Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations
In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fract...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-01-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/6/1/24 |
| _version_ | 1797493891251830784 |
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| author | Muhammad Shakeel Nehad Ali Shah Jae Dong Chung |
| author_facet | Muhammad Shakeel Nehad Ali Shah Jae Dong Chung |
| author_sort | Muhammad Shakeel |
| collection | DOAJ |
| description | In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (<i>G′</i>/<i>G</i><sup>2</sup>)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order <i>α</i> travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters <i>H</i>. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions. |
| first_indexed | 2024-03-10T01:26:29Z |
| format | Article |
| id | doaj.art-17fa1f2729fb4d8d88bf17a64857e406 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T01:26:29Z |
| publishDate | 2022-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-17fa1f2729fb4d8d88bf17a64857e4062023-11-23T13:48:49ZengMDPI AGFractal and Fractional2504-31102022-01-01612410.3390/fractalfract6010024Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential EquationsMuhammad Shakeel0Nehad Ali Shah1Jae Dong Chung2Department of Mathematics, University of Wah, Wah Cantt 47040, PakistanDepartment of Mechanical Engineering, Sejong University, Seoul 05006, KoreaDepartment of Mechanical Engineering, Sejong University, Seoul 05006, KoreaIn this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (<i>G′</i>/<i>G</i><sup>2</sup>)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order <i>α</i> travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters <i>H</i>. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions.https://www.mdpi.com/2504-3110/6/1/24novel (<i>G</i>’/<i>G</i><sup>2</sup>)-expansion methodtime fractional MCH equationsolitary wave solutionshomogeneous balance principleexact solutions |
| spellingShingle | Muhammad Shakeel Nehad Ali Shah Jae Dong Chung Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations Fractal and Fractional novel (<i>G</i>’/<i>G</i><sup>2</sup>)-expansion method time fractional MCH equation solitary wave solutions homogeneous balance principle exact solutions |
| title | Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations |
| title_full | Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations |
| title_fullStr | Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations |
| title_full_unstemmed | Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations |
| title_short | Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations |
| title_sort | novel analytical technique to find closed form solutions of time fractional partial differential equations |
| topic | novel (<i>G</i>’/<i>G</i><sup>2</sup>)-expansion method time fractional MCH equation solitary wave solutions homogeneous balance principle exact solutions |
| url | https://www.mdpi.com/2504-3110/6/1/24 |
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