A New Computational Method for Solving Weakly Singular Fredholm Integral Equations of the First Kind

2018 ◽  
Author(s):  
E. S. Shoukralla ◽  
M. Kamel ◽  
M. A. Markos
Keyword(s):  
Integral Equations ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Weakly Singular

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.

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.

scholarly journals FAST SOLVERS OF WEAKLY SINGULAR INTEGRAL EQUATIONS OF THE SECOND KIND

2018 ◽  
Vol 23 (4) ◽  
pp. 639-664 ◽  
Author(s):  
Sumaira Rehman ◽  
Arvet Pedas ◽  
Gennadi Vainikko
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Fredholm Integral Equations ◽  
Free Term ◽  
Fast Solvers ◽  
Weakly Singular ◽  
Weakly Singular Integral Equations ◽  
Derivatives Of ◽  
Smooth Data

We discuss the bounds of fast solving weakly singular Fredholm integral equations of the second kind with a possible diagonal singularity of the kernel and certain boundary singularities of the derivatives of the free term when the information about the smooth coefficient functions in the kernel and about the free term is restricted to a given number of sample values. In this situation, a fast/quasifast solver is constructed. Thus the complexity of weakly singular integral equations occurs to be close to that of equations with smooth data without singularities. Our construction of fast/quasifast solvers is based on the periodization of the problem.

A Computational Method for Solving Fredholm Integral Equations of the Second Kind

2019 ◽  
Vol 28 (1) ◽  
pp. 280-285
Author(s):  
Shoukralla, E. S. ◽  
EL-Serafi, S. A. ◽  
Elgohary, H. ◽  
Morgan, M.
Keyword(s):  
Integral Equations ◽  
Computational Method ◽  
Fredholm Integral Equations
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