Computational Techniques for Solving Mixed (1 + 1) Dimensional Integral Equations with Strongly Symmetric Singular Kernel
This paper describes an effective strategy based on Lerch polynomial method for solving mixed integral equations (MIE) in position and time with a strongly symmetric singular kernel in the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline&...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-06-01
|
| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/15/6/1284 |
| _version_ | 1797592522343579648 |
|---|---|
| author | Sharifah E. Alhazmi Amr M. S. Mahdy Mohamed A. Abdou Doaa Sh. Mohamed |
| author_facet | Sharifah E. Alhazmi Amr M. S. Mahdy Mohamed A. Abdou Doaa Sh. Mohamed |
| author_sort | Sharifah E. Alhazmi |
| collection | DOAJ |
| description | This paper describes an effective strategy based on Lerch polynomial method for solving mixed integral equations (MIE) in position and time with a strongly symmetric singular kernel in the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mo>×</mo><mi>C</mi><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo></mrow><mo>,</mo><mrow><mo stretchy="false">(</mo><mi>T</mi><mo><</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> The Quadratic numerical method (QNM) was applied to obtain a system of Fredholm integral equations (SFIE), then the Lerch polynomials method (LPM) was applied to transform SFIE into a system of linear algebraic equations (SLAE). The existence and uniqueness of the integral equation’s solution are discussed using Banach’s fixed point theory. Also, the convergence and stability of the solution and the stability of the error are discussed. Several examples are given to illustrate the applicability of the presented method. The Maple program obtains all the results. A numerical simulation is carried out to determine the efficacy of the methodology, and the results are given in symmetrical forms. From the numerical results, it is noted that there is a symmetry utterly identical to the kernel used when replacing each <i>x</i> with <i>y</i>. |
| first_indexed | 2024-03-11T01:52:18Z |
| format | Article |
| id | doaj.art-9f2a520c53b9404b891a44e9f3f61dcf |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T01:52:18Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-9f2a520c53b9404b891a44e9f3f61dcf2023-11-18T12:52:03ZengMDPI AGSymmetry2073-89942023-06-01156128410.3390/sym15061284Computational Techniques for Solving Mixed (1 + 1) Dimensional Integral Equations with Strongly Symmetric Singular KernelSharifah E. Alhazmi0Amr M. S. Mahdy1Mohamed A. Abdou2Doaa Sh. Mohamed3Mathematics Department, Al-Qunfudah University College, Umm Al-Qura University, Al-Qunfudhah 21955, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, EgyptDepartment of Mathematics, Faculty of Education, Alexandria University, Alexandria 21526, EgyptDepartment of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, EgyptThis paper describes an effective strategy based on Lerch polynomial method for solving mixed integral equations (MIE) in position and time with a strongly symmetric singular kernel in the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mo>×</mo><mi>C</mi><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo></mrow><mo>,</mo><mrow><mo stretchy="false">(</mo><mi>T</mi><mo><</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> The Quadratic numerical method (QNM) was applied to obtain a system of Fredholm integral equations (SFIE), then the Lerch polynomials method (LPM) was applied to transform SFIE into a system of linear algebraic equations (SLAE). The existence and uniqueness of the integral equation’s solution are discussed using Banach’s fixed point theory. Also, the convergence and stability of the solution and the stability of the error are discussed. Several examples are given to illustrate the applicability of the presented method. The Maple program obtains all the results. A numerical simulation is carried out to determine the efficacy of the methodology, and the results are given in symmetrical forms. From the numerical results, it is noted that there is a symmetry utterly identical to the kernel used when replacing each <i>x</i> with <i>y</i>.https://www.mdpi.com/2073-8994/15/6/1284Quadratic numerical methodLerch polynomial methodstrongly symmetric singular kernelmixed integral equations |
| spellingShingle | Sharifah E. Alhazmi Amr M. S. Mahdy Mohamed A. Abdou Doaa Sh. Mohamed Computational Techniques for Solving Mixed (1 + 1) Dimensional Integral Equations with Strongly Symmetric Singular Kernel Symmetry Quadratic numerical method Lerch polynomial method strongly symmetric singular kernel mixed integral equations |
| title | Computational Techniques for Solving Mixed (1 + 1) Dimensional Integral Equations with Strongly Symmetric Singular Kernel |
| title_full | Computational Techniques for Solving Mixed (1 + 1) Dimensional Integral Equations with Strongly Symmetric Singular Kernel |
| title_fullStr | Computational Techniques for Solving Mixed (1 + 1) Dimensional Integral Equations with Strongly Symmetric Singular Kernel |
| title_full_unstemmed | Computational Techniques for Solving Mixed (1 + 1) Dimensional Integral Equations with Strongly Symmetric Singular Kernel |
| title_short | Computational Techniques for Solving Mixed (1 + 1) Dimensional Integral Equations with Strongly Symmetric Singular Kernel |
| title_sort | computational techniques for solving mixed 1 1 dimensional integral equations with strongly symmetric singular kernel |
| topic | Quadratic numerical method Lerch polynomial method strongly symmetric singular kernel mixed integral equations |
| url | https://www.mdpi.com/2073-8994/15/6/1284 |
| work_keys_str_mv | AT sharifahealhazmi computationaltechniquesforsolvingmixed11dimensionalintegralequationswithstronglysymmetricsingularkernel AT amrmsmahdy computationaltechniquesforsolvingmixed11dimensionalintegralequationswithstronglysymmetricsingularkernel AT mohamedaabdou computationaltechniquesforsolvingmixed11dimensionalintegralequationswithstronglysymmetricsingularkernel AT doaashmohamed computationaltechniquesforsolvingmixed11dimensionalintegralequationswithstronglysymmetricsingularkernel |