Method of Particular Solutions for Second-Order Differential Equation with Variable Coefficients via Orthogonal Polynomials
In this paper, with classic Legendre polynomials, a method of particular solutions (MPS, for short) is proposed to solve a kind of second-order differential equations with a variable coefficient on a unit interval. The particular solutions, satisfying the natural Dirichlet boundary conditions, are c...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2023-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2023/9748605 |
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| _version_ | 1832548025993527296 |
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| author | Huantian Xie Zhaozhong Zhang Ziwu Jiang Jianwei Zhou |
| author_facet | Huantian Xie Zhaozhong Zhang Ziwu Jiang Jianwei Zhou |
| author_sort | Huantian Xie |
| collection | DOAJ |
| description | In this paper, with classic Legendre polynomials, a method of particular solutions (MPS, for short) is proposed to solve a kind of second-order differential equations with a variable coefficient on a unit interval. The particular solutions, satisfying the natural Dirichlet boundary conditions, are constructed with orthogonal Legendre polynomials for the variable coefficient case. Meanwhile, we investigate the a-priori error estimates of the MPS approximations. Two a-priori error estimations in H1- and L∞-norms are shown to depict the convergence order of numerical approximations, respectively. Some numerical examples and convergence rates are provided to validate the merits of our proposed meshless method. |
| format | Article |
| id | doaj-art-f3efe6a2e8804f6793c1f4e798fe0012 |
| institution | Kabale University |
| issn | 2314-8888 |
| language | English |
| publishDate | 2023-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-f3efe6a2e8804f6793c1f4e798fe00122025-02-03T06:42:45ZengWileyJournal of Function Spaces2314-88882023-01-01202310.1155/2023/9748605Method of Particular Solutions for Second-Order Differential Equation with Variable Coefficients via Orthogonal PolynomialsHuantian Xie0Zhaozhong Zhang1Ziwu Jiang2Jianwei Zhou3School of Mathematics and StatisticsSchool of Mathematics and StatisticsSchool of Mathematics and StatisticsSchool of Mathematics and StatisticsIn this paper, with classic Legendre polynomials, a method of particular solutions (MPS, for short) is proposed to solve a kind of second-order differential equations with a variable coefficient on a unit interval. The particular solutions, satisfying the natural Dirichlet boundary conditions, are constructed with orthogonal Legendre polynomials for the variable coefficient case. Meanwhile, we investigate the a-priori error estimates of the MPS approximations. Two a-priori error estimations in H1- and L∞-norms are shown to depict the convergence order of numerical approximations, respectively. Some numerical examples and convergence rates are provided to validate the merits of our proposed meshless method.http://dx.doi.org/10.1155/2023/9748605 |
| spellingShingle | Huantian Xie Zhaozhong Zhang Ziwu Jiang Jianwei Zhou Method of Particular Solutions for Second-Order Differential Equation with Variable Coefficients via Orthogonal Polynomials Journal of Function Spaces |
| title | Method of Particular Solutions for Second-Order Differential Equation with Variable Coefficients via Orthogonal Polynomials |
| title_full | Method of Particular Solutions for Second-Order Differential Equation with Variable Coefficients via Orthogonal Polynomials |
| title_fullStr | Method of Particular Solutions for Second-Order Differential Equation with Variable Coefficients via Orthogonal Polynomials |
| title_full_unstemmed | Method of Particular Solutions for Second-Order Differential Equation with Variable Coefficients via Orthogonal Polynomials |
| title_short | Method of Particular Solutions for Second-Order Differential Equation with Variable Coefficients via Orthogonal Polynomials |
| title_sort | method of particular solutions for second order differential equation with variable coefficients via orthogonal polynomials |
| url | http://dx.doi.org/10.1155/2023/9748605 |
| work_keys_str_mv | AT huantianxie methodofparticularsolutionsforsecondorderdifferentialequationwithvariablecoefficientsviaorthogonalpolynomials AT zhaozhongzhang methodofparticularsolutionsforsecondorderdifferentialequationwithvariablecoefficientsviaorthogonalpolynomials AT ziwujiang methodofparticularsolutionsforsecondorderdifferentialequationwithvariablecoefficientsviaorthogonalpolynomials AT jianweizhou methodofparticularsolutionsforsecondorderdifferentialequationwithvariablecoefficientsviaorthogonalpolynomials |