Solutions Of Hammerstein Integral Equations Research Articles

Spectral Methods For Hammerstein Integral Equations with Nonsmooth Kernels

In this paper, polynomial-based projection type and modified projection-type methods for approximating the solution of Hammerstein integral equations with a kernel of Green’s function type are proposed. The projection is either an orthogonal projection or an interpolatory projection using Legendre polynomial basis. The orders of convergence of these methods and those of superconvergence of the iterated modified projection-type methods are analyzed. A numerical example is given to illustrate the theoretical estimates.

On the existence of the solution of Hammerstein integral equations and fractional differential equations

On the existence of the solution of Hammerstein integral equations and fractional differential equations

On the numerical treatment and analysis of Hammerstein integral equation

In this paper, we study the quadratic rules for the numerical solution of Hammerstein integral equation based on spline quasi-interpolant. Also the convergence analysis of the methods are given. The theoretical behavior is tested on examples and it is shown that the numerical results confirm theoretical part.

Explicit algorithms for J-fixed points of some non linear mappings in certain Banach spaces

Let E be a real normed linear space and E∗ its dual. In a recent work, Chidume et al. [Chidume, C. E. and Idu, K. O., Approximation of zeros of bounded maximal monotone mappings, solutions of hammerstein integral equations and convex minimizations problems, Fixed Point Theory and Applications, 97 (2016)] introduced the new concepts of J-fixed points and J-pseudocontractive mappings and they shown that a mapping A : E → 2 E∗ is monotone if and only if the map T := (J −A) : E → 2 E∗ is J-pseudocontractive, where J is the normalized duality mapping of E. It is our purpose in this work to introduce an algorithm for approximating J-fixed points of J-pseudocontractive mappings. Our results are applied to approximate zeros of monotone mappings in certain Banach spaces. The results obtained here, extend and unify some recent results in this direction for the class of maximal monotone mappings in uniformly smooth and strictly convex real Banach spaces. Our proof is of independent interest.

Superconvergent product integration method for Hammerstein integral equations

In this paper, we define a superconvergent projection method for approximating the solution of Hammerstein integral equations of the second kind. The projection is chosen either to be the orthogonal or an interpolatory projection at Gauss points onto the space of discontinuous piecewise polynomials. For a smooth kernel or one less smooth along the diagonal, the order of convergence of the proposed method improves upon the classical product integration method. Several numerical examples are given to demonstrate the effectiveness of the current method.

B-spline Wavelet Method for Solving Fredholm Hammerstein Integral Equation Arising from Chemical Reactor Theory

AbstractMathematical model for an adiabatic tubular chemical reactor which processes an irreversible exothermic chemical reaction has been considered. For steady state solution for an adiabatic tubular chemical reactor, the model can be reduced to ordinary differential equation with a parameter in the boundary conditions. Again the ordinary differential equation has been converted into a Hammerstein integral equation which can be solved numerically. B-spline wavelet method has been developed to approximate the solution of Hammerstein integral equation. This method reduces the integral equation to a system of algebraic equations. The numerical results obtained by the present method have been compared with the available results.

Viscosity iterative algorithm for the zero point of monotone mappings in Banach spaces

Inspired by the work of Zegeye (J. Math. Anal. Appl. 343:663–671, 2008) and the recent papers of Chidume et al. (Fixed Point Theory Appl. 2016:97, 2016; Br. J. Math. Comput. Sci. 18:1–14, 2016), we devise a viscosity iterative algorithm without involving the resolvent operator for approximating the zero of a monotone mapping in the setting of uniformly convex Banach spaces. Under concise parameter conditions we establish strong convergence of the proposed algorithm. Moreover, applications to constrained convex minimization problems and solution of Hammerstein integral equations are included. Finally, the performances and computational examples and a comparison with related algorithms are presented to illustrate the efficiency and applicability of our new algorithm.

New semi-implicit midpoint rule for zero of monotone mappings in Banach spaces

Let E be a nonempty closed uniformly convex and 2-uniformly smooth Banach space with dual E∗ and A : E∗ → E be Lipschitz continuous monotone mapping with A− 1(0) ≠ ∅. A new semi-implicit midpoint rule (SIMR) with the general contraction for monotone mappings in Banach spaces is established and proved to converge strongly to x∗ ∈ E, where Jx∗ ∈ A− 1(0). Moreover, applications to convex minimization problems, solution of Hammerstein integral equations, and semi-fixed point of a cluster of semi-pseudo mappings are included.

Algorithm for approximating solutions of Hammerstein integral equations with maximal monotone operators

Let X be a uniformly convex and uniformly smooth real Banach space with dual space X*. Let F: X → X* and K: X* → X be bounded monotone mappings such that the Hammerstein equation u + KFu = 0 has a solution. An explicit iteration sequence is constructed and proved to converge strongly to a solution of this equation.

