Interpolation method for solving Volterra integral equations with weakly singular kernel using an advanced barycentric Lagrange formula
We presented an interpolation method for solving weakly singular Volterra integral equations of the second kind (SVK2). The method based on the barycentric Lagrange interpolation.. For the chosen nodes of the two singular kernel variables, we created two rules that confirm that the denominator of th...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Elsevier
2022-09-01
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| Series: | Ain Shams Engineering Journal |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2090447922000545 |
| _version_ | 1797985989368479744 |
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| author | E.S. Shoukralla B.M. Ahmed M. Sayed Ahmed Saeed |
| author_facet | E.S. Shoukralla B.M. Ahmed M. Sayed Ahmed Saeed |
| author_sort | E.S. Shoukralla |
| collection | DOAJ |
| description | We presented an interpolation method for solving weakly singular Volterra integral equations of the second kind (SVK2). The method based on the barycentric Lagrange interpolation.. For the chosen nodes of the two singular kernel variables, we created two rules that confirm that the denominator of the kernel will never become zero or have an imaginary value. The product of four matrices, one of which is the square coefficients matrix of the barycentric functions, expresses both the unknown and data functions. The singular kernel is interpolated twice with to its two variables. Furthermore, the kernel is represented by five matrices: two of which are the monomial basis of the kernel's variables, and one matrix expresses the kernel's values at the two sets of nodes for the kernel's variables. Without using the collocation approach, we get the matrix of the unknown coefficients by substituting the interpolated solution into the integral equation to obtain an equivalent algebraic system. By solving the algebraic system, we get the interpolated solution. We solved five examples, two for non-singular equations and three for weakly singular equations. By adopting lower interpolation degrees, the interpolated solutions for non-singular equations are shown to be equal to the exact ones, whereas, for singular equations, they are found to strongly converge to the exact ones and are superior to those obtained by other cited methods. This ensures the proposed method's uniqueness and high accuracy. |
| first_indexed | 2024-04-11T07:25:45Z |
| format | Article |
| id | doaj.art-ca6e019172084112b919f33522d98b48 |
| institution | Directory Open Access Journal |
| issn | 2090-4479 |
| language | English |
| last_indexed | 2024-04-11T07:25:45Z |
| publishDate | 2022-09-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Ain Shams Engineering Journal |
| spelling | doaj.art-ca6e019172084112b919f33522d98b482022-12-22T04:37:04ZengElsevierAin Shams Engineering Journal2090-44792022-09-01135101743Interpolation method for solving Volterra integral equations with weakly singular kernel using an advanced barycentric Lagrange formulaE.S. Shoukralla0B.M. Ahmed1M. Sayed2Ahmed Saeed3Physics and Engineering Mathematics Department, Faculty of Electronic Engineering, Menouf, EgyptEngineering Mathematics and Physics Department, Future University in Egypt, Cairo, EgyptPhysics and Engineering Mathematics Department, Faculty of Electronic Engineering, Menouf, EgyptElectrical Engineering Department, Future University in Egypt, Cairo, EgyptWe presented an interpolation method for solving weakly singular Volterra integral equations of the second kind (SVK2). The method based on the barycentric Lagrange interpolation.. For the chosen nodes of the two singular kernel variables, we created two rules that confirm that the denominator of the kernel will never become zero or have an imaginary value. The product of four matrices, one of which is the square coefficients matrix of the barycentric functions, expresses both the unknown and data functions. The singular kernel is interpolated twice with to its two variables. Furthermore, the kernel is represented by five matrices: two of which are the monomial basis of the kernel's variables, and one matrix expresses the kernel's values at the two sets of nodes for the kernel's variables. Without using the collocation approach, we get the matrix of the unknown coefficients by substituting the interpolated solution into the integral equation to obtain an equivalent algebraic system. By solving the algebraic system, we get the interpolated solution. We solved five examples, two for non-singular equations and three for weakly singular equations. By adopting lower interpolation degrees, the interpolated solutions for non-singular equations are shown to be equal to the exact ones, whereas, for singular equations, they are found to strongly converge to the exact ones and are superior to those obtained by other cited methods. This ensures the proposed method's uniqueness and high accuracy.http://www.sciencedirect.com/science/article/pii/S2090447922000545Barycentric Lagrange interpolationWeakly singular Volterra kernelsComputational methodsScatteringRadiationImage processing |
| spellingShingle | E.S. Shoukralla B.M. Ahmed M. Sayed Ahmed Saeed Interpolation method for solving Volterra integral equations with weakly singular kernel using an advanced barycentric Lagrange formula Ain Shams Engineering Journal Barycentric Lagrange interpolation Weakly singular Volterra kernels Computational methods Scattering Radiation Image processing |
| title | Interpolation method for solving Volterra integral equations with weakly singular kernel using an advanced barycentric Lagrange formula |
| title_full | Interpolation method for solving Volterra integral equations with weakly singular kernel using an advanced barycentric Lagrange formula |
| title_fullStr | Interpolation method for solving Volterra integral equations with weakly singular kernel using an advanced barycentric Lagrange formula |
| title_full_unstemmed | Interpolation method for solving Volterra integral equations with weakly singular kernel using an advanced barycentric Lagrange formula |
| title_short | Interpolation method for solving Volterra integral equations with weakly singular kernel using an advanced barycentric Lagrange formula |
| title_sort | interpolation method for solving volterra integral equations with weakly singular kernel using an advanced barycentric lagrange formula |
| topic | Barycentric Lagrange interpolation Weakly singular Volterra kernels Computational methods Scattering Radiation Image processing |
| url | http://www.sciencedirect.com/science/article/pii/S2090447922000545 |
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