scholarly journals Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
C. E. Chidume ◽  
N. Djitté
Keyword(s):  
Hilbert Space ◽  
Galerkin Method ◽  
Integral Equations ◽  
Real Hilbert Space ◽  
Iteration Process ◽  
Nonlinear Integral Equations ◽  
Hammerstein Equation ◽  
Monotone Maps ◽  
Continuity Assumption ◽  
The Galerkin Method

Suppose that H is a real Hilbert space and F,K:H→H are bounded monotone maps with D(K)=D(F)=H. Let u* denote a solution of the Hammerstein equation u+KFu=0. An explicit iteration process is shown to converge strongly to u*. No invertibility or continuity assumption is imposed on K and the operator F is not restricted to be angle-bounded. Our result is a significant improvement on the Galerkin method of Brézis and Browder.

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) 

scholarly journals Convergence of approximate solution of mixed Hammerstein type integral equations

2018 ◽  
Vol 38 (2) ◽  
pp. 61-74
Author(s):  
Monireh Nosrati Sahlan
Keyword(s):  
Galerkin Method ◽  
Integral Equations ◽  
Basis Functions ◽  
Computational Method ◽  
Operational Matrices ◽  
B Spline ◽  
Spline Wavelets ◽  
Type Integral ◽  
The Galerkin Method ◽  
Dual Functions

In the present paper, a computational method for solving nonlinear Volterra-Fredholm Hammerestein integral equations is proposed by using compactly supported semiorthogonal cubic B-spline wavelets as basis functions. Dual functions and Operational matrices of B-spline wavelets via Galerkin method are utilized to reduce the computation of integral equations to some algebraic system, where in the Galerkin method dual of B-spline wavelets are applied as weighting functions. The method is computationally attractive, and applications are demonstrated through illustrative examples.

scholarly journals Implicit iteration process of nonexpansive non-self-mappings

2005 ◽  
Vol 2005 (19) ◽  
pp. 3103-3110
Author(s):  
Somyot Plubtieng ◽  
Rattanaporn Punpaeng
Keyword(s):  
Hilbert Space ◽  
Convex Subset ◽  
Real Hilbert Space ◽  
Iteration Process ◽  
Nonempty Closed Convex Subset ◽  
Implicit Iteration Process ◽  
Nearest Point ◽  
Nearest Point Projection ◽  
Point Projection ◽  
Implicit Iteration

SupposeCis a nonempty closed convex subset of real Hilbert spaceH. LetT:C→Hbe a nonexpansive non-self-mapping andPis the nearest point projection ofHontoC. In this paper, we study the convergence of the sequences{xn},{yn},{zn}satisfyingxn=(1−αn)u+αnT[(1−βn)xn+βnTxn],yn=(1−αn)u+αnPT[(1−βn)yn+βnPTyn], andzn=P[(1−αn)u+αnTP[(1−βn)zn+βnTzn]], where{αn}⊆(0,1),0≤βn≤β<1andαn→1asn→∞. Our results extend and improve the recent ones announced by Xu and Yin, and many others.

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