scholarly journals Galerkin Approximations for the Solution of Fredholm Volterra Integral Equation of Second Kind

2021 ◽  
Vol 41 (1) ◽  
pp. 1-14
Author(s):  
Asma Akter Akhia ◽  
Goutam Saha
Keyword(s):  
Integral Equation ◽  
Galerkin Method ◽  
Volterra Integral Equation ◽  
Numerical Solutions ◽  
Laguerre Polynomials ◽  
Hermite Polynomials ◽  
Approximate Solutions ◽  
Linear Combinations ◽  
Galerkin Approximations ◽  
Error Estimations

In this research, we have introduced Galerkin method for finding approximate solutions of Fredholm Volterra Integral Equation (FVIE) of 2nd kind, and this method shows the result in respect of the linear combinations of basis polynomials. Here, BF (product of Bernstein and Fibonacci polynomials), CH (product of Chebyshev and Hermite polynomials), CL (product of Chebyshev and Laguerre polynomials), FL (product of Fibonacci and Laguerre polynomials) and LLE (product of Legendre and Laguerre polynomials) polynomials are established and considered as basis function in Galerkin method. Also, we have tried to observe the behavior of all these approximate solutions finding from Galerkin method for different problems and then a comparison is shown using some standard error estimations. In addition, we observe the error graphs of numerical solutions in Galerkin method for different problems of FVIE of second kind. GANITJ. Bangladesh Math. Soc.41.1 (2021) 1–14

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) 

On the numerical solutions of Fredholm–Volterra integral equation

2003 ◽  
Vol 146 (2-3) ◽  
pp. 713-728 ◽  
Author(s):  
M.A. Abdou ◽  
Khamis I. Mohamed ◽  
A.S. Ismail
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Numerical Solutions

Jacobi Spectral Galerkin and Iterated Methods for Nonlinear Volterra Integral Equation

2016 ◽  
Vol 11 (4) ◽  
Author(s):  
Yin Yang ◽  
Yanping Chen
Keyword(s):  
Integral Equation ◽  
Galerkin Method ◽  
Rate Of Convergence ◽  
Volterra Integral Equation ◽  
Galerkin Methods ◽  
Nonlinear Volterra Integral Equation ◽  
L2 Norm ◽  
Spectral Galerkin Method ◽  
Global Superconvergence ◽  
Methods Numerical

In this paper, a Jacobi spectral Galerkin method is developed for nonlinear Volterra integral equations (VIEs) of the second kind. The spectral rate of convergence for the proposed method is established in the L∞-norm and the weighted L2-norm. Global superconvergence properties are discussed by iterated Galerkin methods. Numerical results are presented to demonstrate the effectiveness of the proposed method.

Calculated Pressure Distributions and Shock Shapes on Thick Conical Wings at High Supersonic Speeds

1967 ◽  
Vol 18 (2) ◽  
pp. 185-206 ◽  
Author(s):  
L. C. Squire
Keyword(s):  
Integral Equation ◽  
Cross Sections ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Poor Agreement ◽  
Delta Wings ◽  
Newtonian Theory ◽  
Pressure Distributions ◽  
First Order Correction ◽  
Good Agreement

SummaryIn recent papers Messiter and Hida have proposed a first-order correction to simple Newtonian theory for the pressure distributions on the lower surfaces of lifting conical bodies with detached shocks. The method involves the solution of an integral equation which Messiter solved numerically for thin delta wings, while Hida gave an approximate solution for thick wings with diamond and bi-convex cross-sections. It is shown in the present paper that Hida’s approximate solutions give poor agreement with experiment, and a series of more precise numerical solutions of the equation are given for wings with diamond cross-sections. The pressures, and shock shapes, obtained from these solutions are in very good agreement with experiment at Mach numbers as low as 4·0.The method has also been extended to Nonweiler wings at off-design when the shock wave is detached from the leading edges. Again the agreement with experiment is good provided the integral equation is solved numerically.

scholarly journals On the Analysis of Numerical Methods for Nonstandard Volterra Integral Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
H. S. Mamba ◽  
M. Khumalo
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Existence And Uniqueness ◽  
Numerical Experiments ◽  
Numerical Solutions ◽  
Trapezoidal Rule ◽  
Collocation Methods ◽  
Point Collocation ◽  
The One ◽  
Theoretical Results

We consider the numerical solutions of a class of nonlinear (nonstandard) Volterra integral equation. We prove the existence and uniqueness of the one point collocation solutions and the solution by the repeated trapezoidal rule for the nonlinear Volterra integral equation. We analyze the convergence of the collocation methods and the repeated trapezoidal rule. Numerical experiments are used to illustrate theoretical results.

scholarly journals On the Stability of Numerical Methods for Nonlinear Volterra Integral Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-18 ◽  
Author(s):  
E. Messina ◽  
Y. Muroya ◽  
E. Russo ◽  
A. Vecchio
Keyword(s):  
Integral Equation ◽  
Numerical Methods ◽  
Numerical Solution ◽  
Volterra Integral Equation ◽  
Linear Part ◽  
Approximate Solutions ◽  
Constant Sign ◽  
Quadrature Methods ◽  
Decay Of Solutions ◽  
The Stability

Here we investigate the behavior of the analytical and numerical solution of a nonlinear second kind Volterra integral equation where the linear part of the kernel has a constant sign and we provide conditions for the boundedness or decay of solutions and approximate solutions obtained by Volterra Runge-Kutta and Direct Quadrature methods.

scholarly journals On the Use of Piecewise Standard Polynomials in the Numerical Solutions of Fourth Order Boundary Value Problems

2014 ◽  
Vol 33 ◽  
pp. 53-64 ◽  
Author(s):  
Md. Shafiqul Islam ◽  
Md. Bellal Hossain
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Fourth Order ◽  
Legendre Polynomials ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Linear Combinations ◽  
Piecewise Continuous ◽  
The Given

This paper is devoted to find the numerical solutions of the fourth order linear and nonlinear differential equations using piecewise continuous and differentiable polynomials, such as Bernstein, Bernoulli and Legendre polynomials with specified boundary conditions. We derive rigorous matrix formulations for solving linear and non-linear fourth order BVP and special care is taken about how the polynomials satisfy the given boundary conditions. The linear combinations of each polynomial are exploited in the Galerkin weighted residual approximation. The derived formulation is illustrated through various numerical examples. Our approximate solutions are compared with the exact solutions, and also with the solutions of the existing methods. The approximate solutions converge to the exact solutions monotonically even with desired large significant digits. GANIT J. Bangladesh Math. Soc. Vol. 33 (2013) 53-64 DOI: http://dx.doi.org/10.3329/ganit.v33i0.17659

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