Fast and Accurate Numerical Algorithm with Performance Assessment for Nonlinear Functional Volterra Equations
An efficient numerical algorithm is developed for solving nonlinear functional Volterra integral equations. The core idea is to define an appropriate operator, then combine the Krasnoselskij iterative scheme with collocation at discrete points and the Newton–Cotes quadrature rule. This results in an...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-04-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/4/333 |
| _version_ | 1797605356504875008 |
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| author | Chinedu Nwaigwe Sanda Micula |
| author_facet | Chinedu Nwaigwe Sanda Micula |
| author_sort | Chinedu Nwaigwe |
| collection | DOAJ |
| description | An efficient numerical algorithm is developed for solving nonlinear functional Volterra integral equations. The core idea is to define an appropriate operator, then combine the Krasnoselskij iterative scheme with collocation at discrete points and the Newton–Cotes quadrature rule. This results in an explicit scheme that does not require solving a nonlinear or linear algebraic system. For the convergence analysis, the discretization error is estimated and proved to converge via a recurrence relation. The discretization error is combined with the Krasnoselskij iteration error to estimate the total approximation error, hence establishing the convergence of the method. Then, numerical experiments are provided, first, to demonstrate the second order convergence of the proposed method, and secondly, to show the better performance of the scheme over the existing nonlinear-based approach. |
| first_indexed | 2024-03-11T04:59:57Z |
| format | Article |
| id | doaj.art-4a2a4441f4c941d0a1733fd7760a5fac |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-11T04:59:57Z |
| publishDate | 2023-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-4a2a4441f4c941d0a1733fd7760a5fac2023-11-17T19:19:43ZengMDPI AGFractal and Fractional2504-31102023-04-017433310.3390/fractalfract7040333Fast and Accurate Numerical Algorithm with Performance Assessment for Nonlinear Functional Volterra EquationsChinedu Nwaigwe0Sanda Micula1Department of Mathematics, Rivers State University, Port Harcourt 5080, NigeriaDepartment of Mathematics, Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, 1 M. Kogălniceanu Street, 400084 Cluj-Napoca, RomaniaAn efficient numerical algorithm is developed for solving nonlinear functional Volterra integral equations. The core idea is to define an appropriate operator, then combine the Krasnoselskij iterative scheme with collocation at discrete points and the Newton–Cotes quadrature rule. This results in an explicit scheme that does not require solving a nonlinear or linear algebraic system. For the convergence analysis, the discretization error is estimated and proved to converge via a recurrence relation. The discretization error is combined with the Krasnoselskij iteration error to estimate the total approximation error, hence establishing the convergence of the method. Then, numerical experiments are provided, first, to demonstrate the second order convergence of the proposed method, and secondly, to show the better performance of the scheme over the existing nonlinear-based approach.https://www.mdpi.com/2504-3110/7/4/333Krasnoselskij iterationtrapezoidal rulegeneralized Banach contraction principlecollocation methodconvergence analysis |
| spellingShingle | Chinedu Nwaigwe Sanda Micula Fast and Accurate Numerical Algorithm with Performance Assessment for Nonlinear Functional Volterra Equations Fractal and Fractional Krasnoselskij iteration trapezoidal rule generalized Banach contraction principle collocation method convergence analysis |
| title | Fast and Accurate Numerical Algorithm with Performance Assessment for Nonlinear Functional Volterra Equations |
| title_full | Fast and Accurate Numerical Algorithm with Performance Assessment for Nonlinear Functional Volterra Equations |
| title_fullStr | Fast and Accurate Numerical Algorithm with Performance Assessment for Nonlinear Functional Volterra Equations |
| title_full_unstemmed | Fast and Accurate Numerical Algorithm with Performance Assessment for Nonlinear Functional Volterra Equations |
| title_short | Fast and Accurate Numerical Algorithm with Performance Assessment for Nonlinear Functional Volterra Equations |
| title_sort | fast and accurate numerical algorithm with performance assessment for nonlinear functional volterra equations |
| topic | Krasnoselskij iteration trapezoidal rule generalized Banach contraction principle collocation method convergence analysis |
| url | https://www.mdpi.com/2504-3110/7/4/333 |
| work_keys_str_mv | AT chinedunwaigwe fastandaccuratenumericalalgorithmwithperformanceassessmentfornonlinearfunctionalvolterraequations AT sandamicula fastandaccuratenumericalalgorithmwithperformanceassessmentfornonlinearfunctionalvolterraequations |