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In this study, an efficient numerical solution based on the Newton-PKSOR method with Simpson's 1/3 rule has been proposed to determine the approximate solution of nonlinear Fredholm integral equations. For this purpose, the discretization process is performed using Simpson's 1/3 rule to generate an approximation equation which leads to a system of nonlinear algebraic equations. Then, the generated nonlinear system is converted into a linear form using Newton's method so it can be solved by the PKSOR iterative method. Several numerical examples are used in this study to illustrate the efficiency and accuracy of the Newton-PKSOR method. Furthermore, the numerical results of this study have been compared to Newton-GS and Newton-KSOR methods to validate the findings. The results show that the Newton-PKSOR method with Simpson's 1/3 on certain cases can form a more precise approximate solution in solving nonlinear Fredholm integral equations. Also, the implementation of the Newton-PKSOR method can produce a lower number of iterations and iteration time compared to Newton-GS and Newton-KSOR methods. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical strategy to address nonlinear volterra indispensable conditions in view of a progressive estimation procedure is thought of. An arrangement of capacity is delivered which merges to the solution. Some models are introduced to show techniques. To observe a complete bound of the mistake, we examine blunder. At long last, a few models are introduced to results create the impression that this technique is exceptionally successful and helpful to settle these conditions. [ABSTRACT FROM AUTHOR]

In this work, we compare two approaches of using Adomian Decomposition Method to solve nonlinear nonhomogeneous volterra integral equation with known exact algebraic solutions and proving the existence and uniqueness of such solutions. A procedure for solving nonlinear Fuzzy volterra integral equation Numerically of second kind with some different kernel and also we study and prove the existe and uniqueness for our formulation give some a numerica applications to prove how this method is give a convergence to the exact solution, in this work we will use maple language. [ABSTRACT FROM AUTHOR]

In this paper, we will consider an approximate solution of nonlinear integral Hammerstein equation by using Legendre wavelets in order to solve by Bubnov-Galerkin method. There are will be described the procedure of obtaining Legendre wavelets, its application as a basis for projection method. Theorem about solution of system of nonlinear equations will be considered, which was obtained after using Bubnov-Galerkin method. Finally numerical calculations and visualization of results will be made to show the efficiency of this method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the two-dimensional nonlinear Volterra-Fredholm fuzzy integral equation (2D-NVFFIE). The homotopy analysis method (HAM) is used for determining the approximated solution of the investigated equation. We convert 2D-NVFFIE to a system of nonlinear Volterra-Fredholm integral equation in crisp case. Hence, we obtain an approximate solutions of this system and consequently obtain an approximation for the fuzzy solution of the Volterra-Fredholm fuzzy integral equation. We proof the convergence of the proposed method and error estimation between the exact and the approximate solution. A numerical example is given to demonstrate the validity and applicability of the proposed technique. [ABSTRACT FROM AUTHOR]

In this paper, we present some fixed point results for self-mappings satisfying an implicit relation in partial b-metric spaces which are generalization of several previously fixed point theorems in the literature. As an application, we obtained solution of non-linear integral equation. [ABSTRACT FROM AUTHOR]

A classic two-parameter epidemiological SIR-model of the coronavirus propagation is considered. The first integrals of the system of non-linear equations are obtained. The Painlevé test shows that the system of equations is not integrable in the general case. Using the first integrals of the system of equations, analytical dependencies for the number of infected patients I(R) depends on recovered patients R. It is demonstrated that data on morbidity in Hubei (China), Italy, Austria, South Korea, Moscow (Russia) as well some Australian territories are satisfactorily described by the expressions obtained for I(R). [ABSTRACT FROM AUTHOR]

