An Algorithm for the Solution of Nonlinear Volterra–Fredholm Integral Equations with a Singular Kernel
The nonlinear Volterra–Fredholm integral Equation (NVFIE) with a singular kernel is discussed such that the kernel of position can take the Hilbert kernel form, Carleman function, logarithmic form, or Cauchy kernel. Using the quadrature method, the NVFIE with a singular kernel leads to a system of n...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-10-01
|
| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/7/10/730 |
| _version_ | 1827720903431028736 |
|---|---|
| author | Sahar M. Abusalim Mohamed A. Abdou Mohamed E. Nasr Mohamed A. Abdel-Aty |
| author_facet | Sahar M. Abusalim Mohamed A. Abdou Mohamed E. Nasr Mohamed A. Abdel-Aty |
| author_sort | Sahar M. Abusalim |
| collection | DOAJ |
| description | The nonlinear Volterra–Fredholm integral Equation (NVFIE) with a singular kernel is discussed such that the kernel of position can take the Hilbert kernel form, Carleman function, logarithmic form, or Cauchy kernel. Using the quadrature method, the NVFIE with a singular kernel leads to a system of nonlinear integral equations. The existence and unique numerical solution of this system is discussed, as is the truncation error of the numerical solution. The solution of the nonlinear integral equation system is obtained using the spectral relations and techniques of the Chebyshev polynomial method. Finally, we will discuss examples of when the kernel takes various forms to demonstrate this technique’s high accuracy and simplicity. Some numerical results and estimating errors are calculated and plotted using the program Wolfram Mathematica 10. |
| first_indexed | 2024-03-10T21:13:48Z |
| format | Article |
| id | doaj.art-74838c8840124c478078e22f6b45ec24 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T21:13:48Z |
| publishDate | 2023-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-74838c8840124c478078e22f6b45ec242023-11-19T16:34:28ZengMDPI AGFractal and Fractional2504-31102023-10-0171073010.3390/fractalfract7100730An Algorithm for the Solution of Nonlinear Volterra–Fredholm Integral Equations with a Singular KernelSahar M. Abusalim0Mohamed A. Abdou1Mohamed E. Nasr2Mohamed A. Abdel-Aty3Department of Mathematics, College of Science and Arts, Jouf University, Al-Qurayyat 77455, Saudi ArabiaDepartment of Mathematics, Faculty of Education, Alexandria University, Alexandria 21511, EgyptDepartment of Mathematics, College of Science and Arts, Jouf University, Al-Qurayyat 77455, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Benha University, Benha 13518, EgyptThe nonlinear Volterra–Fredholm integral Equation (NVFIE) with a singular kernel is discussed such that the kernel of position can take the Hilbert kernel form, Carleman function, logarithmic form, or Cauchy kernel. Using the quadrature method, the NVFIE with a singular kernel leads to a system of nonlinear integral equations. The existence and unique numerical solution of this system is discussed, as is the truncation error of the numerical solution. The solution of the nonlinear integral equation system is obtained using the spectral relations and techniques of the Chebyshev polynomial method. Finally, we will discuss examples of when the kernel takes various forms to demonstrate this technique’s high accuracy and simplicity. Some numerical results and estimating errors are calculated and plotted using the program Wolfram Mathematica 10.https://www.mdpi.com/2504-3110/7/10/730singular nonlinear Volterra–Fredholm integral equationalgebraic systemBanach fixed point theoremsingular kernelChebyshev polynomials |
| spellingShingle | Sahar M. Abusalim Mohamed A. Abdou Mohamed E. Nasr Mohamed A. Abdel-Aty An Algorithm for the Solution of Nonlinear Volterra–Fredholm Integral Equations with a Singular Kernel Fractal and Fractional singular nonlinear Volterra–Fredholm integral equation algebraic system Banach fixed point theorem singular kernel Chebyshev polynomials |
| title | An Algorithm for the Solution of Nonlinear Volterra–Fredholm Integral Equations with a Singular Kernel |
| title_full | An Algorithm for the Solution of Nonlinear Volterra–Fredholm Integral Equations with a Singular Kernel |
| title_fullStr | An Algorithm for the Solution of Nonlinear Volterra–Fredholm Integral Equations with a Singular Kernel |
| title_full_unstemmed | An Algorithm for the Solution of Nonlinear Volterra–Fredholm Integral Equations with a Singular Kernel |
| title_short | An Algorithm for the Solution of Nonlinear Volterra–Fredholm Integral Equations with a Singular Kernel |
| title_sort | algorithm for the solution of nonlinear volterra fredholm integral equations with a singular kernel |
| topic | singular nonlinear Volterra–Fredholm integral equation algebraic system Banach fixed point theorem singular kernel Chebyshev polynomials |
| url | https://www.mdpi.com/2504-3110/7/10/730 |
| work_keys_str_mv | AT saharmabusalim analgorithmforthesolutionofnonlinearvolterrafredholmintegralequationswithasingularkernel AT mohamedaabdou analgorithmforthesolutionofnonlinearvolterrafredholmintegralequationswithasingularkernel AT mohamedenasr analgorithmforthesolutionofnonlinearvolterrafredholmintegralequationswithasingularkernel AT mohamedaabdelaty analgorithmforthesolutionofnonlinearvolterrafredholmintegralequationswithasingularkernel AT saharmabusalim algorithmforthesolutionofnonlinearvolterrafredholmintegralequationswithasingularkernel AT mohamedaabdou algorithmforthesolutionofnonlinearvolterrafredholmintegralequationswithasingularkernel AT mohamedenasr algorithmforthesolutionofnonlinearvolterrafredholmintegralequationswithasingularkernel AT mohamedaabdelaty algorithmforthesolutionofnonlinearvolterrafredholmintegralequationswithasingularkernel |