Spline Collocation for Nonlinear Fredholm Integral Equations

Journal of Al-Qadisiyah for Computer Science and Mathematics

The second order non-polynomial spline function for solving system of two nonlinear Volterra integral equations is proposed in this paper. An algorithm introduced as well to numerical examples to illustrate carry out of this method. Also, we compare the absolute error of quadratic non-polynomial spline method with absolute error of linear non-polynomial spline method and the exact solution.

2006

In this paper, we find numerical solution of xðtÞ þ k

CERN European Organization for Nuclear Research - Zenodo, 2022

This study determines the numerical solution of linear mixed Volterra-Fredholm integral equations of the second kind using the linear spline function. The proposed method is based on using the unknown function's linear spline function at an arbitrary point and converting the Volterra-Fredholm integral equation into a system of linear equations with regard to the unknown function using the integration method. By solving the given system, an approximate solution can be easily established. This is done with the help of a computer program that uses the Python code program version 3.9. Furthermore, we demonstrated theoretical results on the method's uniqueness and convergence analyses.

International Journal of Computer Mathematics, 2019

In our knowledge so far, the Non-polynomial Spline functions (NPS) have not been yet applied for approximating the integral equations. In this article, we want to use such functions for obtaining numerical solutions of Fredholm integral equations of the second kind. In our approach, the coefficients of the non polynomial spline are obtained by solving a system of linear equations. Finally, these functions are utilized to reduce the Fredholm integral equations to the solution of algebraic equations. Analysis of convergence is investigated. At the end, our test examples are presented to show the effectiveness of the method.

Journal of Physics: Conference Series

The aim of this paper,we offereda new numerical methodwhich is Touchard Polynomials (T-Ps) for solving Linear Fredholm Integral Equation of the Second Kind (LFIE2-K), to find approximating Numerical Solution (N-S). At the beginning, we demonstrate (T-Ps) andconstruct the operational matrix which is a matrix representation for solution. The algorithm and someexamples are given; comparing the numerical results of proposed method with the numerical results of the other numerical method which is Bernstein Polynomials (B-Ps).Wewill show the high resolution of results by proposed method.The comparison between the Exact Solution(E-S) and the results of two methods are given by calculating absolute value of error and the Least Square Error (L.S.E).The results are calculated in Matlabcode.

Journal of Al-Qadisiyah for Computer Science and Mathematics

Using the quadratic spline function, this paper finds the numerical solution of mixed Volterra-Fredholm integral equations of the second kind. The proposed method is based on employing the quadratic spline function of the unknown function at an arbitrary point and using the integration method to turn the Volterra-Fredholm integral equation into a system of linear equations with respect to the unknown function. An approximate solution can be easily established by solving the given system. This is accomplished with the help of a computer program that runs on Python 3.9..

Mathematics of Computation, 1978

An interpolation scheme based on piecewise cubic polynomials with Gaussian points as interpolation points is analyzed and applied to the solution of Fredholm equations of the second kind Introduction. We consider an interpolation scheme

Mathematics and Statistics, 2022

This paper is concerned with producing an efficient numerical method to solve nonlinear Fredholm integral equations using Half-Sweep Newton-PKSOR (HSNPKSOR) iteration. The computation of numerical methods in solving nonlinear equations usually requires immense amounts of computational complexity. By implementing a Half-Sweep approach, the complexity of the calculation is tried to be reduced to produce a more efficient method. For this purpose, the steps of the solution process are discussed beginning with the derivation of nonlinear Fredholm integral equations using a quadrature scheme to get the half-sweep approximation equation. Then, the generated approximation equation is used to develop a nonlinear system. Following that, the formulation of the HSNPKSOR iterative method is constructed to solve nonlinear Fredholm integral equations. To verify the performance of the proposed method, the experimental results were compared with the Full-Sweep Newton-KSOR (FSNKSOR), Half-Sweep Newton-KSOR (HSNKSOR), and Full-Sweep Newton-PKSOR (FSNPKSOR) using three parameters: number of iteration, iteration time, and maximum absolute error. Several examples are used in this study to illustrate the efficiency of the tested methods. Based on the numerical experiment, the results appear that the HSNPKSOR method is effective in solving nonlinear Fredholm integral equations mainly in terms of iteration time compared to rest tested methods.

Fuzzy Information and Engineering

In this paper, Bernstein polynomials are utilized to present an approximateanalytical solution for fuzzy Fredholm integral equations of the second kind. A sequence which coincides with the exact solution is precisely generated by the proposed method. The method is explained with illustrative examples.

2019

The main purpose of this work is to use the Chelyshkov-collocation method for the solution of threedimensional Fredholm integral equations. The method is based on the approximate solution in terms of Chelyshkov polynomials with unknown coefficients. This method transforms the integral equation to a system of linear algebraic equations by means of collocation points. Finally, numerical results are included to show the validity and applicability of the method and comparisons are made with existing results.