Solving Boundary Value Problems for a Class of Differential Equations Based on Elastic Transformation and Similar Construction Methods
To address the boundary value problem associated with a class of third-order nonlinear differential equations with variable coefficients, this study integrates three key methods: the elastic transformation method (ETM), the similar construction method (SCM), and the elastic inverse transformation me...
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MDPI AG
2025-04-01
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| Series: | AppliedMath |
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| author | Jinfeng Liu Pengshe Zheng Jiajia Xie |
| author_facet | Jinfeng Liu Pengshe Zheng Jiajia Xie |
| author_sort | Jinfeng Liu |
| collection | DOAJ |
| description | To address the boundary value problem associated with a class of third-order nonlinear differential equations with variable coefficients, this study integrates three key methods: the elastic transformation method (ETM), the similar construction method (SCM), and the elastic inverse transformation method (EITM). Firstly, ETM is employed to transform the original high-order nonlinear differential equations into the Tschebycheff equation, successfully reducing the order of the problem. Subsequently, SCM is applied to determine the general solution of the Tschebycheff equation under boundary conditions, thereby ensuring a structured and systematic approach. Ultimately, the EITM is used to reconstruct the solution of the original third-order nonlinear differential equation. The accuracy of the obtained solution is further validated by analyzing the corresponding solution curves. The synergy of these methods introduces a novel approach to solving nonlinear differential equations and extends the application of Tschebycheff equations in nonlinear systems. |
| format | Article |
| id | doaj-art-0dcce6de06764a17a1181bc7c8b5595f |
| institution | Kabale University |
| issn | 2673-9909 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | AppliedMath |
| spelling | doaj-art-0dcce6de06764a17a1181bc7c8b5595f2025-08-20T03:32:28ZengMDPI AGAppliedMath2673-99092025-04-01524110.3390/appliedmath5020041Solving Boundary Value Problems for a Class of Differential Equations Based on Elastic Transformation and Similar Construction MethodsJinfeng Liu0Pengshe Zheng1Jiajia Xie2School of Science, Xihua University, Chengdu 610039, ChinaSchool of Science, Xihua University, Chengdu 610039, ChinaSchool of Science, Xihua University, Chengdu 610039, ChinaTo address the boundary value problem associated with a class of third-order nonlinear differential equations with variable coefficients, this study integrates three key methods: the elastic transformation method (ETM), the similar construction method (SCM), and the elastic inverse transformation method (EITM). Firstly, ETM is employed to transform the original high-order nonlinear differential equations into the Tschebycheff equation, successfully reducing the order of the problem. Subsequently, SCM is applied to determine the general solution of the Tschebycheff equation under boundary conditions, thereby ensuring a structured and systematic approach. Ultimately, the EITM is used to reconstruct the solution of the original third-order nonlinear differential equation. The accuracy of the obtained solution is further validated by analyzing the corresponding solution curves. The synergy of these methods introduces a novel approach to solving nonlinear differential equations and extends the application of Tschebycheff equations in nonlinear systems.https://www.mdpi.com/2673-9909/5/2/41elastic transformation methodsimilar construction methodelastic inverse transformation methodTschebycheff equation |
| spellingShingle | Jinfeng Liu Pengshe Zheng Jiajia Xie Solving Boundary Value Problems for a Class of Differential Equations Based on Elastic Transformation and Similar Construction Methods AppliedMath elastic transformation method similar construction method elastic inverse transformation method Tschebycheff equation |
| title | Solving Boundary Value Problems for a Class of Differential Equations Based on Elastic Transformation and Similar Construction Methods |
| title_full | Solving Boundary Value Problems for a Class of Differential Equations Based on Elastic Transformation and Similar Construction Methods |
| title_fullStr | Solving Boundary Value Problems for a Class of Differential Equations Based on Elastic Transformation and Similar Construction Methods |
| title_full_unstemmed | Solving Boundary Value Problems for a Class of Differential Equations Based on Elastic Transformation and Similar Construction Methods |
| title_short | Solving Boundary Value Problems for a Class of Differential Equations Based on Elastic Transformation and Similar Construction Methods |
| title_sort | solving boundary value problems for a class of differential equations based on elastic transformation and similar construction methods |
| topic | elastic transformation method similar construction method elastic inverse transformation method Tschebycheff equation |
| url | https://www.mdpi.com/2673-9909/5/2/41 |
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