Direct Power Series Approach for Solving Nonlinear Initial Value Problems
In this research, a new approach for solving fractional initial value problems is presented. The main goal of this study focuses on establishing direct formulas to find the coefficients of approximate series solutions of target problems. The new method provides analytical series solutions for both f...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-01-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/2/111 |
| _version_ | 1797622457803210752 |
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| author | Emad Salah Rania Saadeh Ahmad Qazza Raed Hatamleh |
| author_facet | Emad Salah Rania Saadeh Ahmad Qazza Raed Hatamleh |
| author_sort | Emad Salah |
| collection | DOAJ |
| description | In this research, a new approach for solving fractional initial value problems is presented. The main goal of this study focuses on establishing direct formulas to find the coefficients of approximate series solutions of target problems. The new method provides analytical series solutions for both fractional and ordinary differential equations or systems directly, without complicated computations. To show the reliability and efficiency of the presented technique, interesting examples of systems and fractional linear and nonlinear differential equations of ordinary and fractional orders are presented and solved directly by the new approach. This new method is faster and better than other analytical methods in establishing many terms of analytic solutions. The main motivation of this work is to introduce general new formulas that express the series solutions of some types of differential equations in a simple way and with less calculations compared to other numerical power series methods, that is, there is no need for differentiation, discretization, or taking limits while constructing the approximate solution. |
| first_indexed | 2024-03-11T09:10:34Z |
| format | Article |
| id | doaj.art-f27093a7cbd74780bd258187e4261288 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T09:10:34Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-f27093a7cbd74780bd258187e42612882023-11-16T19:05:24ZengMDPI AGAxioms2075-16802023-01-0112211110.3390/axioms12020111Direct Power Series Approach for Solving Nonlinear Initial Value ProblemsEmad Salah0Rania Saadeh1Ahmad Qazza2Raed Hatamleh3Department of Mathematics, Zarqa University, Zarqa 13110, JordanDepartment of Mathematics, Zarqa University, Zarqa 13110, JordanDepartment of Mathematics, Zarqa University, Zarqa 13110, JordanDepartment of Mathematics, Jadara University, Irbid 21111, JordanIn this research, a new approach for solving fractional initial value problems is presented. The main goal of this study focuses on establishing direct formulas to find the coefficients of approximate series solutions of target problems. The new method provides analytical series solutions for both fractional and ordinary differential equations or systems directly, without complicated computations. To show the reliability and efficiency of the presented technique, interesting examples of systems and fractional linear and nonlinear differential equations of ordinary and fractional orders are presented and solved directly by the new approach. This new method is faster and better than other analytical methods in establishing many terms of analytic solutions. The main motivation of this work is to introduce general new formulas that express the series solutions of some types of differential equations in a simple way and with less calculations compared to other numerical power series methods, that is, there is no need for differentiation, discretization, or taking limits while constructing the approximate solution.https://www.mdpi.com/2075-1680/12/2/111fractional differential equationsanalytical solutionpower seriesCaputo derivative |
| spellingShingle | Emad Salah Rania Saadeh Ahmad Qazza Raed Hatamleh Direct Power Series Approach for Solving Nonlinear Initial Value Problems Axioms fractional differential equations analytical solution power series Caputo derivative |
| title | Direct Power Series Approach for Solving Nonlinear Initial Value Problems |
| title_full | Direct Power Series Approach for Solving Nonlinear Initial Value Problems |
| title_fullStr | Direct Power Series Approach for Solving Nonlinear Initial Value Problems |
| title_full_unstemmed | Direct Power Series Approach for Solving Nonlinear Initial Value Problems |
| title_short | Direct Power Series Approach for Solving Nonlinear Initial Value Problems |
| title_sort | direct power series approach for solving nonlinear initial value problems |
| topic | fractional differential equations analytical solution power series Caputo derivative |
| url | https://www.mdpi.com/2075-1680/12/2/111 |
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