Integral Equations Of Hammerstein Type Research Articles

Analysis of efficient discretization technique for nonlinear integral equations of Hammerstein type

Purpose This study focuses on investigating the numerical solution of second-kind nonlinear Volterra–Fredholm–Hammerstein integral equations (NVFHIEs) by discretization technique. The purpose of this paper is to develop an efficient and accurate method for solving NVFHIEs, which are crucial for modeling systems with memory and cumulative effects, integrating past and present influences with nonlinear interactions. They are widely applied in control theory, population dynamics and physics. These equations are essential for solving complex real-world problems. Design/methodology/approach Demonstrating the solution’s existence and uniqueness in the equation is accomplished by using the Picard iterative method as a key technique. Using the trapezoidal discretization method is the chosen approach for numerically approximating the solution, yielding a nonlinear system of algebraic equations. The trapezoidal method (TM) exhibits quadratic convergence to the solution, supported by the application of a discrete Grönwall inequality. A novel Grönwall inequality is introduced to demonstrate the convergence of the considered method. This approach enables a detailed analysis of the equation’s behavior and facilitates the development of a robust solution method. Findings The numerical results conclusively show that the proposed method is highly efficacious in solving NVFHIEs, significantly reducing computational effort. Numerical examples and comparisons underscore the method’s practicality, effectiveness and reliability, confirming its outstanding performance compared to the referenced method. Originality/value Unlike existing approaches that rely on a combination of methods to tackle different aspects of the complex problems, especially nonlinear integral equations, the current approach presents a significant single-method solution, providing a comprehensive approach to solving the entire problem. Furthermore, the present work introduces the first numerical approaches for the considered integral equation, which has not been previously explored in the existing literature. To the best of the authors’ knowledge, the work is the first to address this equation, providing a foundational contribution for future research and applications. This innovative strategy not only simplifies the computational process but also offers a more comprehensive understanding of the problem’s dynamics.

Superconvergent method for weakly singular Fredholm-Hammerstein integral equations with non-smooth solutions and its application

In this article, we propose shifted Jacobi spectral Galerkin method (SJSGM) and iterated SJSGM to solve nonlinear Fredholm integral equations of Hammerstein type with weakly singular kernel. We have rigorously studied convergence analysis of the proposed methods. Even though the exact solution exhibits non-smooth behaviour, we manage to achieve superconvergence order for the iterated SJSGM. Further, using smoothing transformation, we improve the regularity of the exact solution, which enhances the convergence order of the SJSGM and iterated SJSGM. We have also shown the applicability of our proposed methods to high-order nonlinear weakly singular integro-differential equations and achieved superconvergence. Several numerical examples have been implemented to demonstrate the theoretical results.

Asymptotic Behavior of the Solution for One Class of Nonlinear Integral Equations of Hammerstein Type on the Whole Axis

Asymptotic Behavior of the Solution for One Class of Nonlinear Integral Equations of Hammerstein Type on the Whole Axis

Location, separation and approximation of solutions of nonlinear Hammerstein-type integral equations

From Newton's method, we construct a Newton-type iterative method that allows studying a class of nonlinear Hammerstein-type integral equations. This method is reduced to Newton's method if the kernel of the integral equation is separable and, unlike Newton's method, can be applied to approximate a solution if the kernel is nonseparable. In addition, from an analysis of the global convergence of the method, we can locate and separate solutions of the nonlinear Hammerstein-type integral equations involved. For this study of the global convergence, we use auxiliary functions and obtain restricted global convergence domains that are usually balls.

Solution of Hammerstein type integral equation with two variables via a new fixed point theorem

Solution of Hammerstein type integral equation with two variables via a new fixed point theorem

Kurchatov-type methods for non-differentiable Hammerstein-type integral equations

We consider a generic type of nonlinear Hammerstein-type integral equations with the particularity of having non-differentiable kernel of Nemystkii type. So, in order to solve it we consider a uniparametric family of iterative processes derivative free, with the main advantage that for a special value of the involved parameter the iterative method obtained coincides with Newton’s method, that is due to the fact of evaluating the divided difference operator when the two values are the same. We perform a qualitative convergence study by choosing an auxiliary point, that allow us to obtain the existence and separation of solutions of the given equation, that is, local and semilocal convergence balls can be obtained.

Asymptotic Behavior of the Solution for One Class of Nonlinear Integral Equations of Hammerstein Type on the Whole Axis

A class of nonlinear integral equations on the whole axis with a noncompact integral operator of Hammerstein type is investigated. This class of equations has applications in various fields of natural science. In particular, such equations are found in mathematical biology, in the kinetic theory of gases, in the theory of radiation transfer, etc. The existence of a nonnegative nontrivial and bounded solution is proved. The asymptotic behavior of the constructed solution on ±∞ is studied. In one important special case, the uniqueness of the constructed solution in a certain weighted space is established. At the end of the work, specific applied examples of the equations under study are given.

