{"title":"Numerical solutions of multidimensional nonlinear hyperbolic partial integro-differential equations using a redefined cubic B-spline-based differential quadrature method","authors":"Raziyeh Mirzahashemi, Mohammad Heydari","doi":"10.1016/j.camwa.2025.08.018","DOIUrl":null,"url":null,"abstract":"<div><div>The primary objective of this work is to present an efficient numerical method for solving multidimensional nonlinear hyperbolic partial integro-differential equations (HPIDEs). To implement this method, the following steps are sequentially followed. First, by integrating both sides of the HPIDE, we transform it into a new form of a partial integro-differential equation with a time derivative of first order. This new formulation allows the proposed method to ultimately reduce the problem of finding an approximate solution to a system of linear algebraic equations without employing linearization techniques. Next, for the discretization of the newly obtained form in temporal direction, a combination of the Crank–Nicolson finite difference technique and numerical integration methods, including the trapezoidal and rectangle integration rules, is utilized. This process results in a finite difference scheme with second-order convergence, and its stability and convergence are thoroughly examined using energy method. The Richardson extrapolation technique is also utilized to improve the convergence order in the temporal dimension. Furthermore, a differential quadrature method (DQM) based on a redefined structure of cubic B-splines is employed for the spatial discretization of the problem. Finally, some numerical examples in various dimensions are provided to evaluate the accuracy and effectiveness of the proposed method.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"198 ","pages":"Pages 214-238"},"PeriodicalIF":2.5000,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computers & Mathematics with Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0898122125003505","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
The primary objective of this work is to present an efficient numerical method for solving multidimensional nonlinear hyperbolic partial integro-differential equations (HPIDEs). To implement this method, the following steps are sequentially followed. First, by integrating both sides of the HPIDE, we transform it into a new form of a partial integro-differential equation with a time derivative of first order. This new formulation allows the proposed method to ultimately reduce the problem of finding an approximate solution to a system of linear algebraic equations without employing linearization techniques. Next, for the discretization of the newly obtained form in temporal direction, a combination of the Crank–Nicolson finite difference technique and numerical integration methods, including the trapezoidal and rectangle integration rules, is utilized. This process results in a finite difference scheme with second-order convergence, and its stability and convergence are thoroughly examined using energy method. The Richardson extrapolation technique is also utilized to improve the convergence order in the temporal dimension. Furthermore, a differential quadrature method (DQM) based on a redefined structure of cubic B-splines is employed for the spatial discretization of the problem. Finally, some numerical examples in various dimensions are provided to evaluate the accuracy and effectiveness of the proposed method.
期刊介绍:
Computers & Mathematics with Applications provides a medium of exchange for those engaged in fields contributing to building successful simulations for science and engineering using Partial Differential Equations (PDEs).