An Algorithm for the Solution of Integro-Fractional Differential Equations with a Generalized Symmetric Singular Kernel
This work studies an integro-fractional differential equation (<b>I-FrDE</b>) with a generalized symmetric singular kernel. The scientific approach in this study was to transform the integro-differential equation (<b>I-DE</b>) into a mixed integral equation (<b>MIE</...
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MDPI AG
2024-10-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/8/11/644 |
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| author | Sameeha A. Raad Mohammed A. Abdou |
| author_facet | Sameeha A. Raad Mohammed A. Abdou |
| author_sort | Sameeha A. Raad |
| collection | DOAJ |
| description | This work studies an integro-fractional differential equation (<b>I-FrDE</b>) with a generalized symmetric singular kernel. The scientific approach in this study was to transform the integro-differential equation (<b>I-DE</b>) into a mixed integral equation (<b>MIE</b>) with an Able kernel in fractional time and a generalized symmetric singular kernel in position. Additionally, the authors first set conditions on the singular kernels, whether related to time or position, and then transform the integral equation into an integral operator. Secondly, the solution is unique, which is proven by means of fixed-point theorems. In combination with the solution rules, the convergence of the solution is studied, and the error equation resulting from the solution is a stable error-integral influencer equation. Next, to solve this <b>MIE</b>, the authors apply a special technique to separate the variables and produce an integral equation in position with coefficients, in the form of an integral operator in time. As the most effective technique for resolving singular integral equations, the Toeplitz matrix method (<b>TMM</b>) is utilized to convert the integral equation into an algebraic system for the purpose of solving the position problem. The existence of a solution to the linear algebraic system in Banach space is then demonstrated. Lastly, certain applications where the functions of the generalized symmetric kernel are cubic or exponential and it assumes the logarithmic, Cauchy, or Carleman form are discussed. In each case, Maple 18 is also used to compute the error estimate. |
| format | Article |
| id | doaj-art-bafd40522cdf4ceb820012a2843f1c83 |
| institution | OA Journals |
| issn | 2504-3110 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-bafd40522cdf4ceb820012a2843f1c832025-08-20T01:53:41ZengMDPI AGFractal and Fractional2504-31102024-10-0181164410.3390/fractalfract8110644An Algorithm for the Solution of Integro-Fractional Differential Equations with a Generalized Symmetric Singular KernelSameeha A. Raad0Mohammed A. Abdou1Mathematics Department, Faculty of Science, Umm Al-Qura University, Makkah 21955, Saudi ArabiaDepartment of Mathematics, Faculty of Education, Alexandria University, Alexandria 21526, EgyptThis work studies an integro-fractional differential equation (<b>I-FrDE</b>) with a generalized symmetric singular kernel. The scientific approach in this study was to transform the integro-differential equation (<b>I-DE</b>) into a mixed integral equation (<b>MIE</b>) with an Able kernel in fractional time and a generalized symmetric singular kernel in position. Additionally, the authors first set conditions on the singular kernels, whether related to time or position, and then transform the integral equation into an integral operator. Secondly, the solution is unique, which is proven by means of fixed-point theorems. In combination with the solution rules, the convergence of the solution is studied, and the error equation resulting from the solution is a stable error-integral influencer equation. Next, to solve this <b>MIE</b>, the authors apply a special technique to separate the variables and produce an integral equation in position with coefficients, in the form of an integral operator in time. As the most effective technique for resolving singular integral equations, the Toeplitz matrix method (<b>TMM</b>) is utilized to convert the integral equation into an algebraic system for the purpose of solving the position problem. The existence of a solution to the linear algebraic system in Banach space is then demonstrated. Lastly, certain applications where the functions of the generalized symmetric kernel are cubic or exponential and it assumes the logarithmic, Cauchy, or Carleman form are discussed. In each case, Maple 18 is also used to compute the error estimate.https://www.mdpi.com/2504-3110/8/11/644fractional differential equationmixed integral equationseparation of variableslinear algebraic systemToeplitz matrixsingular kernel |
| spellingShingle | Sameeha A. Raad Mohammed A. Abdou An Algorithm for the Solution of Integro-Fractional Differential Equations with a Generalized Symmetric Singular Kernel Fractal and Fractional fractional differential equation mixed integral equation separation of variables linear algebraic system Toeplitz matrix singular kernel |
| title | An Algorithm for the Solution of Integro-Fractional Differential Equations with a Generalized Symmetric Singular Kernel |
| title_full | An Algorithm for the Solution of Integro-Fractional Differential Equations with a Generalized Symmetric Singular Kernel |
| title_fullStr | An Algorithm for the Solution of Integro-Fractional Differential Equations with a Generalized Symmetric Singular Kernel |
| title_full_unstemmed | An Algorithm for the Solution of Integro-Fractional Differential Equations with a Generalized Symmetric Singular Kernel |
| title_short | An Algorithm for the Solution of Integro-Fractional Differential Equations with a Generalized Symmetric Singular Kernel |
| title_sort | algorithm for the solution of integro fractional differential equations with a generalized symmetric singular kernel |
| topic | fractional differential equation mixed integral equation separation of variables linear algebraic system Toeplitz matrix singular kernel |
| url | https://www.mdpi.com/2504-3110/8/11/644 |
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