Positive Almost Periodic Solutions for Some Nonlinear Integral Equations

This paper discussed some existence theorems for nonlinear functional integral equations in the space L^1 of Lebesgue integrable functions,by using the Darbo fixed point theorem associated with the Hausdorff measure of noncompactness. Also, as an application, we discuss the existence of solutions for some nonlinear integral equations with fractional order.

The Numerical Solution of Integral Equations on the Half-Line

Numerial Methods for Volterra Integral Equations with Singular Kernels

An iterative solution of a particular non-linear Fredholm integral equation is considered. The method is based on the reduction of the equation for the residual at any stage of the iteration to a linear integral equation for the required perturbation, using Newton' s method. This equation is then solved by using a low order expansion in terms of Chebyshev polynomials. The numerical integrations required in the basic iterative process are carried out by the powerful Clenshaw-Curtis quadrature prescription. The non-linear integral equation considered arises from the unstable two-point boundary value problem for a tubular chemical reactor. The capacity of the present method in dealing with such numerical difficulties is illustrated in representative calculations. Comparisons are made with conventional solutions of the parent differential equation and also with attempts at direct iterative solution of the integral equation by standard Neumann-Liouville series and finally with the direct algebraic approach.

This paper establishes a comprehensive framework for fixed point theory in MR-metric spaces,a generalization of standard metric spaces that incorporates three-point relations. We present fourfundamental theorems:(1) A Banach contraction principle with optimal contraction constant k < 1/3R(2) A solvability theorem for Fredholm-type integral equations(3) A Krasnoselskii-type hybrid fixed point theorem(4) A Leray-Schauder alternative for generalized contractionsThe theoretical results are applied to:• Nonlinear integral equations in neutron transport theory• Optimization problems in neural networks• Boundary value problems for nonlinear ODEsKey innovations include the development of error estimates in the MR-metric framework and the derivation of precise existence conditions for operator equations. The work bridges theoretical mathematics with practical applications in physics and machine learning.

Many different problems in mathematics, physics, engineering can be expressed in the form of integral equations.Among these are diffraction problems, population growth, heat transfer, particle transport problems, electrical engineering, elasticity, control, elastic waves, diffusion problems, quantum mechanics, heat radiation, electrostatics and contact problems.Therefore, the solutions which are obtained by the mathematical methods play an important role in these fields.The most two basic types of integral equations are called Fredholm (FIEs) and Volterra (VIEs).In many instances, the ordinary and partial differential equations can be converted into Fredhom and Volterra integral equations that are solved more effectively.We aim through this research to present an improved Adomian decomposition method based on modified Bernstein polynomials (ADM-MBP) to solve nonlinear integral equations of the second kind.We introduced efficient method, constructed on modified Bernstein polynomials.The formulation is developed to solve nonlinear Fredholm and Volterra integral equations of second kind.This method is tested for some examples from nonlinear integral equations.Maple software was used to obtain the solutions of these examples.The results demonstrate reliability of the proposed method.Generally, the proposed method is very convenient to apply to find the solutions of Fredholm and Volterra integral equations of second kind.

Solving linear and nonlinear Abel fuzzy integral equations by homotopy analysis method

Numerical solution of nonlinear mixed Volterra-Fredholm integral equations in complex plane via PQWs

The proof of convergence of the series solution to a class of nonlinear two-dimensional Hammerstein integral equation (NTHIE), including the necessary and su¢ cient conditions that guarantee a unique solution, is introduced. Adomian Decomposition Method (ADM) and Homotopy Analysis Method (HAM) are used to solve the NTHIE. It was found that, when using the traditional Adomian polynomials (4), ADM and HAM are exactly the same. But, when using the proposed accelerated Adomian polynomials formula (5), ADM converges faster than HAM. The proposed accelerated Adomian polynomials formula is used directly to prove the convergence of the series solution. Convergence approach is reliable enough to estimate the maximum absolute truncated error.

