scholarly journals Numerical Solutions Of Volterra Integral Equations Using Legendre Polynomials

2013 ◽  
Vol 32 ◽  
pp. 29-35
Author(s):  
Md Azizur Rahman ◽  
Md Shafiqul Islam
Keyword(s):  
Integral Equations ◽  
Legendre Polynomials ◽  
Numerical Solutions ◽  
Volterra Integral Equations ◽  
Matrix Formulation ◽  
Numerical Examples ◽  
Piecewise Polynomials ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Linear Volterra Integral Equations

In this paper, Legendre piecewise polynomials are used to approximate the solutions of linear Volterra integral equations. Both second and first kind integral equations with regular as well as weakly singular kernels are considered. A matrix formulation is given for linear Volterra integral equations by the technique of Galerkin method. Numerical examples are considered to verify the accuracy of the proposed derivations, and the numerical solutions in this paper are also compared with the existing methods in the published literature. DOI: http://dx.doi.org/10.3329/ganit.v32i0.13643 GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 32 (2012) 29 – 35  

scholarly journals Numerical Solutions of Volterra Integral Equations Using Laguerre Polynomials

2012 ◽  
Vol 4 (2) ◽  
pp. 357 ◽  
Author(s):  
M. A. Rahman ◽  
M. S. Islam ◽  
M. M. Alam
Keyword(s):  
Integral Equations ◽  
Present Method ◽  
Numerical Solutions ◽  
Laguerre Polynomials ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Weighted Residual Method ◽  
Matrix Formulation ◽  
Numerical Examples ◽  
Singular Kernels

We solve numerically Volterra integral equations, of first and second kind with regular and singular kernels, by the well known Galerkin weighted residual method. For this, we derive a simple and efficient matrix formulation using Laguerre polynomials as trial functions. Several numerical examples are tested. The approximate solutions of some examples coincide with the exact solutions on using a very few Laguerre polynomials. The approximate results, obtained by the present method, confirm the convergence of numerical solutions and are compared with the existing methods available in the literature.Keywords: Volterra integral equations; Galerkin method; Laguerre polynomials.© 2012 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi: http://dx.doi.org/10.3329/jsr.v4i2.9407    J. Sci. Res. 4 (2), 357-364 (2012)

scholarly journals NUMERICAL SOLUTION OF VOLTERRA INTEGRAL EQUATIONS WITH WEAKLY SINGULAR KERNELS WHICH MAY HAVE A BOUNDARY SINGULARITY

2009 ◽  
Vol 14 (1) ◽  
pp. 79-89 ◽  
Author(s):  
Marek Kolk ◽  
Arvet Pedas
Keyword(s):  
Integral Equations ◽  
Volterra Integral Equations ◽  
Piecewise Polynomial ◽  
Boundary Singularity ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Convergence Estimates ◽  
Linear Volterra Integral Equations ◽  
Piecewise Polynomial Collocation Method ◽  
Polynomial Collocation

We propose a piecewise polynomial collocation method for solving linear Volterra integral equations of the second kind with kernels which, in addition to a weak diagonal singularity, may have a weak boundary singularity. Global convergence estimates are derived and a collection of numerical results is given.

scholarly journals Singularly perturbed Volterra integral equations with weakly singular kernels

2002 ◽  
Vol 30 (3) ◽  
pp. 129-143 ◽  
Author(s):  
Angelina Bijura
Keyword(s):  
Integral Equations ◽  
Special Function ◽  
Initial Boundary ◽  
Volterra Integral Equations ◽  
Singularly Perturbed ◽  
Interesting Aspect ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Linear Volterra Integral Equations

We consider finding asymptotic solutions of the singularly perturbed linear Volterra integral equations with weakly singular kernels. An interesting aspect of these problems is that the discontinuity of the kernel causes layer solutions to decay algebraically rather than exponentially within the initial (boundary) layer. To analyse this phenomenon, the paper demonstrates the similarity that these solutions have to a special function called the Mittag-Leffler function.

scholarly journals Solutions of Linear and Nonlinear Volterra Integral Equations Using Hermite and Chebyshev Polynomials

2013 ◽  
Vol 11 (8) ◽  
pp. 2910-2920
Author(s):  
Md. Shafiqul Islam ◽  
Md. Azizur Rahman
Keyword(s):  
Integral Equations ◽  
Good Accuracy ◽  
Numerical Approach ◽  
Volterra Integral Equations ◽  
Basis Functions ◽  
Matrix Formulation ◽  
Numerical Examples ◽  
Singular Kernels ◽  
Piecewise Continuous ◽  
Linear And Nonlinear

The purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on Galerkin weighted residual approximation. In this method Hermite and Chebyshev piecewise, continuous and differentiable polynomials are exploited as basis functions. A rigorous effective matrix formulation is proposed to solve the linear and nonlinear Volterra integral equations of the first and second kind with regular and singular kernels. The algorithm is simple and can be coded easily. The efficiency of the proposed method is tested on several numerical examples to get the desired and reliable good accuracy.

A stochastic operational matrix method for numerical solutions of multi-dimensional stochastic Itô–Volterra integral equations

2020 ◽  
Vol 28 (3) ◽  
pp. 209-216
Author(s):  
S. Singh ◽  
S. Saha Ray
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Numerical Examples ◽  
Operational Matrix Method ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix ◽  
Block Pulse Function

AbstractIn this article, hybrid Legendre block-pulse functions are implemented in determining the approximate solutions for multi-dimensional stochastic Itô–Volterra integral equations. The block-pulse function and the proposed scheme are used for deriving a methodology to obtain the stochastic operational matrix. Error and convergence analysis of the scheme is discussed. A brief discussion including numerical examples has been provided to justify the efficiency of the mentioned method.

scholarly journals Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials

2010 ◽  
Vol 2 (2) ◽  
pp. 264-272 ◽  
Author(s):  
A. Shirin ◽  
M. S. Islam
Keyword(s):  
Integral Equation ◽  
Galerkin Method ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Fredholm Integral Equations ◽  
Matrix Formulation ◽  
Piecewise Polynomials ◽  
The Galerkin Method

In this paper, Bernstein piecewise polynomials are used to solve the integral equations numerically. A matrix formulation is given for a non-singular linear Fredholm Integral Equation by the technique of Galerkin method. In the Galerkin method, the Bernstein polynomials are used as the approximation of basis functions. Examples are considered to verify the effectiveness of the proposed derivations, and the numerical solutions guarantee the desired accuracy.  Keywords: Fredholm integral equation; Galerkin method; Bernstein polynomials. © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v2i2.4483               J. Sci. Res. 2 (2), 264-272 (2010) 

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