# Hyperbolic B-Spline Function-Based Differential Quadrature Method for the Approximation of 3D Wave Equations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Hyperbolic B-Spline Differential Quadrature Method

## 3. Stability Analysis

## 4. Computational Results

**Example**

**1.**

**Example**

**2.**

^{−19}, which shows that the proposed method provides very accurate results.

**Example**

**3.**

#### Computational Complexity

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Chen, H.; Zhou, H.; Jiang, S.; Rao, Y. Fractional Laplacian viscoacoustic wave equation low-rank temporal extrapolation. IEEE Access
**2019**, 99, 1–11. [Google Scholar] [CrossRef] - Li, P.-W.; Fan, C.-M. Generalized finite difference method for two-dimensional shallow water equations. Eng. Anal. Bound. Elem.
**2017**, 80, 58–71. [Google Scholar] [CrossRef] - Baccouch, M.; Temimi, H. A high-order space-time ultra-weak discontinuous Galerkin method for the second-order wave equation in one space dimension. J. Comput. Appl. Math.
**2021**, 389, 113331. [Google Scholar] [CrossRef] - Yang, S.P.; Liu, F.W.; Feng, L.B.; Turner, I. A novel finite volume method for the nonlinear two-sided space distributed-order diffusion equation with variable coefficients. J. Comput. Appl. Math.
**2021**, 388, 113337. [Google Scholar] [CrossRef] - Xie, X.; Liu, Y.J. An adaptive model order reduction method for boundary element-based multi-frequency acoustic wave problems. Comput. Meth. Appl. Mech. Engin.
**2021**, 373, 113532. [Google Scholar] [CrossRef] - Takekawa, J.; Mikada, H. A mesh-free finite-difference method for elastic wave propagation in the frequency-domain. Comput. Geosci.
**2018**, 118, 65–78. [Google Scholar] [CrossRef] - Gao, L.F.; Keyes, D. Combining finite element and finite difference methods for isotropic elastic wave simulations in an energy-conserving manner. J. Comput. Phys.
**2019**, 378, 665–685. [Google Scholar] [CrossRef] [Green Version] - Ranocha, H.; Mitsotakis, D.; Ketcheson, D.I. A broad class of conservative numerical methods for dispersive wave equations. Commun. Comput. Phys.
**2021**, 29, 979–1029. [Google Scholar] [CrossRef] - Wang, F.; Zhang, J.; Ahmad, I.; Farooq, A.; Ahmad, H. A novel meshfree strategy for a viscous wave equation with variable coefficients. Front. Phys.
**2021**, 9, 359. [Google Scholar] - Bakushinsky, A.B.; Leonov, A.S. Numerical solution of a three-dimensional coefficient inverse problem for the wave equation with integral data in a cylindrical domain. Numer. Analys. Appl.
**2019**, 12, 311–325. [Google Scholar] [CrossRef] - Dehghan, M. On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation. Numer. Meth. Part. Diff. Eq.
**2005**, 21, 24–40. [Google Scholar] [CrossRef] - Dehghan, M. Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math. Comput. Simul.
**2006**, 71, 16–30. [Google Scholar] [CrossRef] - Mohanty, R.K.; Gopal, V. A new off-step high order approximation for the solution of three-space dimensional nonlinear wave equations. Appl. Math. Model.
**2013**, 37, 2802–2815. [Google Scholar] [CrossRef] - Titarev, V.A.; Toro, E.F. ADER schemes for three-dimensional non-linear hyperbolic systems. J. Comput. Phys.
**2005**, 204, 715–736. [Google Scholar] [CrossRef] - Zhang, Z.; Li, D.; Cheng, Y.; Liew, K. The improved element-free Galerkin method for three-dimensional wave equation. Acta Mech. Sin.
**2012**, 28, 808–818. [Google Scholar] [CrossRef] - Shivanian, E. Meshless local Petrov-Galerkin (MLPG) method for three-dimensional nonlinear wave equations via moving least squares approximation. Eng. Anal. Bound. Elem.
**2015**, 50, 249–257. [Google Scholar] [CrossRef] - Shukla, H.S.; Tamsir, M.; Jiwari, R.; Srivastava, V.K. A numerical algorithm for computation modelling of 3D nonlinear wave equations based on exponential modified cubic B-spline differential quadrature method. Int. J. Comput. Math.
**2018**, 95, 752–766. [Google Scholar] [CrossRef] - Bellman, R.; Kashef, B.G.; Casti, J. Differential quadrature: A technique for the rapid solution of nonlinear differential equations. J. Comput. Phy.
**1972**, 10, 40–52. [Google Scholar] [CrossRef] - Korkmaz, A.; Dag, I. Shock wave simulations using sinc differential quadrature method. Eng. Comput.
**2011**, 28, 654–674. [Google Scholar] [CrossRef] - Shu, C.; Chew, Y.T. Fourier expansion-based differential quadrature and its application to Helmholtz eigenvalue problems. Commun. Numer. Methods Eng.
