scholarly journals Computational method for solving weakly singular Fredholm integral equations of the second kind using an advanced barycentric Lagrange interpolation formula

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
E. S. Shoukralla ◽  
Nermin Saber ◽  
Ahmed Y. Sayed
Keyword(s):  
Integral Equations ◽  
Lagrange Interpolation ◽  
Interpolation Formula ◽  
Good Choice ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Error Norm ◽  
Monomial Basis ◽  
Weakly Singular ◽  
Lagrange Interpolation Formula

AbstractIn this study, we applied an advanced barycentric Lagrange interpolation formula to find the interpolate solutions of weakly singular Fredholm integral equations of the second kind. The kernel is interpolated twice concerning both variables and then is transformed into the product of five matrices; two of them are monomial basis matrices. To isolate the singularity of the kernel, we developed two techniques based on a good choice of different two sets of nodes to be distributed over the integration domain. Each set is specific to one of the kernel arguments so that the kernel values never become zero or imaginary. The significant advantage of thetwo presented techniques is the ability to gain access to an algebraic linear system equivalent to the interpolant solution without applying the collocation method. Moreover, the convergence in the mean of the interpolant solution and the maximum error norm estimation are studied. The interpolate solutions of the illustrated four examples are found strongly converging uniformly to the exact solutions.

scholarly journals Computational Method for Solving Weakly Singular Fredholm Integral Equations of the Second Kind using an Advanced Barycentric Lagrange Interpolation Formula

2021 ◽  
Author(s):  
Emil Shoukralla ◽  
Nermin Saber ◽  
Ahmed Yehia Sayed
Keyword(s):  
Integral Equations ◽  
Lagrange Interpolation ◽  
Interpolation Formula ◽  
Good Choice ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Error Norm ◽  
Monomial Basis ◽  
Weakly Singular ◽  
Lagrange Interpolation Formula

Abstract In this study, we applied an advanced barycentric Lagrange interpolation formula to find the interpolate solutions of weakly singular ‎Fredholm integral equations of the ‎second kind. The kernel is ‎interpolated twice concerning ‎both variables and then is transformed into the product of five ‎matrices; two of them are monomial basis ‎matrices. To isolate the singularity of the kernel, we ‎developed two techniques based on a good choice of ‎different two sets of nodes to be distributed ‎over the integration domain. Each set is specific to one of the ‎kernel arguments so that the kernel ‎values never become zero or imaginary. The significant advantage of the ‎two presented ‎techniques is the ability to gain ‎‎access to an algebraic linear system equivalent to the interpolant solution without applying the collocation method. Moreover, the convergence in the ‎mean of the interpolant solution ‎and the maximum error norm estimation are studied. The ‎interpolate solutions of the illustrated four ‎examples are found strongly converging uniformly to the ‎exact solutions.

scholarly journals Approximate Solution Technique for Singular Fredholm Integral Equations of the First Kind with Oscillatory Kernels

2019 ◽  
pp. 1-9
Author(s):  
Vivian Ndfutu Nfor ◽  
George Emese Okecka
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Legendre Polynomials ◽  
Lagrange Interpolation ◽  
Interpolation Formula ◽  
Verification And Validation ◽  
Solution Technique ◽  
Fredholm Integral Equations ◽  
Lagrange Interpolation Formula ◽  
Interpolation Nodes

An efficient quadrature formula was developed for evaluating numerically certain singular Fredholm integral equations of the first kind with oscillatory trigonometric kernels.  The method is based on the Lagrange interpolation formula and the orthogonal polynomial considered are the Legendre polynomials whose zeros served as interpolation nodes. A test example was provided for the verification and validation of the rule developed. The results showed the convergence of the solution and can be improved by increasing n.

Numerical study of stochastic Volterra–Fredholm integral equations by using second kind Chebyshev wavelets

2016 ◽  
Vol 24 (2) ◽  
Author(s):  
Fakhrodin Mohammadi ◽  
Parastoo Adhami
Keyword(s):  
Integral Equations ◽  
Numerical Study ◽  
Operational Matrix ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Wavelet Method ◽  
Chebyshev Wavelets ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix ◽  
Efficiency And Reliability

AbstractIn this paper, we present a computational method for solving stochastic Volterra–Fredholm integral equations which is based on the second kind Chebyshev wavelets and their stochastic operational matrix. Convergence and error analysis of the proposed method are investigated. Numerical results are compared with the block pulse functions method for some non-trivial examples. The obtained results reveal efficiency and reliability of the proposed wavelet method.

Numerical Algorithms of Optimal Complexity for Weakly Singular Volterra Integral Equations

2006 ◽  
Vol 6 (4) ◽  
pp. 436-442 ◽  
Author(s):  
A.N. Tynda
Keyword(s):  
Numerical Methods ◽  
Integral Equations ◽  
Numerical Algorithms ◽  
Volterra Integral Equations ◽  
Volterra Equations ◽  
Fredholm Integral Equations ◽  
Optimal Complexity ◽  
Weakly Singular ◽  
Different Types ◽  
Singular Kernels

AbstractIn this paper we construct complexity order optimal numerical methods for Volterra integral equations with different types of weakly singular kernels. We show that for Volterra equations (in contrast to Fredholm integral equations) using the ”block-by-block” technique it is not necessary to employ the additional iterations to construct complexity optimal methods.

scholarly journals Solving of Linear Volterra-Fredholm Integral Equations via Modification of Block Pulse Functions

2021 ◽  
Vol 17 (1) ◽  
pp. 33
Author(s):  
Ayyubi Ahmad
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Triangular System ◽  
Algebraic Equations ◽  
Block Pulse Functions ◽  
Linear Algebraic Equations

A computational method based on modification of block pulse functions is proposed for solving numerically the linear Volterra-Fredholm integral equations. We obtain integration operational matrix of modification of block pulse functions on interval [0,T). A modification of block pulse functions and their integration operational matrix can be reduced to a linear upper triangular system. Then, the problem under study is transformed to a system of linear algebraic equations which can be used to obtain an approximate solution of  linear Volterra-Fredholm integral equations. Furthermore, the rate of convergence is  O(h) and error analysis of the proposed method are investigated. The results show that the approximate solutions have a good of efficiency and accuracy.

Sign in / Sign up

Export Citation Format

Share Document