Efficient Solution of Burgers’, Modified Burgers’ and KdV–Burgers’ Equations Using B-Spline Approximation Functions
This paper discusses the application of the orthogonal collocation on finite elements (OCFE) method using quadratic and cubic B-spline basis functions on partial differential equations. Collocation is performed at Gaussian points to obtain an optimal solution, hence the name orthogonal collocation....
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MDPI AG
2023-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/8/1847 |
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| author | Nabendra Parumasur Rasheed A. Adetona Pravin Singh |
| author_facet | Nabendra Parumasur Rasheed A. Adetona Pravin Singh |
| author_sort | Nabendra Parumasur |
| collection | DOAJ |
| description | This paper discusses the application of the orthogonal collocation on finite elements (OCFE) method using quadratic and cubic B-spline basis functions on partial differential equations. Collocation is performed at Gaussian points to obtain an optimal solution, hence the name orthogonal collocation. The method is used to solve various cases of Burgers’ equations, including the modified Burgers’ equation. The KdV–Burgers’ equation is considered as a test case for the OCFE method using cubic splines. The results compare favourably with existing results. The stability and convergence of the method are also given consideration. The method is unconditionally stable and second-order accurate in time and space. |
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| format | Article |
| id | doaj.art-43ca72cf9c584d0588ed12ba143e9d9a |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T04:46:40Z |
| publishDate | 2023-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-43ca72cf9c584d0588ed12ba143e9d9a2023-11-17T20:17:24ZengMDPI AGMathematics2227-73902023-04-01118184710.3390/math11081847Efficient Solution of Burgers’, Modified Burgers’ and KdV–Burgers’ Equations Using B-Spline Approximation FunctionsNabendra Parumasur0Rasheed A. Adetona1Pravin Singh2School of Mathematics Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South AfricaSchool of Mathematics Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South AfricaSchool of Mathematics Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South AfricaThis paper discusses the application of the orthogonal collocation on finite elements (OCFE) method using quadratic and cubic B-spline basis functions on partial differential equations. Collocation is performed at Gaussian points to obtain an optimal solution, hence the name orthogonal collocation. The method is used to solve various cases of Burgers’ equations, including the modified Burgers’ equation. The KdV–Burgers’ equation is considered as a test case for the OCFE method using cubic splines. The results compare favourably with existing results. The stability and convergence of the method are also given consideration. The method is unconditionally stable and second-order accurate in time and space.https://www.mdpi.com/2227-7390/11/8/1847B-splineorthogonal collocationfinite elementBurgers’ equation |
| spellingShingle | Nabendra Parumasur Rasheed A. Adetona Pravin Singh Efficient Solution of Burgers’, Modified Burgers’ and KdV–Burgers’ Equations Using B-Spline Approximation Functions Mathematics B-spline orthogonal collocation finite element Burgers’ equation |
| title | Efficient Solution of Burgers’, Modified Burgers’ and KdV–Burgers’ Equations Using B-Spline Approximation Functions |
| title_full | Efficient Solution of Burgers’, Modified Burgers’ and KdV–Burgers’ Equations Using B-Spline Approximation Functions |
| title_fullStr | Efficient Solution of Burgers’, Modified Burgers’ and KdV–Burgers’ Equations Using B-Spline Approximation Functions |
| title_full_unstemmed | Efficient Solution of Burgers’, Modified Burgers’ and KdV–Burgers’ Equations Using B-Spline Approximation Functions |
| title_short | Efficient Solution of Burgers’, Modified Burgers’ and KdV–Burgers’ Equations Using B-Spline Approximation Functions |
| title_sort | efficient solution of burgers modified burgers and kdv burgers equations using b spline approximation functions |
| topic | B-spline orthogonal collocation finite element Burgers’ equation |
| url | https://www.mdpi.com/2227-7390/11/8/1847 |
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