**1997**, 13, 643–653. [Google Scholar] [CrossRef] - Shu, C.; Xue, H. Explicit computation of weighting coefficients in the harmonic differential quadrature. J. Sound Vib.
**1997**, 204, 549–555. [Google Scholar] [CrossRef] - Bashan, A.; Karakoc, S.B.G.; Geyikli, T. Approximation of the KdVB equation by the quintic B-spline differential quadrature method. Kuwait J. Sci.
**2015**, 42, 67–92. [Google Scholar] - Korkmaz, A.; Dag, I. Cubic B-spline differential quadrature methods and stability for Burgers equation. Eng. Comput.
**2013**, 30, 320–344. [Google Scholar] [CrossRef] - Shukla, H.S.; Tamsir, M.; Srivastava, V.K.; Kumar, J. Numerical solution of two dimensional coupled viscous Burger equation using modified cubic B-spline differential quadrature method. AIP Adv.
**2014**, 4, 117134. [Google Scholar] [CrossRef] - Shukla, H.S.; Tamsir, M.; Srivastava, V.K. Numerical simulation of two dimensional sine-Gordon solitons using modified cubic B-spline differential quadrature method. AIP Adv.
**2015**, 5, 017121. [Google Scholar] [CrossRef] - Korkmaz, A.; Dag, I. Cubic B-spline differential quadrature methods for the advection-diffusion equation. Int. J. Numer. Meth. Heat Fluid Flow
**2012**, 22, 1021–1036. [Google Scholar] [CrossRef] - Tamsir, M.; Srivastava, V.K.; Jiwari, R. An algorithm based on exponential modified cubic B-spline differential quadrature method for nonlinear Burgers’ equation. Appl. Math. Comput.
**2016**, 290, 111–124. [Google Scholar] [CrossRef] - Jiwari, R.; Pandit, S.; Mittal, R.C. Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method. Comput. Phys. Commun.
**2012**, 183, 600–616. [Google Scholar] [CrossRef] - Korkmaz, A.; Dag, I. Numerical simulations of boundary-forced RLW equation with cubic B-spline-based differential quadrature methods. Arab. J. Sci. Eng.
**2013**, 38, 1151–1160. [Google Scholar] [CrossRef] - Korkmaz, A.; Dag, I. Quartic and quintic B-spline methods for advection diffusion equation. Appl. Math. Comput.
**2016**, 274, 208–219. [Google Scholar] [CrossRef] - Jiwari, R. Lagrange interpolation and modified cubic B-spline differential quadrature methods for solving hyperbolic partial differential equations with Dirichlet and Neumann boundary conditions. Comput. Phys. Commun.
**2015**, 193, 55–65. [Google Scholar] [CrossRef] - Lin, J.; Reutskiy, S. A cubic B-spline semi-analytical algorithm for simulation of 3D steady-state convection-diffusion-reaction problems. Appl. Math. Comput.
**2020**, 371, 124944. [Google Scholar] [CrossRef] - Ali, I.; Seadawy, A.R.; Rizvi, S.T.R.; Younis, M.; Ali, K. Conserved quantities along with Painlevé analysis and optical solitons for the nonlinear dynamics of Heisenberg ferromagnetic spin chains model. Int. J. Mod. Phys. B
**2020**, 34, 2050283. [Google Scholar] [CrossRef] - Lu, D.; Seadwy, A.R.; Iqbal, M. Mathematical methods via construction of traveling and solitary wave solutions of three coupled system of nonlinear partial differential equations and their applications. Res. Phys.
**2018**, 11, 1161–1171. [Google Scholar] [CrossRef] - Seadawy, A.R.; Khalid, K.A.; Nuruddeen, R.I. A variety of soliton solutions for the fractional Wazwaz-Benjamin-Bona-Mahony equations. Res. Phys.
**2019**, 12, 2234–2241. [Google Scholar] - Akram, U.; Seadawy, A.R.; Rizvi, S.T.R.; Younis, M.; Althobaiti, S.; Sayed, S. Traveling wave solutions for the fractional Wazwaz–Benjamin–Bona–Mahony model in arising shallow water waves. Res. Phys.
**2021**, 20, 103725. [Google Scholar] [CrossRef] - Ahlberg, J.H.; Nilson, E.N.; Walsh, J.L. The Theory of Splines and Their Applications; Academic Press: New York, NY, USA, 1967. [Google Scholar]
- Lu, C. Error analysis for interpolating complex cubic splines with deficiency 2. J. Approx. Theory
**1982**, 36, 183–196. [Google Scholar] [CrossRef] [Green Version] - Kapoor, M.; Joshi, V. Numerical approximation of 1D and 2D non-linear Schrödinger equations by implementing modified cubic Hyperbolic B-spline based DQM. Part. Diff. Eq. Appl. Math.
**2021**, 4, 100076. [Google Scholar] [CrossRef] - Shu, C. Differential Quadrature and its Application in Engineering, 1st ed.; Athenaeum Press Ltd.: Newcastle Upon Tyne, UK, 2000. [Google Scholar]
- Gottlieb, S.; Ketcheson, D.I.; Shu, C.W. High order strong stability preserving time discretizations. J. Sci. Comput.
**2009**, 38, 251–289. [Google Scholar] [CrossRef]

