8.3287 × 10^{−7} | 2.4000 × 10^{−6} |

0.05 | 2.0757 × 10^{−6} | 5.9211 × 10^{−6} |

0.06 | 4.3945 × 10^{−6} | 1.2409 × 10^{−5} |

0.07 | 8.3134 × 10^{−6} | 2.3235 × 10^{−5} |

0.08 | 1.4484 × 10^{−5} | 4.0070 × 10^{−5} |

0.09 | 2.3698 × 10^{−5} | 6.4889 × 10^{−5} |

0.10 | 3.6899 × 10^{−5} | 1.0000 × 10^{−4} |

**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 ($\alpha $, $\beta $, z) = (0.1, 0.1, 0.1) and B = 3 at different $\tau $.

$\mathit{\tau}$ | Exact Solutions | ||
---|---|---|---|

0.2 | 1.12500000 | 7.3742 × 10^{−4} | 1.8000 × 10^{−3} |

0.3 | 1.28571428 | 4.7493 × 10^{−3} | 1.0414 × 10^{−2} |

0.4 | 1.50000000 | 1.9535 × 10^{−2} | 3.8400 × 10^{−2} |

0.5 | 1.80000000 | 6.3951 × 10^{−2} | 1.1250 × 10^{−1} |

0.6 | 2.25000000 | 1.8545 × 10^{−1} | 2.9160 × 10^{−1} |

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**MDPI and ACS Style**

Singh, G.; Singh, I.; AlDerea, A.M.; Alanzi, A.M.; Khalifa, H.A.E.-W.
Solutions of (2+1)-D & (3+1)-D Burgers Equations by New Laplace Variational Iteration Technique. *Axioms* **2023**, *12*, 647.
https://doi.org/10.3390/axioms12070647

**AMA Style**

Singh G, Singh I, AlDerea AM, Alanzi AM, Khalifa HAE-W.
Solutions of (2+1)-D & (3+1)-D Burgers Equations by New Laplace Variational Iteration Technique. *Axioms*. 2023; 12(7):647.
https://doi.org/10.3390/axioms12070647

**Chicago/Turabian Style**

Singh, Gurpreet, Inderdeep Singh, Afrah M. AlDerea, Agaeb Mahal Alanzi, and Hamiden Abd El-Wahed Khalifa.
2023. "Solutions of (2+1)-D & (3+1)-D Burgers Equations by New Laplace Variational Iteration Technique" *Axioms* 12, no. 7: 647.
https://doi.org/10.3390/axioms12070647