scholarly journals Approximate Solution and its Convergence Analysis for Hypersingular Integral Equations

2019 ◽  
Vol 69 (2) ◽  
pp. 173-178
Author(s):  
Vaishali Sharma ◽  
Amit Setia
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Analytic Solution ◽  
Singular Integral Equations ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Hypersingular Integral ◽  
Hypersingular Integral Equations ◽  
Flight Vehicles ◽  
Theoretical Results

This paper propose a residual based Galerkin method with Legendre polynomial as a basis functions to find the approximate solution of hypersingular integral equations. These equations   occur quite naturally in the field of aeronautics  such as  problem of aerodynamics of flight vehicles  and during mathematical modeling  of  vortex  wakes  behind  aircraft.  The analytic solution of these kind of equations is known only for a particular case ( m(x,t) =0 in Eqn (1)).  Also,   in these  singular integral equations which occur during the formulation of many boundary value problems, the known function  m(x,t) in (Eqn (1)) is not always   zero.  Our proposed method find the approximate solution by converting the integral equations into a linear system of algebraic equations which is easy to solve. The convergence of sequence of approximate solutions is proved and error bound is obtained theoretically. The validation of derived theoretical results and implementation of method is also shown with the aid of numerical illustrations.                            

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.

An approximate solution of nonlinear hypersingular integral equations

2014 ◽  
Vol 86 ◽  
pp. 1-21 ◽  
Author(s):  
I.V. Boykov ◽  
E.S. Ventsel ◽  
V.A. Roudnev ◽  
A.I. Boykova
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Hypersingular Integral ◽  
Hypersingular Integral Equations

Orthogonal Approximate Solution of Cauchy-type Singular Integral Equations

2003 ◽  
Vol 3 (2) ◽  
pp. 330-356
Author(s):  
R. Smarzewski ◽  
M. A. Sheshko
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Singular Integral ◽  
Chebyshev Polynomials ◽  
Singular Integral Equations ◽  
Approximate Solutions ◽  
Cauchy Type

AbstractChebyshev polynomials of the first and second kind are used to derive approximate solutions of the Cauchy-type singular integral equations.

An Iterative Algorithm of Hypersingular Integral Equations for Crack Problems

2008 ◽  
Vol 385-387 ◽  
pp. 793-796
Author(s):  
Kazuhiro Oda ◽  
Naoaki Noda
Keyword(s):  
Integral Equations ◽  
Body Force ◽  
The Body ◽  
Algebraic Equations ◽  
Hypersingular Integral ◽  
Force Method ◽  
Hypersingular Integral Equations ◽  
Body Force Method ◽  
Linear Algebraic Equations ◽  
Crack Problems

Crack problems are reducible to singular integral equations with strongly singular kernels by means of the body force method. In the ordinary method, the integral equations are reduced to a system of linear algebraic equations. In this paper, an iterative method for the numerical solution of the hypersingular integral equations of the body force method is proposed. This method is based on the Gauss- Chebyshev numerical integration rule and is very simple to program. The solution is achieved without solving the system of linear algebraic equations. The proposed method is applied to some plane elasticity crack problems and is seen to give convergent results.

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