# Approximation Solution of Fuzzy Singular Volterra Integral Equation by Non-Polynomial Spline

## Abstract

A non-polynomial spline (NPS) is an approximation method that relies on the triangular and polynomial parts, so the method has infinite derivatives of the triangular part of the NPS to compensate for the loss of smoothness inherited by the polynomial. In this paper, we propose polynomial-free linear and quadratic spline types to solve fuzzy Volterra integral equations (FVIE) of the 2nd kind with the weakly singular kernel (FVIEWSK) and Abel's type kernel. The linear type algorithm gives four parameters to form a linear spline. In comparison, the quadratic type algorithm gives five parameters to create a quadratic spline, which is more of a credit for the exact solution. These algorithms process kernel singularities with a simple technique. Illustrative examples use MathCad software to deal with upper and lower-bound solutions to fuzzy problems. The method provides a reliable way to ensure that an exact solution is approximated. Also, figures and tables show the potential of the method.

## Article Details

How to Cite
Ibrahim, Z. A. ., & Hasan, N. N. . (2023). Approximation Solution of Fuzzy Singular Volterra Integral Equation by Non-Polynomial Spline. Ibn AL-Haitham Journal For Pure and Applied Sciences, 36(1), 407–414. https://doi.org/10.30526/36.1.2860
Issue
Section
mathematics
Author Biographies

## References

Zadeh, L. Fuzzy Sets Information and Control, 1965, 8, 338-353.

Mahmoud P.; Farshid M.and Mohammad K. Solving linear and Nonlinear Abel Fuzzy Integral Equation by fuzzy Laplace Transforms. Mathematical Inverse Problems, 2014, 1(2).

Bushnaq, S.; Ullah, Z.; Ullah, A. and Shah, K. Solution of Integral Equation with Abel's Type Kernel using a Novel Hybrid Method. Advances in Difference Equation, Springer Open Journal, 2020, 165.

Zahra, A. Collocation Method for Fuzzy Volterra Integral Equation of the Second Kind with Weakly Singular Kernels, Research Square, 2021.

Elgaili, A.I.; Abdel radi, A.A.; Zakieldeen A.M. ; Reem, H.I. Analytical and Numerical Solution of Linear Volterra Integral Equations of the Second Kind with Weakly Singular Kernel by using the Sixth Order of Non-Polynomial Spline Functions by matlab.Turkish Journal of Computer and Mathematics Education, 2021, 12(14).

Biswas, S.; Roy, T.K. Fuzzy Linear Integral Equation and its Application Inbiomathematical Model. Advances in Fuzzy Mathematics, 2017, 12(5), 1137–1157.

Waffa, A.I. Numerical Method for Solving Integral Equations, 2016, M.Sc. Thesis Mustansiriayah University.

Ali, F.J; Anakira, R. N.; Alomari, A.K. ; Noraziah, M. Solution and Analysis of the Fuzzy Volterra Integral Equation via Homotopy Analysis Method. Computer Modeling in Engineering and Sciences, 2021,127(3).

Muna, M. ; Sarah, H. Solution of Second Kind Volterra Integrals Equations using Non- Polynomial Spline Functions. Baghdad Science Journal, 2014, 11(2).

Rawaa, I.E.; Atefa, J.S. Numerical Treatment of First Order Volterra Integro-Differential Equation using Non-Polynomial spline function. Iraqi Journal of Science ,2020, 114-121.

Sara, H.H. Volterra Integral Equation using Non-Polynomial Spline Functions, 2013, M.Sc. Thesis College of Science Baghdad University.