A New Technique for Solving a Nonlinear Integro-Differential Equation with Fractional Order in Complex Space
This work aims to explore the solution of a nonlinear fractional integro-differential equation in the complex domain through the utilization of both analytical and numerical approaches. The demonstration of the existence and uniqueness of a solution is established under certain appropriate condition...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-10-01
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| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/7/11/796 |
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| author | Amnah E. Shammaky Eslam M. Youssef Mohamed A. Abdou Mahmoud M. ElBorai Wagdy G. ElSayed Mai Taha |
| author_facet | Amnah E. Shammaky Eslam M. Youssef Mohamed A. Abdou Mahmoud M. ElBorai Wagdy G. ElSayed Mai Taha |
| author_sort | Amnah E. Shammaky |
| collection | DOAJ |
| description | This work aims to explore the solution of a nonlinear fractional integro-differential equation in the complex domain through the utilization of both analytical and numerical approaches. The demonstration of the existence and uniqueness of a solution is established under certain appropriate conditions with the use of Banach fixed point theorems. To date, no research effort has been undertaken to look into the solution of this integro equation, particularly due to its fractional order specification within the complex plane. The validation of the proposed methodology was performed by utilizing a novel strategy that involves implementing the Rationalized Haar wavelet numerical method with the application of the Bernoulli polynomial technique. The primary reason for choosing the proposed technique lies in its ability to transform the solution of the given nonlinear fractional integro-differential equation into a representation that corresponds to a linear system of algebraic equations. Furthermore, we conduct a comparative analysis between the outcomes obtained from the suggested method and those derived from the rationalized Haar wavelet method without employing any shared mathematical methodologies. In order to evaluate the precision and effectiveness of the proposed method, a series of numerical examples have been developed. |
| first_indexed | 2024-03-09T16:49:20Z |
| format | Article |
| id | doaj.art-df222a089da84d7f8c8365257f1c86db |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T16:49:20Z |
| publishDate | 2023-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-df222a089da84d7f8c8365257f1c86db2023-11-24T14:43:00ZengMDPI AGFractal and Fractional2504-31102023-10-0171179610.3390/fractalfract7110796A New Technique for Solving a Nonlinear Integro-Differential Equation with Fractional Order in Complex SpaceAmnah E. Shammaky0Eslam M. Youssef1Mohamed A. Abdou2Mahmoud M. ElBorai3Wagdy G. ElSayed4Mai Taha5Department of Mathematics, Faculty of Science, Jazan University, Jazan 45142, Saudi ArabiaDepartment of Mathematics, Faculty of Education, Alexandria University, Alexandria 21256, EgyptDepartment of Mathematics, Faculty of Education, Alexandria University, Alexandria 21256, EgyptDepartment of Mathematics, Faculty of Science, Alexandria University, Alexandria 21515, EgyptDepartment of Mathematics, Faculty of Science, Alexandria University, Alexandria 21515, EgyptDepartment of Mathematics, Faculty of Education, Alexandria University, Alexandria 21256, EgyptThis work aims to explore the solution of a nonlinear fractional integro-differential equation in the complex domain through the utilization of both analytical and numerical approaches. The demonstration of the existence and uniqueness of a solution is established under certain appropriate conditions with the use of Banach fixed point theorems. To date, no research effort has been undertaken to look into the solution of this integro equation, particularly due to its fractional order specification within the complex plane. The validation of the proposed methodology was performed by utilizing a novel strategy that involves implementing the Rationalized Haar wavelet numerical method with the application of the Bernoulli polynomial technique. The primary reason for choosing the proposed technique lies in its ability to transform the solution of the given nonlinear fractional integro-differential equation into a representation that corresponds to a linear system of algebraic equations. Furthermore, we conduct a comparative analysis between the outcomes obtained from the suggested method and those derived from the rationalized Haar wavelet method without employing any shared mathematical methodologies. In order to evaluate the precision and effectiveness of the proposed method, a series of numerical examples have been developed.https://www.mdpi.com/2504-3110/7/11/796complex planeRiemann–Liouville fractional integral operatorfractional integro-differential equationsfixed point theoremrationalized Haar wavelet |
| spellingShingle | Amnah E. Shammaky Eslam M. Youssef Mohamed A. Abdou Mahmoud M. ElBorai Wagdy G. ElSayed Mai Taha A New Technique for Solving a Nonlinear Integro-Differential Equation with Fractional Order in Complex Space Fractal and Fractional complex plane Riemann–Liouville fractional integral operator fractional integro-differential equations fixed point theorem rationalized Haar wavelet |
| title | A New Technique for Solving a Nonlinear Integro-Differential Equation with Fractional Order in Complex Space |
| title_full | A New Technique for Solving a Nonlinear Integro-Differential Equation with Fractional Order in Complex Space |
| title_fullStr | A New Technique for Solving a Nonlinear Integro-Differential Equation with Fractional Order in Complex Space |
| title_full_unstemmed | A New Technique for Solving a Nonlinear Integro-Differential Equation with Fractional Order in Complex Space |
| title_short | A New Technique for Solving a Nonlinear Integro-Differential Equation with Fractional Order in Complex Space |
| title_sort | new technique for solving a nonlinear integro differential equation with fractional order in complex space |
| topic | complex plane Riemann–Liouville fractional integral operator fractional integro-differential equations fixed point theorem rationalized Haar wavelet |
| url | https://www.mdpi.com/2504-3110/7/11/796 |
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