Semilocal Convergence of the Extension of Chun’s Method

In this work, we use the technique of recurrence relations to prove the semilocal convergence in Banach spaces of the multidimensional extension of Chun’s iterative method. This is an iterative method of fourth order, that can be transferred to the multivariable case by using the divided difference operator. We obtain the domain of existence and uniqueness by taking a suitable starting point and imposing a Lipschitz condition to the first Fréchet derivative in the whole domain. Moreover, we apply the theoretical results obtained to a nonlinear integral equation of Hammerstein type, showing the applicability of our results.

Convergence analysis of Galerkin and multi-Galerkin methods for nonlinear-Hammerstein integral equations on the half-line using Laguerre polynomials

In this paper, we consider Galerkin and multi-Galerkin methods and their iterated versions for solving the nonlinear Hammerstein-type integral equation on the half-line with sufficiently smooth kernels, using Laguerre polynomials as basis functions. We obtain optimal convergence results in iterated-Galerkin method in both infinity and weighted -norms. We also obtain the superconvergence results in both multi-Galerkin and iterated multi-Galerkin methods, respectively, in weighted -norm. Numerical results are presented to validate the theoretical results.

On generalized $Phi$-strongly monotone mappings and algorithms for the solution of equations of Hammerstein type

‎In this paper‎, ‎we consider the class of generalized $Phi$-strongly monotone mappings and the methods of approximating a solution of equations of Hammerstein type‎. ‎Auxiliary mapping is defined for nonlinear integral equations of Hammerstein type‎. ‎The auxiliary mapping is the composition of bounded generalized $Phi$-strongly monotone mappings which satisfy the range condition‎. ‎Suitable conditions are imposed to obtain the boundedness and to show that the auxiliary mapping is a generalized $Phi$-strongly which satisfies the range condition‎. ‎A sequence is constructed and it is shown that it converges strongly to a solution of equations of Hammerstein type‎. ‎The results in this paper improve and extend some recent corresponding results on the approximation of a solution of equations of Hammerstein type‎.

On semilocal convergence of three-step Kurchatov method under weak condition

The purpose of this paper to establish the semilocal convergence analysis of three-step Kurchatov method under weaker conditions in Banach spaces. We construct the recurrence relations under the assumption that involved first-order divided difference operators satisfy the omega condition. Theorems are given for the existence-uniqueness balls enclosing the unique solution. The application of the iterative method is shown by solving nonlinear system of equations and nonlinear Hammerstein-type integral equations. It illustrates the theoretical development of this study.

Cone expansion and cone compression fixed point theorems for sum of two operators and their applications

In this article, we first establish a series of user-friendly versions of fixed point theorems in cones for sum of two operators with one being either contractive or expansive and the other being compact, one of which covers the classical Krasnoselskii’s fixed point theorem concerning cone expansion and compression of norm type. Along the way, we also offer some sufficient conditions which slightly relax the compactness requirement on the one summand operator. Second, as applications to some of our main results, we consider the eigenvalue problems of Krasnoselskii type in critical case and the existence of one positive solution to one parameter operator equations. Finally, to illustrate the usefulness and the applicability of our fixed point results, we study the existence of one nontrivial positive solution to certain integral equations of Hammerstein type and of perturbed Volterra type.

Convergence and dynamics of improved Chebyshev-Secant-type methods for non differentiable operators

In this paper, the convergence and dynamics of improved Chebyshev-Secant-type iterative methods are studied for solving nonlinear equations in Banach space settings. Their semilocal convergence is established using recurrence relations under weaker continuity conditions on first-order divided differences. Convergence theorems are established for the existence-uniqueness of the solutions. Next, center-Lipschitz condition is defined on the first-order divided differences and its influence on the domain of starting iterates is compared with those corresponding to the domain of Lipschitz conditions. Several numerical examples including Automotive Steering problems and nonlinear mixed Hammerstein-type integral equations are analyzed, and the output results are compared with those obtained by some of similar existing iterative methods. It is found that improved results are obtained for all the numerical examples. Further, the dynamical analysis of the iterative method is carried out. It confirms that the proposed iterative method has better stability properties than its competitors.

Galerkin and multi-Galerkin methods for weakly singular Volterra–Hammerstein integral equations and their convergence analysis

In this paper, we consider the Galerkin and iterated Galerkin methods to approximate the solution of the second kind nonlinear Volterra integral equations of Hammerstein type with the weakly singular kernel of algebraic type, using piecewise polynomial basis functions based on graded mesh. We prove that the Galerkin method converges with the order of convergence $${\mathcal {O}}(n^{-m})$$, whereas iterated Galerkin method converges with the order $${\mathcal {O}}(n^{-2m})$$ in uniform norm. We also prove that in iterated multi-Galerkin method the order of convergence is $${\mathcal {O}}(n^{-3m}) $$. Numerical results are provided to justify the theoretical results.

