{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T19:33:33Z","timestamp":1772652813827,"version":"3.50.1"},"reference-count":41,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,10,28]],"date-time":"2022-10-28T00:00:00Z","timestamp":1666915200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We propose a differential quadrature method (DQM) based on cubic hyperbolic B-spline basis functions for computing 3D wave equations. This method converts the problem into a system of ODEs. We use an optimum five-stage and order four SSP Runge-Kutta (SSPRK-(5,4)) scheme to solve the obtained system of ODEs. The matrix stability analysis is also investigated. The accuracy and efficiency of the proposed method are demonstrated via three numerical examples. It has been found that the proposed method gives more accurate results than the existing methods. The main purpose of this work is to present an accurate, economically easy-to-implement, and stable technique for solving hyperbolic partial differential equations.<\/jats:p>","DOI":"10.3390\/axioms11110597","type":"journal-article","created":{"date-parts":[[2022,10,29]],"date-time":"2022-10-29T23:45:00Z","timestamp":1667087100000},"page":"597","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Hyperbolic B-Spline Function-Based Differential Quadrature Method for the Approximation of 3D Wave Equations"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6389-4879","authenticated-orcid":false,"given":"Mohammad","family":"Tamsir","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, Jazan University, Jazan 45142, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9168-5126","authenticated-orcid":false,"given":"Mutum Zico","family":"Meetei","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Jazan University, Jazan 45142, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ahmed H.","family":"Msmali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Jazan University, Jazan 45142, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,28]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"Fractional Laplacian viscoacoustic wave equation low-rank temporal extrapolation","volume":"99","author":"Chen","year":"2019","journal-title":"IEEE Access"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.enganabound.2017.03.012","article-title":"Generalized finite difference method for two-dimensional shallow water equations","volume":"80","author":"Li","year":"2017","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"113331","DOI":"10.1016\/j.cam.2020.113331","article-title":"A high-order space-time ultra-weak discontinuous Galerkin method for the second-order wave equation in one space dimension","volume":"389","author":"Baccouch","year":"2021","journal-title":"J. Comput. Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"113337","DOI":"10.1016\/j.cam.2020.113337","article-title":"A novel finite volume method for the nonlinear two-sided space distributed-order diffusion equation with variable coefficients","volume":"388","author":"Yang","year":"2021","journal-title":"J. Comput. Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"113532","DOI":"10.1016\/j.cma.2020.113532","article-title":"An adaptive model order reduction method for boundary element-based multi-frequency acoustic wave problems","volume":"373","author":"Xie","year":"2021","journal-title":"Comput. Meth. Appl. Mech. Engin."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.cageo.2018.05.011","article-title":"A mesh-free finite-difference method for elastic wave propagation in the frequency-domain","volume":"118","author":"Takekawa","year":"2018","journal-title":"Comput. Geosci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1016\/j.jcp.2018.11.031","article-title":"Combining finite element and finite difference methods for isotropic elastic wave simulations in an energy-conserving manner","volume":"378","author":"Gao","year":"2019","journal-title":"J. Comput. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"979","DOI":"10.4208\/cicp.OA-2020-0119","article-title":"A broad class of conservative numerical methods for dispersive wave equations","volume":"29","author":"Ranocha","year":"2021","journal-title":"Commun. Comput. Phys."},{"key":"ref_9","first-page":"359","article-title":"A novel meshfree strategy for a viscous wave equation with variable coefficients","volume":"9","author":"Wang","year":"2021","journal-title":"Front. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1134\/S1995423919040013","article-title":"Numerical solution of a three-dimensional coefficient inverse problem for the wave equation with integral data in a cylindrical domain","volume":"12","author":"Bakushinsky","year":"2019","journal-title":"Numer. Analys. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1002\/num.20019","article-title":"On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation","volume":"21","author":"Dehghan","year":"2005","journal-title":"Numer. Meth. Part. Diff. Eq."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.matcom.2005.10.001","article-title":"Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices","volume":"71","author":"Dehghan","year":"2006","journal-title":"Math. Comput. Simul."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2802","DOI":"10.1016\/j.apm.2012.06.021","article-title":"A new off-step high order approximation for the solution of three-space dimensional nonlinear wave equations","volume":"37","author":"Mohanty","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1016\/j.jcp.2004.10.028","article-title":"ADER schemes for three-dimensional non-linear hyperbolic systems","volume":"204","author":"Titarev","year":"2005","journal-title":"J. Comput. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"808","DOI":"10.1007\/s10409-012-0083-x","article-title":"The improved element-free Galerkin method for three-dimensional wave equation","volume":"28","author":"Zhang","year":"2012","journal-title":"Acta Mech. Sin."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1016\/j.enganabound.2014.08.014","article-title":"Meshless local Petrov-Galerkin (MLPG) method for three-dimensional nonlinear wave equations via moving least squares approximation","volume":"50","author":"Shivanian","year":"2015","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"752","DOI":"10.1080\/00207160.2017.1296573","article-title":"A numerical algorithm for computation modelling of 3D nonlinear wave equations based on exponential modified cubic B-spline differential quadrature method","volume":"95","author":"Shukla","year":"2018","journal-title":"Int. J. Comput. