Exact differential transform method and semi-analytic Chebyshev collocation method for ordinary differential equations
Abstract This article introduces two algorithms designed for solving differential equations: the differential transform method and the second-kind Chebyshev collocation method. It provides a detailed explanation of the structure and key properties of each method. The differential transform method wo...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2025-03-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13661-025-02019-5 |
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| Summary: | Abstract This article introduces two algorithms designed for solving differential equations: the differential transform method and the second-kind Chebyshev collocation method. It provides a detailed explanation of the structure and key properties of each method. The differential transform method works by converting the differential equation along with the initial or boundary conditions into a series using an iterative process, which ultimately leads to the exact solution of the problem. On the other hand, the Chebyshev collocation method transforms the differential equation into a system of equations involving unknown coefficients, which must be determined by solving the system. This process utilizes the properties of matrices to facilitate the modification. Furthermore, the article carefully analyzes the convergence behavior and error estimation of both methods, ensuring their reliability and precision. Several test problems are solved using these algorithms, and the absolute errors are compared against those of alternative solution methods. The results of these comparisons demonstrate that the proposed algorithms provide the most accurate solutions when compared to other methods currently available. |
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| ISSN: | 1687-2770 |