A novel analytical Aboodh residual power series method for solving linear and nonlinear time-fractional partial differential equations with variable coefficients
The goal of this research is to develop a novel analytic technique for obtaining the approximate and exact solutions of the Caputo time-fractional partial differential equations (PDEs) with variable coefficients. We call this technique as the Aboodh residual power series method (ARPSM), because it a...
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AIMS Press
2022-07-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2022929?viewType=HTML |
| _version_ | 1818041749767979008 |
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| author | Muhammad Imran Liaqat Sina Etemad Shahram Rezapour Choonkil Park |
| author_facet | Muhammad Imran Liaqat Sina Etemad Shahram Rezapour Choonkil Park |
| author_sort | Muhammad Imran Liaqat |
| collection | DOAJ |
| description | The goal of this research is to develop a novel analytic technique for obtaining the approximate and exact solutions of the Caputo time-fractional partial differential equations (PDEs) with variable coefficients. We call this technique as the Aboodh residual power series method (ARPSM), because it apply the Aboodh transform along with the residual power series method (RPSM). It is based on a new version of Taylor's series that generates a convergent series as a solution. Establishing the coefficients for a series, like the RPSM, necessitates the computation of the fractional derivatives each time. As ARPSM just requires the idea of an infinite limit, we simply need a few computations to get the coefficients. This technique solves nonlinear problems without the He's polynomials and Adomian polynomials, so the small size of computation of this technique is the strength of the scheme, which is an advantage over the homotopy perturbation method and the Adomian decomposition method. The absolute and relative errors of five linear and non-linear problems are numerically examined to determine the efficacy and accuracy of ARPSM for time-fractional PDEs with variable coefficients. In addition, numerical results are also compared with other methods such as the RPSM and the natural transform decomposition method (NTDM). Some graphs are also plotted for various values of fractional orders. The results show that our technique is easy to use, accurate, and effective. Mathematica software is used to calculate the numerical and symbolic quantities in the paper. |
| first_indexed | 2024-12-10T08:35:22Z |
| format | Article |
| id | doaj.art-e687e7befd43430b849b917363974b3c |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-10T08:35:22Z |
| publishDate | 2022-07-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-e687e7befd43430b849b917363974b3c2022-12-22T01:55:59ZengAIMS PressAIMS Mathematics2473-69882022-07-0179169171694810.3934/math.2022929A novel analytical Aboodh residual power series method for solving linear and nonlinear time-fractional partial differential equations with variable coefficientsMuhammad Imran Liaqat0Sina Etemad1Shahram Rezapour 2Choonkil Park 31. National College of Business Administration & Economics, Lahore, Pakistan2. Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran2. Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran 3. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan4. Research Institute for Natural Sciences, Hanyang University, Seoul 04763, KoreaThe goal of this research is to develop a novel analytic technique for obtaining the approximate and exact solutions of the Caputo time-fractional partial differential equations (PDEs) with variable coefficients. We call this technique as the Aboodh residual power series method (ARPSM), because it apply the Aboodh transform along with the residual power series method (RPSM). It is based on a new version of Taylor's series that generates a convergent series as a solution. Establishing the coefficients for a series, like the RPSM, necessitates the computation of the fractional derivatives each time. As ARPSM just requires the idea of an infinite limit, we simply need a few computations to get the coefficients. This technique solves nonlinear problems without the He's polynomials and Adomian polynomials, so the small size of computation of this technique is the strength of the scheme, which is an advantage over the homotopy perturbation method and the Adomian decomposition method. The absolute and relative errors of five linear and non-linear problems are numerically examined to determine the efficacy and accuracy of ARPSM for time-fractional PDEs with variable coefficients. In addition, numerical results are also compared with other methods such as the RPSM and the natural transform decomposition method (NTDM). Some graphs are also plotted for various values of fractional orders. The results show that our technique is easy to use, accurate, and effective. Mathematica software is used to calculate the numerical and symbolic quantities in the paper.https://www.aimspress.com/article/doi/10.3934/math.2022929?viewType=HTMLaboodh transformresidual power series methodcaputo fractional derivativeapproximate solutionexact solutionpartial differential equations |
| spellingShingle | Muhammad Imran Liaqat Sina Etemad Shahram Rezapour Choonkil Park A novel analytical Aboodh residual power series method for solving linear and nonlinear time-fractional partial differential equations with variable coefficients AIMS Mathematics aboodh transform residual power series method caputo fractional derivative approximate solution exact solution partial differential equations |
| title | A novel analytical Aboodh residual power series method for solving linear and nonlinear time-fractional partial differential equations with variable coefficients |
| title_full | A novel analytical Aboodh residual power series method for solving linear and nonlinear time-fractional partial differential equations with variable coefficients |
| title_fullStr | A novel analytical Aboodh residual power series method for solving linear and nonlinear time-fractional partial differential equations with variable coefficients |
| title_full_unstemmed | A novel analytical Aboodh residual power series method for solving linear and nonlinear time-fractional partial differential equations with variable coefficients |
| title_short | A novel analytical Aboodh residual power series method for solving linear and nonlinear time-fractional partial differential equations with variable coefficients |
| title_sort | novel analytical aboodh residual power series method for solving linear and nonlinear time fractional partial differential equations with variable coefficients |
| topic | aboodh transform residual power series method caputo fractional derivative approximate solution exact solution partial differential equations |
| url | https://www.aimspress.com/article/doi/10.3934/math.2022929?viewType=HTML |
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