Solving Volterra-Fredholm integral equations by non-polynomial spline function based on weighted residual methods
In this paper, a method that utilizes a non-polynomial spline function based on the weighted residual technique to approximate solutions for linear Volterra-Fredholm integral equations is presented. The approach begins with the selection of a series of knots along the integration interval. We then...
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Academician Ye.A. Buketov Karaganda University
2025-03-01
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| Series: | Қарағанды университетінің хабаршысы. Математика сериясы |
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| Online Access: | https://mts.buketov.edu.kz/index.php/mathematics-vestnik/article/view/768 |
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| author | S.H. Salim R.K. Saeed K.H.F. Jwamer |
| author_facet | S.H. Salim R.K. Saeed K.H.F. Jwamer |
| author_sort | S.H. Salim |
| collection | DOAJ |
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In this paper, a method that utilizes a non-polynomial spline function based on the weighted residual technique to approximate solutions for linear Volterra-Fredholm integral equations is presented. The approach begins with the selection of a series of knots along the integration interval. We then create a set of basis functions, defined as non-polynomial spline functions, between each pair of adjacent knots. The unknown function is expressed as a linear combination of these basis functions to approximate the solution of integral equations. The coefficients of the spline function are calculated by solving a system of linear equations derived from substituting the spline approximation into the integral equation while maintaining continuity and smoothness at the knots. Non-polynomial splines are beneficial for approximating functions with complex shapes and for solving integral equations with non-smooth kernels. However, the solution’s accuracy significantly relies on the selection of knots, and the method may require extensive computational resources for large systems. To illustrate the effectiveness of the method, three examples are presented, implemented using Python version 3.9. The paper also addresses the error analysis theorem relevant to the proposed non-polynomial spline function.
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| format | Article |
| id | doaj-art-e6bfcbcb309245a393f8a40474835f7b |
| institution | Kabale University |
| issn | 2518-7929 2663-5011 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | Academician Ye.A. Buketov Karaganda University |
| record_format | Article |
| series | Қарағанды университетінің хабаршысы. Математика сериясы |
| spelling | doaj-art-e6bfcbcb309245a393f8a40474835f7b2025-08-20T03:42:15ZengAcademician Ye.A. Buketov Karaganda UniversityҚарағанды университетінің хабаршысы. Математика сериясы2518-79292663-50112025-03-01117110.31489/2025m1/155-169Solving Volterra-Fredholm integral equations by non-polynomial spline function based on weighted residual methodsS.H. Salim0R.K. Saeed1https://orcid.org/0000-0001-5165-3333K.H.F. Jwamer2https://orcid.org/0000-0003-4009-0357College of Basic Education, University of Sulaimani, Sulaymaniyah, IraqCollege of Science, Salahaddin University-Erbil, Erbil, Kurdistan Region, IraqCollege of Science, University of Sulaimani, Sulaymaniyah, Iraq In this paper, a method that utilizes a non-polynomial spline function based on the weighted residual technique to approximate solutions for linear Volterra-Fredholm integral equations is presented. The approach begins with the selection of a series of knots along the integration interval. We then create a set of basis functions, defined as non-polynomial spline functions, between each pair of adjacent knots. The unknown function is expressed as a linear combination of these basis functions to approximate the solution of integral equations. The coefficients of the spline function are calculated by solving a system of linear equations derived from substituting the spline approximation into the integral equation while maintaining continuity and smoothness at the knots. Non-polynomial splines are beneficial for approximating functions with complex shapes and for solving integral equations with non-smooth kernels. However, the solution’s accuracy significantly relies on the selection of knots, and the method may require extensive computational resources for large systems. To illustrate the effectiveness of the method, three examples are presented, implemented using Python version 3.9. The paper also addresses the error analysis theorem relevant to the proposed non-polynomial spline function. https://mts.buketov.edu.kz/index.php/mathematics-vestnik/article/view/768Volterra integral equationFredholm integral equationnon-polynomial splineweighted residual methods |
| spellingShingle | S.H. Salim R.K. Saeed K.H.F. Jwamer Solving Volterra-Fredholm integral equations by non-polynomial spline function based on weighted residual methods Қарағанды университетінің хабаршысы. Математика сериясы Volterra integral equation Fredholm integral equation non-polynomial spline weighted residual methods |
| title | Solving Volterra-Fredholm integral equations by non-polynomial spline function based on weighted residual methods |
| title_full | Solving Volterra-Fredholm integral equations by non-polynomial spline function based on weighted residual methods |
| title_fullStr | Solving Volterra-Fredholm integral equations by non-polynomial spline function based on weighted residual methods |
| title_full_unstemmed | Solving Volterra-Fredholm integral equations by non-polynomial spline function based on weighted residual methods |
| title_short | Solving Volterra-Fredholm integral equations by non-polynomial spline function based on weighted residual methods |
| title_sort | solving volterra fredholm integral equations by non polynomial spline function based on weighted residual methods |
| topic | Volterra integral equation Fredholm integral equation non-polynomial spline weighted residual methods |
| url | https://mts.buketov.edu.kz/index.php/mathematics-vestnik/article/view/768 |
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