Bernoulli operational matrix method for the numerical solution of nonlinear two-dimensional Volterra–Fredholm integral equations of Hammerstein type

Two-dimensional Volterra–Fredholm integral equations of Hammerstein type are studied. Using the Banach Fixed Point Theorem, the existence and uniqueness of a solution to these equations in the space $$L^\infty ([0,1]\times [0,1])$$ is proved. Then, the operational matrices of integration and product for two-variable Bernoulli polynomials are derived and utilized to reduce the solution of the considered problem to the solution of a system of nonlinear algebraic equations that can be solved by Newton’s method. The error analysis is given and some examples are provided to illustrate the efficiency and accuracy of the method.

Solvability of Coupled Systems of Generalized Hammerstein-Type Integral Equations in the Real Line

In this work, we consider a generalized coupled system of integral equations of Hammerstein-type with, eventually, discontinuous nonlinearities. The main existence tool is Schauder’s fixed point theorem in the space of bounded and continuous functions with bounded and continuous derivatives on R , combined with the equiconvergence at ± ∞ to recover the compactness of the correspondent operators. To the best of our knowledge, it is the first time where coupled Hammerstein-type integral equations in real line are considered with nonlinearities depending on several derivatives of both variables and, moreover, the derivatives can be of different order on each variable and each equation. On the other hand, we emphasize that the kernel functions can change sign and their derivatives in order to the first variable may be discontinuous. The last section contains an application to a model to study the deflection of a coupled system of infinite beams.

Coupled systems of Hammerstein-type integral equations with sign-changing kernels

In this work we consider a generalized coupled systems of two integral equations of Hammerstein-type where the kernel functions may change sign, as well as remain positive on some subintervals, and the nonlinearities may have discontinuities.Moreover the paper provides other new features:The integral equations contain nonlinearities depending on several derivatives of both variables and, moreover, the derivatives can be of different order on each variable and each equation, which increases the range of applications.A new type of cone is introduced, where some requirements may be satisfied only on some subintervals of the domain.Last section contains an application to a coupled system composed by a fourth and second order equations, which models the bending of the main-road of suspension bridges.

Legendre spectral Galerkin and multi-Galerkin methods for nonlinear Volterra integral equations of Hammerstein type

In this paper, we discuss the superconvergence of the Galerkin solutions for second kind nonlinear integral equations of Volterra–Hammerstein type with a smooth kernel. Using Legendre polynomial bases, we obtain order of convergence \({\mathcal{O}}(n^{-r})\) for the Legendre Galerkin method in both \(L^2\)-norm and infinity norm, where n is the highest degree of the Legendre polynomial employed in the approximation and r is the smoothness of the kernel. The iterated Legendre Galerkin solutions converge with the order \({\mathcal{O}}(n^{-2r}),\) whose convergence order is the same as that of the multi-Galerkin solutions. We also prove that iterated Legendre multi-Galerkin method has order of convergence \({\mathcal{O}}(n^{-3r})\) in both \(L^2\)-norm and infinity norm. Numerical examples are given to demonstrate the efficacy of Galerkin and multi-Galerkin methods.

Projection and multi projection methods for nonlinear integral equations on the half-line

In this article, Galerkin and collocation methods and their iterated versions are discussed to solve the nonlinear Hammerstein type integral equation on the half-line for both convolution and non-convolution kernels using the space of piecewise polynomial subspace. We prove that the approximate and iterated approximate solution in Galerkin and collocation methods converges with the order O ( n − r ) and O ( n − 2 r ) respectively in the infinity norm, where n − 1 is the maximum norm of the partition and r denotes the order of the piecewise polynomial employed in the approximation. We improve these results further by considering multi-projection methods. In fact we show that iterated approximate solution in multi-Galerkin, multi-collocation converges with the order O ( n − 4 r ) . Numerical results are presented to confirm the theoretical results.

On coupled systems of Hammerstein integral equations

In this paper we deal with generalized coupled systems of integral equations of Hammerstein type with nonlinearities depending on several derivatives of both variables and we underline that both equations and both variables can have a different regularity. This detail is very important as it allows for the application to, for example, boundary value problems with coupled systems composed of differential equations of different orders and distinct boundary conditions. This issue will open a new field of applications to phenomena modeled by coupled systems requiring different types of regularity for the unknown functions.The arguments follow Guo–Krasnosel’skiĭ compression–expansion theory on cones and the kernel functions are nonnegative and verifying adequate sign and growth assumptions.The dependence of the derivatives is overcome by the construction of suitable cones taking into account certain conditions of sublinearity/superlinearity at the origin and at +∞.