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/0021-9991(72)90089-7","article-title":"Differential quadrature: A technique for the rapid solution of nonlinear differential equations","volume":"10","author":"Bellman","year":"1972","journal-title":"J. Comput. Phy."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"654","DOI":"10.1108\/02644401111154619","article-title":"Shock wave simulations using sinc differential quadrature method","volume":"28","author":"Korkmaz","year":"2011","journal-title":"Eng. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1002\/(SICI)1099-0887(199708)13:8<643::AID-CNM92>3.0.CO;2-F","article-title":"Fourier expansion-based differential quadrature and its application to Helmholtz eigenvalue problems","volume":"13","author":"Shu","year":"1997","journal-title":"Commun. Numer. Methods Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1006\/jsvi.1996.0894","article-title":"Explicit computation of weighting coefficients in the harmonic differential quadrature","volume":"204","author":"Shu","year":"1997","journal-title":"J. Sound Vib."},{"key":"ref_22","first-page":"67","article-title":"Approximation of the KdVB equation by the quintic B-spline differential quadrature method","volume":"42","author":"Bashan","year":"2015","journal-title":"Kuwait J. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1108\/02644401311314312","article-title":"Cubic B-spline differential quadrature methods and stability for Burgers equation","volume":"30","author":"Korkmaz","year":"2013","journal-title":"Eng. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"117134","DOI":"10.1063\/1.4902507","article-title":"Numerical solution of two dimensional coupled viscous Burger equation using modified cubic B-spline differential quadrature method","volume":"4","author":"Shukla","year":"2014","journal-title":"AIP Adv."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"017121","DOI":"10.1063\/1.4906256","article-title":"Numerical simulation of two dimensional sine-Gordon solitons using modified cubic B-spline differential quadrature method","volume":"5","author":"Shukla","year":"2015","journal-title":"AIP Adv."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1021","DOI":"10.1108\/09615531211271844","article-title":"Cubic B-spline differential quadrature methods for the advection-diffusion equation","volume":"22","author":"Korkmaz","year":"2012","journal-title":"Int. J. Numer. Meth. Heat Fluid Flow"},{"key":"ref_27","first-page":"111","article-title":"An algorithm based on exponential modified cubic B-spline differential quadrature method for nonlinear Burgers\u2019 equation","volume":"290","author":"Tamsir","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"600","DOI":"10.1016\/j.cpc.2011.12.004","article-title":"Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method","volume":"183","author":"Jiwari","year":"2012","journal-title":"Comput. Phys. Commun."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1151","DOI":"10.1007\/s13369-012-0353-8","article-title":"Numerical simulations of boundary-forced RLW equation with cubic B-spline-based differential quadrature methods","volume":"38","author":"Korkmaz","year":"2013","journal-title":"Arab. J. Sci. Eng."},{"key":"ref_30","first-page":"208","article-title":"Quartic and quintic B-spline methods for advection diffusion equation","volume":"274","author":"Korkmaz","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.cpc.2015.03.021","article-title":"Lagrange interpolation and modified cubic B-spline differential quadrature methods for solving hyperbolic partial differential equations with Dirichlet and Neumann boundary conditions","volume":"193","author":"Jiwari","year":"2015","journal-title":"Comput. Phys. Commun."},{"key":"ref_32","first-page":"124944","article-title":"A cubic B-spline semi-analytical algorithm for simulation of 3D steady-state convection-diffusion-reaction problems","volume":"371","author":"Lin","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2050283","DOI":"10.1142\/S0217979220502835","article-title":"Conserved quantities along with Painlev\u00e9 analysis and optical solitons for the nonlinear dynamics of Heisenberg ferromagnetic spin chains model","volume":"34","author":"Ali","year":"2020","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_34","first-page":"1161","article-title":"Mathematical methods via construction of traveling and solitary wave solutions of three coupled system of nonlinear partial differential equations and their applications","volume":"11","author":"Lu","year":"2018","journal-title":"Res. Phys."},{"key":"ref_35","first-page":"2234","article-title":"A variety of soliton solutions for the fractional Wazwaz-Benjamin-Bona-Mahony equations","volume":"12","author":"Seadawy","year":"2019","journal-title":"Res. Phys."},{"key":"ref_36","first-page":"103725","article-title":"Traveling wave solutions for the fractional Wazwaz\u2013Benjamin\u2013Bona\u2013Mahony model in arising shallow water waves","volume":"20","author":"Akram","year":"2021","journal-title":"Res. Phys."},{"key":"ref_37","unstructured":"Ahlberg, J.H., Nilson, E.N., and Walsh, J.L. (1967). The Theory of Splines and Their Applications, Academic Press."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/0021-9045(82)90039-9","article-title":"Error analysis for interpolating complex cubic splines with deficiency 2","volume":"36","author":"Lu","year":"1982","journal-title":"J. Approx. Theory"},{"key":"ref_39","first-page":"100076","article-title":"Numerical approximation of 1D and 2D non-linear Schr\u00f6dinger equations by implementing modified cubic Hyperbolic B-spline based DQM","volume":"4","author":"Kapoor","year":"2021","journal-title":"Part. Diff. Eq. Appl. Math."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Shu, C. (2000). Differential Quadrature and its Application in Engineering, Athenaeum Press Ltd.. [1st ed.].","DOI":"10.1007\/978-1-4471-0407-0"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1007\/s10915-008-9239-z","article-title":"High order strong stability preserving time discretizations","volume":"38","author":"Gottlieb","year":"2009","journal-title":"J. Sci. Comput."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/11\/597\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:04:41Z","timestamp":1760144681000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/11\/597"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,28]]},"references-count":41,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["axioms11110597"],"URL":"https:\/\/doi.org\/10.3390\/axioms11110597","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,28]]}}}