A design algorithm for adaptive control of systems described by nonlinear integral operator equations is considered. Kulikowski's original approach is extended to include bounded input functions. The limited range of performance criteria considered previously is similarly extended making use of the concepts of nonlinear functional analysis. The plant operator is assumed to be only partially known, and the optimization problem is formulated as a conditional-minimization problem. The solution is obtained in the form of an iteration and two such iteration techniques are described. The first one is based on the principle of contraction mapping; the second on Newton's modified method. The identification procedure is briefly mentioned and an analog implementation scheme is presented.

Preface. 1. Introduction and Preliminaries. 2. Existence Theory for Nonlinear Fredholm and Volterra Integrodifferential Equations. 3. Solution Sets of Abstract Volterra Equations. 4. Existence Theory for Nonlinear Fredholm and Volterra Integral Equations on Compact Intervals. 5. Existence Theory for Nonlinear Fredholm and Volterra Integral Equations on Half-Open Intervals. 6. Existence Theory for Nonlinear Nonresonant Operator and Integral Equations. 7. Existence Theory for Nonlinear Resonant Operator and Integral Equations. 8. Integral Inclusions. 9. Approximation of Solutions of Operator Equations on the Half Line. 10. Operator Equations in Banach Spaces Relative to the Weak Topology. 11. Stochastic Integral Equations. 12. Periodic Solutions for Operator Equations. Index.

HE simplified Oswatitsch equation is a one-dimensional nonlinear singular integral equation, occurring in transonic aerodynamics.l It is an approximate version of Oswatitsch's integral equation, whose kernel has a dipole singularity. According to transonic small perturbation theory, the Oswatitsch equation is the basic equation governing the steady inviscid irrotational flow of a perfect gas past a thin symmetric profile at zero incidence, with subsonic freestream Mach number Mx rstrud 4 and Nixon5 deserve special mention. In the present work, the simplified Oswatitsch equation has been solved by two different numerical procedures, viz., the direct iteration scheme (DIS) proposed by Niyogi and Chakraborty6 and by the recent perturbed iterative scheme (PIS) put forward by Dey7 for solving a system of nonlinear algebraic equations. Further, this nonlinear integral equation was used as a test case for studying the global convergence behavior of PIS. From computational results, it has been found that for a parabolic arc profile there exists a range of values for the reduced thickness ratio r (which is a transonic similarity parameter), where both the procedures lead to the same shock-free supercritical solution, and that in this range PIS converges much faster than DIS. However, for higher values of r beyond this range, there exists another range where, contrary to expectations, DIS converges but PIS fails to converge, indicating that DIS has a wider range of convergence.

The random position of a particle in turbulent flow r(t;ω) is a vector random function of time t ≥ 0, ω ∈ Ω, where Ω is the supporting set of the underlying probability space (Ω,A,μ). If the statistics of a Lagrangian velocity field is known, the random position vector r(t;ω) is the solution of a nonlinear stochastic integral equation. Depending on the particular application, this equation may be homogeneous or inhomogeneous and with deterministic or random limits. A theorem is presented on the existence and uniqueness of the solution of this integral equation. The proof is based on Banach's fixed point theorem.

The present paper is a continuation of [1]. By using more general type of random contractor than that defined by Lee and Padgett, we show several existence and uniqueness theorems for solutions to a system of nonlinear random operator equations with stochastic domain. Next, we obtain a random coincidence theorem and a random implicit function theorem, which improves and generalizes the corresponding results of Ding, Reddy and Subrahmanyam, Altman, Eng, Krasnoselskii and others. Finally, some applications of our results are given for the solutions of a system of nonlinear random integral equations.

Methods for solving the amplitude-phase problem for one-dimensional signals are proposed. The amplitude-phase problem is investigated in the case of continuous and discrete signals. In both cases, the amplitude-phase problem is modeled by nonlinear singular integral equations. The study of continuous signals leads to nonlinear singular integral equations defined on the numerical axis; discrete ones to nonlinear singular integral equations defined on a unit circle in the plane of a complex variable. The obtained singular integral equations relate to the exceptional case - the symbols of the Frechet derivatives of singular operators degenerate in the entire domain of their definition.The continuous operator method for solution of nonlinear equations is used for solution these nonlinear singular integral equations. The numerical schemes for solving corresponding singular integral equations are constructed. Solutions of model examples have shown effectiveness of the proposed method and numerical algorithms.