On the numerical solution of Hammerstein integral equations using shifted Chebyshev polynomials of the third kind method

On the numerical solution of Hammerstein integral equations using shifted Chebyshev polynomials of the third kind method

Approximation of zeros of bounded maximal monotone mappings, solutions of Hammerstein integral equations and convex minimization problems

AbstractLet E be a real normed space with dual space $E^{*}$ E ∗ and let $A:E\rightarrow2^{E^{*}}$ A : E → 2 E ∗ be any map. Let $J:E\rightarrow2^{E^{*}}$ J : E → 2 E ∗ be the normalized duality map on E. A new class of mappings, J-pseudocontractive maps, is introduced and the notion of J-fixed points is used to prove that $T:=(J-A)$ T : = ( J − A ) is J-pseudocontractive if and only if A is monotone. In the case that E is a uniformly convex and uniformly smooth real Banach space with dual $E^{*}$ E ∗ , $T: E\rightarrow2^{E^{*}}$ T : E → 2 E ∗ is a bounded J-pseudocontractive map with a nonempty J-fixed point set, and $J-T :E\rightarrow2^{E^{*}}$ J − T : E → 2 E ∗ is maximal monotone, a sequence is constructed which converges strongly to a J-fixed point of T. As an immediate consequence of this result, an analog of a recent important result of Chidume for bounded m-accretive maps is obtained in the case that $A:E\rightarrow2^{E^{*}}$ A : E → 2 E ∗ is bounded maximal monotone, a result which complements the proximal point algorithm of Martinet and Rockafellar. Furthermore, this analog is applied to approximate solutions of Hammerstein integral equations and is also applied to convex optimization problems. Finally, the techniques of the proofs are of independent interest.

Existence and approximation of solution for a nonlinear second-order three-point boundary value problem

A nonlinear second-order ordinary differential equation with four cases of three-point boundary value conditions is studied by investigating the existence and approximation of solutions. First, the integration method is proposed to transform the considered boundary value problems into Hammerstein integral equations. Second, the existence of solutions for the obtained Hammerstein integral equations is analyzed by using the Schauder fixed point theorem. The contraction mapping theorem in Banach spaces is further used to address the uniqueness of solutions. Third, the approximate solution of Hammerstein integral equations is constructed by using a new numerical method, and its convergence and error estimate are analyzed. Finally, some numerical examples are addressed to verify the given theorems and methods.

Iterative Approximation of Solutions of Hammerstein Integral Equations with Maximal Monotone Operators in Banach Spaces

Iterative Approximation of Solutions of Hammerstein Integral Equations with Maximal Monotone Operators in Banach Spaces

An Algorithm for Solutions of Hammerstein Integral Equations with Maximal Monotone Operators in Classical Banach Spaces

An Algorithm for Solutions of Hammerstein Integral Equations with Maximal Monotone Operators in Classical Banach Spaces

Legendre Spectral Projection Methods for Fredholm–Hammerstein Integral Equations

In this paper, we consider the Legendre spectral Galerkin and Legendre spectral collocation methods to approximate the solution of Hammerstein integral equation. The convergence of the approximate solutions to the actual solution is discussed and the rates of convergence are obtained. We are able to obtain similar superconvergence rates for the iterated Legendre Galerkin solution for Hammerstein integral equations with smooth kernel as in the case of piecewise polynomial based Galerkin method.

Convergence analysis of Legendre spectral projection methods for Hammerstein integral equations of mixed type

In this paper, we consider the Legendre spectral Galerkin and Legendre spectral collocation methods to approximate the solution of Hammerstein integral equations of mixed type. We prove that the approximated solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the same orders, $$\mathcal {O}(n^{-r})$$ in $$L^{2}$$ -norm and $$\mathcal {O}(n^{\frac{1}{2}-r})$$ in infinity norm, and the iterated Legendre Galerkin solution converges with the order $$\mathcal {O}(n^{-2r})$$ in both $$L^{2}$$ -norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order $$\mathcal {O}(n^{-r})$$ in both $$L^{2}$$ -norm and infinity norm, n being the highest degree of Legendre polynomial employed in the approximation and r being the smoothness of the kernels.

Numerical solution of Hammerstein integral equations of mixed type using the Sinc-collocation method

In this paper, numerical solution of Hammerstein integral equations of mixed type by means of Sinc collocation method is presented. This proposed approximation reduces these kind of nonlinear Hammerstein integral equations of mixed type to a nonlinear system of algebraic equations. We prove that the Sinc procedure for Hammerstein integral equations of mixed type converges to the real solution at an exponential rate of O(e−kN) where k>0 is a constant. Some numerical examples are provided to illustrate the accuracy and the exponential rate of convergence in the proposed method.

Solutions of Hammerstein equations in the space BV ()

In this paper we study the existence of solutions for Hammerstein integral equations in the space of two-variables bounded variation functions BV ()where a = (a1, a2), b = (b1, b2) ∈ ℝ2, = [a1, b1]×[a2, b2], v : → ℝ, k : × → ℝ, → ℝ, and f : × ℝ → ℝ. Under some conditions on λ, k and f we prove that this equation admits only one solution in the space BV () for each v ∈ BV ().

A numerical approximation for solutions of Hammerstein integral equations in Lp spaces

In this work, we give conditions that guarantee the exis- tence and the uniqueness of the solution of the Hammerstein integral equation in the L p space and under such assumptions the successive ap- proximation converges almost everywhere to the solution of the equa- tion. Finally, we treat numerical exemples to conrm these results.

An Iterative Algorithm for Approximating Solutions of Hammerstein Integral Equations

Suppose E is a uniformly smooth real Banach space and F, K: E → E are bounded accretive maps with D(K) = F(E) = E. Assume that the Hammerstein equation u + KFu = 0 has a solution u*. An explicit simple iteration process is shown to converge strongly to u*. No invertibility or continuity assumption is imposed on K. Our theorems are significant improvements on important recent results (e.g. [11]).