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.

scholarly journals Solving volterra integral equation via nonlinear programming

2016 ◽  
Vol 5 (4) ◽  
pp. 192
Author(s):  
Jamal Othman
Keyword(s):  
Integral Equation ◽  
Nonlinear Programming ◽  
Volterra Integral Equation ◽  
Least Square ◽  
Nonlinear Programming Problem ◽  
Programming Technique ◽  
New Approach ◽  
Nonlinear Volterra Integral Equation ◽  
Limited Memory ◽  
Quasi Newton

In this paper we propose an approach to find approximate solution to the nonlinear Volterra integral equation of the second type through a nonlinear programming technique by firstly converting the integral equation into a least square cost function as an objective function for an unconstrained nonlinear programming problem which solved by a nonlinear programming technique (The preconditioned limited- memory quasi-Newton conjugates, gradient method) and as far as we read this is a new approach in the ways of solving the nonlinear Volterra integral equation. We use Maple 11 software as a tool for performing the suggested steps in solving the examples.

scholarly journals Nonlinear Volterra Integral Equation of the Second Kind and Biorthogonal Systems

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
M. I. Berenguer ◽  
D. Gámez ◽  
A. I. Garralda-Guillem ◽  
M. C. Serrano Pérez
Keyword(s):  
Banach Space ◽  
Integral Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Volterra Integral Equation ◽  
Biorthogonal System ◽  
Banach Fixed Point Theorem ◽  
Nonlinear Volterra Integral Equation ◽  
Numerical Resolution

We obtain an approximation of the solution of the nonlinear Volterra integral equation of the second kind, by means of a new method for its numerical resolution. The main tools used to establish it are the properties of a biorthogonal system in a Banach space and the Banach fixed point theorem.

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