Taylor-Galerkin method for solving higher-order nonlinear complex differential equations
The Galerkin approach for numerically resolving higher-order Complex Differential Equations (CDEs) in a rectangular domain in the complex plane is presented in this work. Taylor polynomial functions are used in this method as basis or weighted functions. The CDE is converted into a matrix equation b...
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| Language: | English |
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Elsevier
2024-12-01
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| Series: | MethodsX |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2215016124005296 |
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| author | Md. Humayun Kabir Md. Shafiqul Islam Md. Kamrujjaman |
| author_facet | Md. Humayun Kabir Md. Shafiqul Islam Md. Kamrujjaman |
| author_sort | Md. Humayun Kabir |
| collection | DOAJ |
| description | The Galerkin approach for numerically resolving higher-order Complex Differential Equations (CDEs) in a rectangular domain in the complex plane is presented in this work. Taylor polynomial functions are used in this method as basis or weighted functions. The CDE is converted into a matrix equation by employing the proposed method. A system of linear and nonlinear equations with unknown Taylor coefficients for linear and nonlinear CDEs, respectively, is represented by the resultant matrix equation. Results pertaining to this method’s error analysis are discussed. The existing Taylor and Bessel Collocation methods are compared with the numerical results of the proposed method for linear CDEs, and the existing exact solutions and numerical results of the proposed method for nonlinear CDEs are also compared. The comparative results are displayed graphically for the real (ℜe) and imaginary (ℑm) parts, respectively, as well as in tabular form containing absolute error E(z) and maximum absolute error L∞norm. The methodology of this study focused on the Galerkin integral domain which is a rectangle shape in the complex plane and Taylor polynomial is the shape function. Matrix formulation procedure and iterative technique are implemented to find out the undetermined Taylor coefficients. |
| format | Article |
| id | doaj-art-e0cbe22b9ad845f09925db45e4bceb20 |
| institution | OA Journals |
| issn | 2215-0161 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Elsevier |
| record_format | Article |
| series | MethodsX |
| spelling | doaj-art-e0cbe22b9ad845f09925db45e4bceb202025-08-20T01:59:34ZengElsevierMethodsX2215-01612024-12-011310307810.1016/j.mex.2024.103078Taylor-Galerkin method for solving higher-order nonlinear complex differential equationsMd. Humayun Kabir0Md. Shafiqul Islam1Md. Kamrujjaman2Department of Mathematics, Bangabandhu Sheikh Mujibur Rahman University, Kishoreganj 2300, BangladeshDepartment of Applied Mathematics, University of Dhaka, Dhaka 1000, BangladeshCorresponding author.; Department of Mathematics, University of Dhaka, Dhaka 1000, BangladeshThe Galerkin approach for numerically resolving higher-order Complex Differential Equations (CDEs) in a rectangular domain in the complex plane is presented in this work. Taylor polynomial functions are used in this method as basis or weighted functions. The CDE is converted into a matrix equation by employing the proposed method. A system of linear and nonlinear equations with unknown Taylor coefficients for linear and nonlinear CDEs, respectively, is represented by the resultant matrix equation. Results pertaining to this method’s error analysis are discussed. The existing Taylor and Bessel Collocation methods are compared with the numerical results of the proposed method for linear CDEs, and the existing exact solutions and numerical results of the proposed method for nonlinear CDEs are also compared. The comparative results are displayed graphically for the real (ℜe) and imaginary (ℑm) parts, respectively, as well as in tabular form containing absolute error E(z) and maximum absolute error L∞norm. The methodology of this study focused on the Galerkin integral domain which is a rectangle shape in the complex plane and Taylor polynomial is the shape function. Matrix formulation procedure and iterative technique are implemented to find out the undetermined Taylor coefficients.http://www.sciencedirect.com/science/article/pii/S2215016124005296Complex differential equationsTaylor polynomialsGalerkin methodResidual error correction |
| spellingShingle | Md. Humayun Kabir Md. Shafiqul Islam Md. Kamrujjaman Taylor-Galerkin method for solving higher-order nonlinear complex differential equations MethodsX Complex differential equations Taylor polynomials Galerkin method Residual error correction |
| title | Taylor-Galerkin method for solving higher-order nonlinear complex differential equations |
| title_full | Taylor-Galerkin method for solving higher-order nonlinear complex differential equations |
| title_fullStr | Taylor-Galerkin method for solving higher-order nonlinear complex differential equations |
| title_full_unstemmed | Taylor-Galerkin method for solving higher-order nonlinear complex differential equations |
| title_short | Taylor-Galerkin method for solving higher-order nonlinear complex differential equations |
| title_sort | taylor galerkin method for solving higher order nonlinear complex differential equations |
| topic | Complex differential equations Taylor polynomials Galerkin method Residual error correction |
| url | http://www.sciencedirect.com/science/article/pii/S2215016124005296 |
| work_keys_str_mv | AT mdhumayunkabir taylorgalerkinmethodforsolvinghigherordernonlinearcomplexdifferentialequations AT mdshafiqulislam taylorgalerkinmethodforsolvinghigherordernonlinearcomplexdifferentialequations AT mdkamrujjaman taylorgalerkinmethodforsolvinghigherordernonlinearcomplexdifferentialequations |