In this work, we investigate two-dimensional nonlinear Hammerstein-Fredholm fuzzy functional integral equation (2D-NHFFFIE. We propose an efficient iterative method of successive approximations by using fuzzy variant of the cubature rule of the rectangle for classes of fuzzy number-valued functions of Lipschitz type for two-dimensional case to approximate the solution. We prove the convergence and numerical stability of the presented method with respect to the choice of the first iteration. Finally, some numerical experiment confirm the theoretical results and illustrate the accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In this article the solvability of the problem of optimal control of oscillatory processes described by the integro- differential equation with the Fredholm operator, with given control limitation is investigated. It is established that the desired control is among the solutions of the nonlinear Fredholm integral equation of the first kind. Sufficient conditions for the existence of a solution of nonlinear optimization problem are found. [ABSTRACT FROM AUTHOR]

The article explores general nonlinear equations with a parameter. Sufficient conditions for the existence of a solution of nonlinear integral equations in the form of the sum of two functions for the individual values of the parameter are found. [ABSTRACT FROM AUTHOR]

The paper is devoted to construction of the stable regularized algorithms for solving the structural inverse gravimetry problem for the case of multiple surfaces. The problem is in finding the surfaces that divide the layers of different constant densities using the known gravitational field. The problem is described by the nonlinear integral equation of the first kind; thus, it is an ill-posed one. After discretization, the problem is reduced to solving the system of nonlinear algebraic equations. To solve it, we propose the regularized variant of the discretized equation and construct stable algorithms based on the steepest descent and conjugate gradients methods. The algorithms are tested on the model problem with quasi-real disturbed data. We show that the proposed algorithms provides better accuracy than the classic ones. [ABSTRACT FROM AUTHOR]

Here, the Laplace transform method has been applied with the help of Adomian polynomial to solve nonlinear integral equations. The proposed method is very different from other analytical method because the nonlinear terms are expressed in Adomian polynomial using Newton Raphson formula. The results are investigated and compared with exact solutions. [ABSTRACT FROM AUTHOR]

A numerical method for solving two-dimensional nonlinear Volterra-Fradholm fuzzy integral equation (2D-NVFFIE) of the second kind will is introduced. We convert a nonlinear Volterra-Fredholm fuzzy integral equation to a nonlinear system of Volterra-Fredholm integral equation in crisp case. We use Adomian decomposition method (ADM) to find the approximate solution of this system and hence obtain an approximation for fuzzy solution of the nonlinear Volterra-Fredholm fuzzy integral equation. Also, the existence and uniqueness of the solution and convergence of the proposed method are proved. Numerical example is included to demonstrate the validity and applicability of the proposed technique. [ABSTRACT FROM AUTHOR]

The paper is devoted to construction of the time efficient algorithm for solving the structural inverse gravimetry problem in the case of multilayered medium. The problem is in finding multiple interfaces between layers with different constant densities using known gravitational data. This problem is described by a nonlinear integral equation of the first kind; it is illposed. After discretization of the area and approximation of the integral operator, the problem is reduced to solving a system of nonlinear equations. An efficient method was constructed on the basis of the nonlinear conjugate gradient method. The algorithm uses the approximation of the Jacobian matrix of the integral operator based on dropping out the lesser elements and utilizing the Toeplitz-block-Toeplitz structure of the matrix. The parallel algorithm was implemented for the multicore processors and graphics processors using OpenMP and CUDA technologies. The structural gravimetry problem of reconstructing three surfaces using quasi-real data was solved. [ABSTRACT FROM AUTHOR]

New nonlinear integral equations for determination of the unified exit function combining the intensities of upgoing and downgoing radiation at arbitrary symmetrical levels of a uniform slab have been obtained. By using the modified principle invariance of Ambarzumian-Chandrasekhar at the slab outside boundaries and applying the mirror reflection principle proposed by the author, the analytical spatial-angular structure of inner radiation fields has been investigated. New structural functions that describe the radiation field within the slab of an arbitrary optical thickness and generalize the classical structural functions of Ambarzumian have been proposed. [ABSTRACT FROM AUTHOR]