**Figure 3.**The analytical (

**a**) and numerical (

**b**) solutions with $h=0.05$ and $\mathsf{\Delta}t=0.01$ for $z=0.5$ at $t=1$.

**Figure 5.**The analytical (

**a**) and numerical (

**b**) solutions with $h=0.05$ and $\mathsf{\Delta}t=0.01$ for $z=1$ at $t=1$.

**Figure 6.**The absolute error norms with $h=$ 0.05 and $\mathsf{\Delta}t=$ 0.01 for $z=$ 0.5 at $t=1$.

**Figure 7.**The numerical (

**a**) and analytical (

**b**) solutions with $h=$ 0.05 and $\mathsf{\Delta}t=$ 0.01 for $z=$ 0.5 at $t=1$.

${x}_{i-2}$ | ${x}_{i-1}$ | ${x}_{i}$ | ${x}_{i+1}$ | ${x}_{i+2}$ | |

$H{f}_{i}\left(x\right)$ | 0 | ${\mathsf{\Upsilon}}_{1}$ | ${\mathsf{\Upsilon}}_{2}$ | ${\mathsf{\Upsilon}}_{3}$ | 0 |

$H{{f}^{\prime}}_{i}\left(x\right)$ | 0 | ${\mathsf{\Upsilon}}_{4}$ | 0 | ${\mathsf{\Upsilon}}_{5}$ | 0 |

**Table 2.**Comparison between the present method and existing methods with $h=0.1$ and $\mathsf{\Delta}t=0.01$ at different values of $t$ for Example 1.

$\mathit{t}$ | Present Method | EFG Method [16] | MLPG Method [16] | Expo-MCBDQM [17] |
---|---|---|---|---|

0.1 | 9.131 × 10^{−7} | 1.361376 × 10^{−1} | 6.389040 × 10^{−4} | 1.013 × 10^{−6} |

0.2 | 9.126 × 10^{−7} | 1.108673 × 10^{−1} | 1.621007 × 10^{−3} | 1.666 × 10^{−6} |

0.3 | 9.751 × 10^{−7} | 9.031794 × 10^{−2} | 2.069397 × 10^{−3} | 1.725 × 10^{−6} |

0.4 | 9.357 × 10^{−7} | 7.555177 × 10^{−2} | 1.851491 × 10^{−3} | 1.498 × 10^{−6} |

0.5 | 9.263 × 10^{−7} | 6.113317 × 10^{−2} | 1.406413 × 10^{−3} | 1.196 × 10^{−6} |

0.6 | 8.105 × 10^{−7} | 5.076050 × 10^{−2} | 1.120239 × 10^{−3} | 9.059 × 10^{−7} |

0.7 | 6.102 × 10^{−7} | 4.276296 × 10^{−2} | 8.762877 × 10^{−4} | 7.061 × 10^{−7} |

0.8 | 4.458 × 10^{−7} | 3.416178 × 10^{−2} | 5.762842 × 10^{−4} | 5.566 × 10^{−7} |

0.9 | 3.614 × 10^{−7} | 3.072394 × 10^{−2} | 7.778958 × 10^{−4} | 4.758 × 10^{−7} |

1.0 | 3.326 × 10^{−7} | 2.562088 × 10^{−2} | 8.638225 × 10^{−4} | 4.417 × 10^{−7} |

**Table 3.**Comparison between the present method and existing methods with $h=$ 0.1 and $\mathsf{\Delta}t=0.01$ at different values of $t$ for Example 2.