Based fitting parametrization and transformations of the exact non-linear integral equation for the unified function E of radiation exit from a homogeneous slab of an arbitrary optical thickness, approximate analytical expressions have been obtained for the corresponding slab brightness coefficients in the case of strongly elongated phase functions. The accuracy of the obtained fitting analytical approximations has been evaluated by using appropriate calculations in the visible spectrum region (λ = 400-800 nm) for Heyney-Greenstein phase functions in dependence of the single scattering albedo, vision angles, solar zenith distance and azimuths. [ABSTRACT FROM AUTHOR]

New nonlinear integral equations for determination of the unified exit function combining the intensities of upgoing and downgoing radiation at arbitrary symmetrical levels of a uniform slab have been obtained. By using the modified principle invariance of Ambarzumian-Chandrasekhar at the slab outside boundaries and applying the mirror reflection principle proposed by the author, the analytical spatial-angular structure of inner radiation fields has been investigated. New structural functions that describe the radiation field within the slab of an arbitrary optical thickness and generalize the classical structural functions of Ambarzumian have been proposed. [ABSTRACT FROM AUTHOR]

Based fitting parametrization and transformations of the exact non-linear integral equation for the unified function E of radiation exit from a homogeneous slab of an arbitrary optical thickness, approximate analytical expressions have been obtained for the corresponding slab brightness coefficients in the case of strongly elongated phase functions. The accuracy of the obtained fitting analytical approximations has been evaluated by using appropriate calculations in the visible spectrum region (λ = 400-800 nm) for Heyney-Greenstein phase functions in dependence of the single scattering albedo, vision angles, solar zenith distance and azimuths. [ABSTRACT FROM AUTHOR]

This short survey presents the essential features of what is called Painlevé analysis, i.e. the set of methods based on the singularities of differential equations in order to perform their explicit integration. Full details can be found in The Painlevé handbook or in various lecture notes posted on arXiv. [ABSTRACT FROM AUTHOR]

We study the first sub-leading correction O((ln s)0) to the cusp anomalous dimension in the high spin expansion of finite twist operators in N = 4 SYM theory. This term is still governed by a linear integral equation which we study in the weak and strong coupling regimes. In the strong coupling regime we find agreement with the string theory computations. [ABSTRACT FROM AUTHOR]

In this study, focusing on nonlinear integer programming problems, we propose an approximate solution method based on particle swarm optimization proposed by Kennedy et al. To be more specific, we develop a new particle swarm optimization method which is applicable to discrete optimization problems by incoporating a new method for generating initial search points, the rounding of values obtained by the move scheme and the revision of move methods. Furthermore, we show the efficiency of the proposed particle swarm optimization method by comparing it with an existing method through the application of them into the numerical examples. [ABSTRACT FROM AUTHOR]

We present the formulation of self-consistent Hartree-Fock theory for non-equilibrium electron transport in nanostructures. The derivations are performed by our method for direct calculations of asymptotic, non-equilibrium steady state averages. We use asymptotic single-particle density matrix to approximate the molecular Hamiltonian by its Hartree-Fock form. Then we obtain the close system of coupled nonlinear integral equations for the transformation matrix, which diagonalizes the Hartree-Fock Hamiltonian, and asymptotic single particle density matrix. [ABSTRACT FROM AUTHOR]

A dense superfluid pure neutron matter with an effective Skyrme interaction, which depends on the density n of the neutrons with spin-triplet p-wave pairing similar to those of 3He-A1 and 3He-A, is studied in a strong uniform static magnetic field H in the framework of a generalized non-relativistic Fermi-liquid theory. We present an analytic solution for a set of previously derived nonlinear integral equations for the components of the order parameter and an effective magnetic field Heff. The obtained general formulas (valid for arbitrary parameterization of Skyrme forces) for the phase transition temperatures Tc1,2 of the neutron matter (from normal to a superfluid state of 3He-A1 type and then to a 3He-A type state, respectively) and the expression for Heff at T=0 are linear functions of strong magnetic field H and nonlinear functions of n. The gap equation is also solved at T=0. © 2006 American Institute of Physics [ABSTRACT FROM AUTHOR]