$\mathit{t}$ | Present Method | EFG Method [16] | MLPG Method [17] | Expo-MCBDQM [17] |
---|---|---|---|---|

0.1 | 4.472 × 10^{−6} | 1.653265 × 10^{0} | 2.777931 × 10^{−3} | 5.667 × 10^{−6} |

0.2 | 8.511 × 10^{−6} | 1.005632 × 10^{0} | 8.477482 × 10^{−3} | 9.701 × 10^{−6} |

0.3 | 2.23 × 10^{−6} | 9.786343 × 10^{−1} | 1.352534 × 10^{−2} | 1.231 × 10^{−5} |

0.4 | 5.02 × 10^{−6} | 7.456237 × 10^{−1} | 1.583307 × 10^{−2} | 1.512 × 10^{−5} |

0.5 | 1.143 × 10^{−5} | 6.213675 × 10^{−1} | 1.550351 × 10^{−2} | 1.824 × 10^{−5} |

0.6 | 1.062 × 10^{−5} | 4.354421 × 10^{−1} | 1.367202 × 10^{−2} | 2.222 × 10^{−5} |

0.7 | 1.231 × 10^{−5} | 1.345213 × 10^{−1} | 1.052578 × 10^{−2} | 2.570 × 10^{−5} |

0.8 | 1.324 × 10^{−5} | 9.973233 × 10^{−2} | 6.216680 × 10^{−3} | 2.866 × 10^{−5} |

0.9 | 2.014 × 10^{−5} | 7.132423 × 10^{−2} | 5.280951 × 10^{−3} | 3.117 × 10^{−5} |

1.0 | 2.1025 × 10^{−5} | 6.124572 × 10^{−2} | 2.276681 × 10^{−3} | 3.329 × 10^{−5} |

**Table 4.**Comparison between the present method and existing methods with $h=0.1$ and $\mathsf{\Delta}t=0.01$ at different values of $t$ for Example 3.

$\mathit{t}$ | Present Method | EFG Method [16] | MLPG Method [16] | Expo-MCBDQM [17] |
---|---|---|---|---|

0.1 | 1.673 × 10^{−7} | 1.435666 × 10^{−3} | 8.903029 × 10^{−5} | 2.887 × 10^{−7} |

0.2 | 2.481 × 10^{−7} | 3.867576 × 10^{−3} | 9.910264 × 10^{−5} | 1.257 × 10^{−6} |

0.3 | 1.522 × 10^{−6} | 5.033494 × 10^{−3} | 1.590358 × 10^{−4} | 2.944 × 10^{−6} |

0.4 | 4.137 × 10^{−6} | 7.655177 × 10^{−3} | 3.776687 × 10^{−4} | 5.348 × 10^{−6} |

0.5 | 7.465 × 10^{−6} | 9.119769 × 10^{−3} | 4.781290 × 10^{−4} | 8.787 × 10^{−6} |

0.6 | 3.304 × 10^{−6} | 1.034540 × 10^{−2} | 6.416380 × 10^{−4} | 1.361 × 10^{−5} |

0.7 | 1.007 × 10^{−5} | 3.279875 × 10^{−2} | 8.809498 × 10^{−4} | 2.029 × 10^{−5} |

0.8 | 1.716 × 10^{−5} | 5.233178 × 10^{−2} | 9.279331 × 10^{−4} | 2.918 × 10^{−5} |

0.9 | 3.014 × 10^{−5} | 6.072234 × 10^{−2} | 1.059260 × 10^{−4} | 4.049 × 10^{−5} |

1.0 | 4.221 × 10^{−5} | 7.545088 × 10^{−2} | 1.529316 × 10^{−3} | 5.432 × 10^{−5} |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tamsir, M.; Meetei, M.Z.; Msmali, A.H.
Hyperbolic B-Spline Function-Based Differential Quadrature Method for the Approximation of 3D Wave Equations. *Axioms* **2022**, *11*, 597.
https://doi.org/10.3390/axioms11110597

**AMA Style**

Tamsir M, Meetei MZ, Msmali AH.
Hyperbolic B-Spline Function-Based Differential Quadrature Method for the Approximation of 3D Wave Equations. *Axioms*. 2022; 11(11):597.
https://doi.org/10.3390/axioms11110597

**Chicago/Turabian Style**

Tamsir, Mohammad, Mutum Zico Meetei, and Ahmed H. Msmali.
2022. "Hyperbolic B-Spline Function-Based Differential Quadrature Method for the Approximation of 3D Wave Equations" *Axioms* 11, no. 11: 597.
https://doi.org/10.3390/axioms11110597